AP Calculus glossary

A series converges absolutely when the series formed by taking the absolute value of every term still converges. Absolute convergence is stronger than ordinary convergence and always implies it.

An absolute extremum is the single highest or lowest value a function reaches on an interval. On a closed interval you find it with the Candidates Test: evaluate the function at every critical point and at both endpoints, then compare the numbers.

The absolute maximum is the largest value a function attains on a given interval. On a closed interval a continuous function is guaranteed one, and you find it with the Candidates Test: evaluate the function at every critical number and at both endpoints, then take the largest output.

The absolute minimum is the smallest value a function attains on a given interval. On a closed interval a continuous function is guaranteed one, and the Candidates Test finds it: evaluate the function at every critical number and at both endpoints, then take the smallest output.

The absolute value function returns the distance of a number from zero. It is continuous everywhere but has a corner at the origin, where the one-sided slopes are negative one and one, so it is not differentiable there.

Acceleration is the derivative of velocity with respect to time, and therefore the second derivative of position. Comparing its sign with the sign of velocity tells you whether the object is speeding up or slowing down.

Accumulated change is the net change in a quantity over an interval, obtained by integrating that quantity's rate of change across the interval. It carries the units of the rate multiplied by the units of the variable integrated over, and it is a net figure, so falls cancel earlier rises.

An accumulation function is a definite integral whose upper bound is the variable, so it defines a new function measuring how much has accumulated from a fixed starting point. Its derivative is the integrand evaluated at the upper bound.

The alternate form defines the derivative at a as the limit of f(x) minus f(a) over x minus a as x approaches a. It is equivalent to the h version, and it is the form to recognize when an exam gives you a limit and asks what derivative it represents.

The alternating harmonic series converges to the natural logarithm of two, even though the harmonic series itself diverges. It is the standard example of conditional convergence: it converges as written but not once you take absolute values.

For an alternating series that satisfies the alternating series test, the error in using a partial sum is less than the absolute value of the first omitted term. It is the simplest error bound in the course.

The alternating series test says that a series whose terms alternate in sign converges if the absolute values of its terms decrease and approach zero. Both conditions are needed, and together they are enough.

An antiderivative of a function is any function whose derivative equals it. Because the derivative of a constant is zero, a function has infinitely many antiderivatives, all differing from one another by a constant.

Arc length is the distance measured along a curve rather than straight across. It comes from adding up infinitesimal hypotenuses, which produces an integral of the square root of one plus the square of the derivative.

The area under a curve is the region trapped between the graph and the x-axis over an interval. Where the function is non-negative it equals the definite integral exactly; where the function dips below the axis you must integrate the absolute value instead.

Average acceleration over a time interval is the change in velocity divided by the elapsed time, which is the slope of the secant line joining the endpoints of the velocity graph. Instantaneous acceleration is a different quantity: it is the derivative of velocity at a single instant.

The average rate of change of a function over an interval is the change in output divided by the change in input, which is the slope of the secant line joining the endpoints. It needs no calculus, unlike the instantaneous rate of change.

The average value of a function on an interval is the definite integral over that interval divided by the length of the interval. It is the constant height a rectangle would need to have the same area as the region under the curve.

Average velocity over a time interval is displacement divided by elapsed time, the change in position from start to end divided by how long it took. Graphically it is the slope of the secant line joining the two endpoints of the position graph.

The axis of revolution is the line a plane region is rotated about to sweep out a solid. It fixes every radius in the volume integral, since a radius is the distance from the curve to that axis: rotating about y = 3 gives 3 minus f of x when the curve lies below the line, and f of x minus 3 when it lies above.

The binomial series is the Maclaurin expansion of one plus x raised to a real power k. When k is a non-negative integer it terminates and reproduces the binomial theorem; otherwise it is an infinite series converging for x between negative one and one.

A bounded function is one whose outputs all stay inside a fixed interval, so there is some number M that f(x) never exceeds in size anywhere on the domain. Bounded is a statement about size and nothing else: sin(1/x) never leaves the range from -1 to 1 and still has no limit at 0.

A sequence is bounded when all its terms lie between two fixed numbers. Boundedness alone does not give convergence, but a sequence that is both bounded and monotonic must converge, which is the Monotone Convergence Theorem.

The candidates test finds absolute extrema on a closed interval by evaluating the function at every critical point and at both endpoints, then comparing the resulting values. The largest is the absolute maximum and the smallest is the absolute minimum.

Carrying capacity is the limiting value a logistic model approaches as time increases. Growth is fastest at half the carrying capacity and slows to nothing as the population nears it, giving the logistic curve its S shape.

The center of a power series is the fixed value the series is built around, the number c in a sum of terms times x minus c to the nth power. The series always converges at its center, and its interval of convergence is symmetric about it, reaching one radius in each direction.

Changing the limits of integration means converting the x values on a definite integral into the matching u values when you substitute. You then evaluate the new antiderivative at the new limits, so there is no need to rewrite the answer in terms of x.

The common ratio is the fixed number you multiply one term by to get the next term in a geometric sequence or series. You find it by dividing any term by the one before it. A geometric series converges exactly when the absolute value of the common ratio is less than one.

The direct comparison test settles a series of positive terms by bounding it against a series whose behaviour is known. If it is term by term smaller than a convergent series it converges, and if it is larger than a divergent series it diverges.

Completing the square rewrites an irreducible quadratic as a perfect square plus a constant. In integration it turns a denominator you cannot factor into the shape of the arctangent formula, making the antiderivative an inverse trigonometric function.

A composite function is one function evaluated inside another, so the output of the inner function becomes the input of the outer one. Spotting a composition is what tells you to use the chain rule.

A graph is concave down on an interval where the second derivative is negative. The slope is decreasing, every tangent line lies above the curve, and a tangent line estimate there is an overestimate. Inflection points separate concave down from concave up.

A graph is concave up on an interval where the second derivative is positive. The slope is increasing, tangent lines lie below the curve, and any tangent line approximation there is an underestimate.

Concavity describes which way a graph bends. A graph is concave up where the second derivative is positive and the slope is increasing, and concave down where the second derivative is negative and the slope is decreasing.

The concavity test says a graph is concave up on any interval where the second derivative is positive and concave down on any interval where the second derivative is negative. It describes intervals, unlike the second derivative test, which classifies a single critical point.

A series converges conditionally when it converges as written but the series of its absolute values diverges. The cancellation between positive and negative terms is what makes it work, so the convergence depends on the order of the terms.

The conjugate method resolves a limit of the form zero over zero that contains a square root. Multiply the numerator and denominator by the conjugate of the radical expression, which turns a difference of square roots into a difference of squares and lets the offending factor cancel.

The constant multiple rule says a constant factor passes straight through a derivative, and the sum rule says the derivative of a sum is the sum of the derivatives. Together they make differentiation linear.

The constant of integration is the arbitrary constant added to an indefinite integral. It is required because differentiating any constant gives zero, so a function has infinitely many antiderivatives that differ only by a constant.

A constraint equation is the extra relationship in an optimization problem that ties the variables together. You solve it for one variable and substitute into the objective function, which turns a two-variable problem into one you can differentiate.

A function is continuous at a point when three things hold: the function is defined there, the limit exists there, and the limit equals the function value. Informally, you can draw the graph through that point without lifting your pencil.

A function is continuous on a closed interval when it is continuous at every interior point and one-sidedly continuous at each endpoint. This is the exact hypothesis required by the Extreme Value Theorem, the Intermediate Value Theorem, and the Mean Value Theorem.

A continuous function is continuous at every point of its domain, so its graph has no hole, jump, or vertical asymptote anywhere it is defined. Polynomials are continuous for every real number, while rational, radical, logarithmic, and trigonometric functions are continuous only on their domains.

A series converges when its sequence of partial sums approaches a finite limit, and diverges when it does not. Divergence covers both partial sums that grow without bound and ones that oscillate without settling.

A sequence is convergent when the limit of its terms, as the index runs to infinity, exists and is finite. Every convergent sequence is bounded, but bounded sequences need not converge. A convergent sequence of terms does not make the series built from them converge.

A convergent series is one whose partial sums settle on a finite limit, and that limit is its sum. Where the partial sum has a closed form, as it does for telescoping and geometric series, the sum comes straight out of it. Dropping finitely many terms changes the sum but never changes convergence.

A corner is a point where a graph changes direction abruptly, with left and right derivatives that are different finite numbers. The function is continuous there but not differentiable, because no single tangent slope exists.

A critical number is an x-value in the domain of a function where its derivative equals zero or fails to exist. Critical numbers are the only candidates for a relative extremum, but a candidate is not a guarantee. The critical number is the x-value itself, not the point on the graph.

A critical point of a function is an interior point of its domain where the derivative is either zero or undefined. Every local maximum and minimum occurs at a critical point, though not every critical point is an extremum.

A cross-section is the two-dimensional shape you get by slicing a solid perpendicular to an axis. If you know the area of each cross-section as a function of position, integrating that area function along the axis gives the volume.

Curve sketching is rebuilding the shape of a graph from the signs of its first and second derivatives.

A cusp is a sharp point on a graph where the one-sided slopes grow without bound in opposite directions. The function stays continuous there but is not differentiable, because the tangent slope is not a finite number.

A function is decreasing on an interval when larger inputs give smaller outputs. On an interval where the function is differentiable, this happens exactly when the derivative is negative, and the endpoints of such intervals are where the derivative changes sign.

A definite integral is a number defined as the limit of Riemann sums as the subintervals shrink to zero width. It represents the net signed accumulation of a quantity between two bounds, with area below the axis counting as negative.

The derivative of a function at a point is its instantaneous rate of change there, defined as the limit of the difference quotient as the gap shrinks to zero. Geometrically it is the slope of the tangent line at that point.

The derivative at a point is the limit of the difference quotient as the interval shrinks to zero. It is a single number: the slope of the tangent line to the graph at that point, and the instantaneous rate of change of the function there.

The derivative of any constant is 0, because the graph of a constant function is a horizontal line and a horizontal line has slope 0. Differentiation erases the constant term, which is why every antiderivative carries a constant of integration to record what was lost.

The derivative of a piecewise function is the derivative of whichever piece the input falls in, which settles every point but the break. There the function must be continuous first, and the one sided derivatives from the left and right must agree on a finite value, or no derivative exists.

The difference quotient is the slope of the secant line joining two points on a curve, written as the change in output over the change in input. Taking its limit as the gap shrinks to zero produces the derivative.

A function is differentiable at a point when the limit defining the derivative exists there, which requires the graph to be locally smooth with a single well-defined tangent slope. Differentiability always implies continuity, but continuity does not imply differentiability.

If a function is differentiable at a point then it is continuous there. The converse is false: a function can be continuous at a point and still fail to be differentiable, which is what happens at a corner, a cusp, or a vertical tangent.

A function is differentiable on an open interval when its derivative exists at every point of it, and on a closed interval when it exists at every interior point and as a one-sided derivative at each endpoint. Polynomials, sine, cosine, and e to the x qualify everywhere.

The differential dy is the change in output along the tangent line corresponding to a small change dx in the input. It approximates the true change in the function and is the notation that makes substitution and separable equations work.

A differential equation is an equation that relates an unknown function to its own derivatives. Its solution is a function rather than a number, and there is usually a whole family of them until an initial condition picks one out.

Direct substitution evaluates a limit by putting the value straight into the function. It is valid exactly when the function is continuous at that point, which makes it the first thing to try on any limit.

A discontinuity is a point where continuity fails: the value is missing, the limit is missing, or the two disagree. The CED names three types, removable, jump, and infinite, meaning a vertical asymptote, and textbooks add oscillating as a fourth. The list is a naming convention, not an exhaustive classification.

The disk method computes the volume of a solid of revolution by slicing it into circular disks perpendicular to the axis of rotation and integrating the area of each. It applies when the region being rotated touches the axis, leaving no hole.

Displacement is the net change in position over a time interval, found by integrating velocity. It is signed, so a round trip that returns to the start has a displacement of zero no matter how far the object travelled.

A series diverges when its partial sums fail to approach a finite limit. That can mean the sums grow without bound, as in the harmonic series, or that they never settle at all, as when the terms do not shrink to zero.

No single finite value to settle on: that is what divergence means for a sequence. The terms may run to infinity, or they may oscillate forever, the way the sequence alternating between 1 and negative 1 does, staying inside a bounded range and diverging all the same.

A series is divergent when its partial sums never reach a finite limit, either running off to infinity or oscillating forever without settling. Shrinking terms prevent neither: grouped into blocks that each total at least one half, the harmonic series has partial sums clearing every bound.

The domain of a function is the set of inputs for which it produces a value. Domain restrictions determine where a function can be continuous or differentiable, and they are what make some answers to calculus problems invalid.

A double-angle identity writes a trig function of 2u in terms of functions of u. Read backwards, it writes a squared function as a first power of cos 2u, and those power-reducing forms are the ones calculus needs, because no substitution touches the integral of cosine squared until the square is gone.

Doubling time is how long an exponentially growing quantity takes to reach twice its current size. It equals the natural log of 2 divided by the growth constant. The starting amount cancels out of the equation, so the same interval doubles the quantity again from any moment onward.

Eliminating the parameter converts a pair of parametric equations into one equation relating x and y directly. Solve one equation for the parameter and substitute into the other, or use an identity when the equations are trigonometric.

End behavior is what happens to a function's output as the input runs off to positive or negative infinity. It is stated as two limits at infinity, and for a rational function it is decided entirely by the highest-degree term on the top and the bottom.

Endpoint analysis is the step that turns a radius of convergence into an interval of convergence. The ratio test gives no information at the two endpoints, so each one must be substituted in and tested separately with another convergence test.

An endpoint of an interval of convergence is one of the two inputs exactly one radius from the centre of a power series. At the right endpoint the powers keep a fixed sign and at the left endpoint they alternate, so the same series can converge at one end and diverge at the other.

The epsilon-delta definition states that a limit equals L if, for every tolerance around L, there is a distance around the input point that keeps all outputs within that tolerance. It is the precise version of the informal idea of approaching a value.

An equilibrium solution is a constant function that satisfies a differential equation, found by setting the derivative expression to zero and solving. On a slope field it shows up as a horizontal line that other solution curves approach or run away from.

Estimating a derivative from a table means approximating f prime at a point with a difference quotient built from the nearest table values. Divide the change in the output by the change in the input over the tightest interval that brackets the point, taking a row from each side when the table offers one.

Euler's method approximates the solution of a differential equation by starting at a known point and repeatedly stepping along the tangent line for a small increment. Each step uses the differential equation to compute the slope at the current point.

Euler's number e is the constant approximately equal to 2.71828. It is the unique base for which the exponential function is its own derivative: the slope of e to the x at every point equals the height there. That self-matching property is why e, and its logarithm, are called natural.

An even function satisfies f of negative x equals f of x, making its graph symmetric about the y-axis. Over an interval symmetric about zero, its integral equals twice the integral over the right half, which often saves real work.

An explicit function is one written with the output isolated, in the form y = f(x), so each input produces its value directly. An implicit relation instead mixes x and y in a single equation.

Exponential decay describes a quantity falling at a rate proportional to how much is left, which makes the derivative equal to a negative constant times the amount. The solution is an exponential with a negative exponent, and it has a constant half-life.

An exponential function has the variable in the exponent and a constant base. Its defining property in calculus is that its rate of change is proportional to its current value, which is why e to the x is its own derivative.

Exponential growth describes a quantity whose rate of change is proportional to its current amount. The differential equation says the derivative equals a constant times the quantity, and its solution is a constant multiple of e raised to that constant times time.

An exponential model sets a quantity's rate of change proportional to the amount present, so its solution is the starting value times e to the kt. Two measurements pin it down: the natural log of their ratio divided by the time between them gives k, and either one then gives the starting value.

The Extreme Value Theorem says that a function continuous on a closed, bounded interval must attain both an absolute maximum and an absolute minimum somewhere on that interval. It guarantees the extrema exist without saying where they are.

An extremum is a maximum or minimum value of a function. A local extremum is largest or smallest compared only to nearby points, while an absolute extremum is largest or smallest across the entire domain or interval under consideration.

The factorial n! is the product of the positive integers from 1 up to n, and 0! is defined to be 1. The fact that settles most series questions is that (n+1)! equals (n+1) times n!, which is why factorials cancel down to a single factor in the ratio test.

The first derivative test classifies a critical point by examining the sign of the derivative on both sides. A change from positive to negative gives a local maximum, negative to positive gives a local minimum, and no sign change means neither.

Part 1 of the Fundamental Theorem of Calculus says that differentiating an integral with a variable upper bound returns the integrand evaluated at that bound. It is the statement that differentiation and integration are inverse operations.

Part 2 of the Fundamental Theorem of Calculus says a definite integral equals any antiderivative evaluated at the upper bound minus the same antiderivative at the lower bound. It is what makes definite integrals computable without Riemann sums.

The general solution of a differential equation is the entire family of functions that satisfy it, expressed with an arbitrary constant. Every specific solution is obtained by choosing a value for that constant.

A geometric sequence has a constant ratio r between consecutive terms, so the nth term is the first term times r to the power n minus 1. It converges when the absolute value of r is less than 1, and also when r equals 1, where every term is the same number.

A geometric series is one where each term is a fixed multiple of the previous term. It converges exactly when the absolute value of that common ratio is less than one, and then its sum is the first term divided by one minus the ratio.

The graph of the derivative, f prime, describes the original function f. Where f prime is positive, f is increasing; where f prime is negative, f is decreasing. Zeros of f prime mark critical numbers, and where f prime is itself increasing, f is concave up. The height of f prime is the slope of f.

The greatest integer function returns the largest integer less than or equal to x, so it is a staircase that steps up by one at every integer. It is the standard example of a function with a jump discontinuity at each integer, where the one-sided limits differ by exactly one.

A half-life is the time an exponentially decaying quantity takes to fall to half of whatever it currently is. Setting the amount equal to half the starting amount makes the starting value cancel, leaving the half-life equal to the natural log of 2 divided by the size of the decay constant.

The harmonic series is the sum of the reciprocals of the positive integers. Its terms shrink to zero, yet the series diverges, which makes it the standard proof that shrinking terms are not enough for convergence.

A higher-order derivative is what you get by differentiating repeatedly. The second derivative is the derivative of the derivative and measures how the rate of change is itself changing, which is what determines concavity and acceleration.

A horizontal asymptote is a horizontal line that a graph approaches as the input grows without bound in either direction. A function has one exactly when its limit at infinity or at negative infinity is a finite number.

A horizontal tangent is a tangent line of slope 0. It occurs at an input c where f'(c) = 0, and the line itself is y = f(c). Every horizontal tangent at an interior point sits at a critical point, but not every critical point has one, and a flat tangent is not automatically a maximum or minimum.

An implicit function is one defined by an equation relating the variables without the output isolated, such as the equation of a circle. Its derivative is found by differentiating both sides and solving for the derivative rather than by rearranging first.

The implicit second derivative is the result of differentiating dy/dx implicitly a second time. The expression that comes out still contains dy/dx, so the first derivative gets substituted back in and simplified until the answer is written only in terms of x and y.

An improper integral is one with an infinite bound of integration or an integrand that grows without bound somewhere on the interval. It is defined as a limit of ordinary definite integrals, and it converges if that limit is finite and diverges otherwise.

A function is increasing on an interval where its derivative is positive and decreasing where its derivative is negative. Increasing behaviour is a property of an interval, not of a single point.

The increasing/decreasing test states that a function increases where its derivative is positive and decreases where it is negative. You apply it by checking the sign of the derivative on each interval between the critical numbers and any points where the function or its derivative is undefined.

An indefinite integral is the collection of all antiderivatives of a function, written with an integral sign and no limits. Its result is a family of functions plus a constant of integration, which is what distinguishes it from a definite integral.

An indeterminate form is a limit expression whose value cannot be determined from its form alone, such as zero over zero or infinity minus infinity. It is not an answer but a signal that you need algebra or L'Hopital's rule to resolve the limit.

The index of summation is the counter variable in sigma notation together with its starting value below the sigma. It is a placeholder, so renaming it changes nothing, and shifting where it starts is allowed provided the terms are rewritten to compensate, leaving the sum unchanged.

An infinite discontinuity occurs where at least one one-sided limit is infinite, so the function grows without bound near the point. It always corresponds to a vertical asymptote and can never be removed by redefining a value.

An infinite limit means the function grows without bound as the input approaches a point, written as equal to positive or negative infinity. Because infinity is not a number, an infinite limit is a description of how the limit fails to exist, not a value.

An inflection point is a point on a graph where the concavity changes from up to down or from down to up. Finding one requires the second derivative to change sign there, which is a stronger condition than simply equalling zero.

An initial condition is a single known value of the solution, usually given as the output at a starting input. It is what pins down the constant of integration and turns a general solution into one particular curve.

An initial value problem is a differential equation packaged with an initial condition. Solving it means finding the single solution curve that passes through the given point, rather than the whole family of curves the equation allows.

Initial velocity is an object's velocity at time zero. It is the initial condition that pins down the constant of integration when you antidifferentiate acceleration, so velocity at any later time equals the initial velocity plus the accumulated change in velocity since time zero.

In a composition, the inner function is the one applied first, sitting inside the parentheses. The chain rule differentiates the outer function while holding the inner one fixed, then multiplies by the derivative of the inner function.

The instantaneous rate of change is how fast a quantity is changing at one exact instant, and it is precisely the derivative. It is the limit of average rates of change over intervals that shrink to nothing around that instant.

The integral test says that for a positive, decreasing, continuous function, the series of its values and the improper integral of the function either both converge or both diverge. It is what proves the p-series rule.

The integrand is the function being integrated, written between the integral sign and the differential. Identifying it clearly is what tells you which integration technique applies.

The Intermediate Value Theorem says that if a function is continuous on a closed interval, it attains every value between the two endpoint values somewhere on that interval. It is the standard tool for proving a solution exists without finding it.

Interval notation describes a set of real numbers by its two endpoints: a square bracket includes the endpoint and a parenthesis excludes it. The interval from 2 to 5 with brackets contains both 2 and 5, the same interval with parentheses contains neither, and infinity always takes a parenthesis.

The interval of convergence is the complete set of inputs for which a power series converges. You find the radius with the ratio test, then test each endpoint separately, because the ratio test is inconclusive exactly there.

An inverse function undoes another function: if g is the inverse of f, then f of g of x equals x and g of f of x equals x, so each undoes the other. A function has an inverse only when it is one-to-one. Arcsine, arctangent, and the natural log are defined this way.

The derivative of an inverse function at a point equals the reciprocal of the original function's derivative, evaluated at the matching point. You never need a formula for the inverse itself, only the point correspondence.

The derivatives of the inverse trigonometric functions are algebraic, not trigonometric. Arcsin x has derivative one over the square root of one minus x squared, and arctan x has derivative one over one plus x squared. Arccos and arccot are the negatives of their partners.

An inverse trigonometric function undoes a trig function on a domain restricted to make it one to one. Arcsine returns angles from negative pi over two to pi over two, arccosine returns angles from zero to pi, and arctangent returns angles strictly between negative pi over two and pi over two.

Inverse trigonometric integrals are the antiderivative patterns that produce arcsine and arctangent. With only a constant multiple of du on top, the denominator decides: a squared minus u squared under a root gives arcsine, while a plain sum a squared plus u squared gives arctangent.

A jump discontinuity occurs at a point where both one-sided limits exist but disagree, so the graph steps abruptly from one value to another. Because the one-sided limits differ, the two-sided limit does not exist and no redefinition can repair the break.

A justification is the sentence of reasoning that supports an answer on free response. A correct conclusion without one routinely earns no credit, because the point is awarded for the reason, and the reason must name the specific fact you used.

The Lagrange error bound limits how far a Taylor polynomial can be from the function it approximates. It uses the maximum size of the next derivative on the interval between the centre and the point of interest.

A left Riemann sum approximates a definite integral using the left endpoint of each subinterval as the rectangle height. For an increasing function every rectangle sits below the curve, so the sum underestimates the integral; for a decreasing function it overestimates.

The left-hand limit is the value a function approaches as the input moves toward a point through values below it. It is written with a minus superscript on the target. The two-sided limit exists only when the left-hand and right-hand limits are equal.

Leibniz notation writes a derivative as dy over dx, which names both the changing quantity and the variable it changes with respect to. That explicitness makes it the standard notation for related rates, implicit differentiation, and any problem with several variables.

A limit is the single value a function approaches as its input approaches a given number. The limit describes where the function is heading, not where it lands, so a limit can exist at a point where the function is undefined.

A limit at infinity is the value a function approaches as its input grows without bound in the positive or negative direction. It describes the end behavior of the graph, and a finite limit at infinity is exactly what produces a horizontal asymptote.

The limit comparison test compares two series of positive terms by taking the limit of their ratio. If that limit is finite and positive, both series converge or both diverge together.

The limit definition of the derivative sets f prime of x equal to the limit, as h approaches zero, of the difference quotient: f of x plus h minus f of x, all over h. It turns the average rate of change over a shrinking interval into the instantaneous rate of change at a point.

A limit does not exist when the function fails to approach a single finite value. There are three standard causes: the left and right limits disagree, the function grows without bound, or the function oscillates infinitely often near the point.

The limit laws let you distribute a limit across sums, differences, products, quotients, and powers. Each law requires the individual limits to exist, and the quotient law additionally requires the denominator's limit to be nonzero.

To find the limit of a composite function, take the inner limit first, call it L, then apply the outer function to L. That step is guaranteed valid when the outer function is continuous at L. If it is discontinuous at L, the composite limit can differ from its value there or fail to exist.

The limit of a sequence is the single value its terms approach as the index n grows without bound. If no such value exists, the sequence diverges. This limit matters for series too: the terms of any convergent series must approach zero, which is exactly what the nth-term test checks.

The limits of integration are the two numbers attached to a definite integral, marking where the accumulation starts and stops. When you substitute, either convert them to the new variable or convert the antiderivative back before evaluating.

Linear approximation estimates a function near a point by using its tangent line there. It is accurate close to the point of tangency and degrades as you move away, and concavity tells you the direction of the error.

Linearization approximates a function near a point by using its tangent line there. Because a smooth curve and its tangent line are nearly identical close to the point of tangency, the line gives a good estimate for inputs near that point.

Local linearity means that a differentiable function, zoomed in near a point, looks almost exactly like its tangent line there. That is why the tangent line closely estimates function values near the point of tangency, and it is the basis of linear approximation.

The logarithm properties say the log of a product is the sum of the logs, the log of a quotient is the difference, and the log of a power moves the exponent out front as a coefficient. In calculus they are applied before differentiating, to turn one hard derivative into several easy ones.

Logarithmic differentiation is a technique where you take the natural logarithm of both sides of an equation before differentiating implicitly. It converts products into sums and exponents into coefficients, which is what makes a variable base raised to a variable power differentiable.

The logistic differential equation, dP/dt = kP times the quantity 1 minus P over L, has two equilibrium solutions: P = 0 and P = L. Their signs settle the rest. A population below L rises toward it, one above L falls toward it, and no solution curve ever crosses L. See logistic growth for the model itself.

Logistic growth models a quantity that grows nearly exponentially when small but levels off as it approaches a maximum called the carrying capacity. Its growth rate is fastest when the quantity is exactly half the carrying capacity.

A Maclaurin series is a Taylor series centred at zero. It is not a different object, just the most common special case, and the standard ones for the exponential, sine, and cosine functions are worth knowing by heart.

The Mean Value Theorem says that if a function is continuous on a closed interval and differentiable on the open interval, then somewhere inside there is a point where the instantaneous rate of change equals the average rate of change across the whole interval.

The Mean Value Theorem for Integrals says that if a function is continuous on a closed interval, then at some interior point the function equals its average value on that interval. So the definite integral equals that one height multiplied by the width of the interval.

A midpoint Riemann sum uses the height of the function at the centre of each subinterval. It is normally far more accurate than a left or right sum, and where concavity does not change it errs in the opposite direction to the trapezoidal rule.

A function is monotonic on an interval when it is either increasing throughout or decreasing throughout, never both. Equivalently the derivative does not change sign on that interval, which also guarantees the function is one-to-one there.

Motion in the plane describes a particle whose position is a vector, x of t and y of t. Velocity is the vector x prime of t and y prime of t, speed is its magnitude, square root of x prime squared plus y prime squared, and total distance is the integral of speed over the time interval.

The natural exponential function is e to the x, where e is about 2.71828. It is the unique exponential function that is its own derivative, so at every point its slope equals its height. Its antiderivative is itself plus a constant.

The natural logarithm is the logarithm to base e. Its derivative is one over x, and reading that backwards makes it the antiderivative of one over x, which is the single case the power rule cannot handle.

The net change theorem says that integrating a rate of change over an interval gives the net change in the underlying quantity. It is the interpretation of the Fundamental Theorem of Calculus that applied problems actually use.

A definite integral computes net signed area, counting regions above the x-axis as positive and regions below as negative. That is why an integral can come out zero or negative even though geometric area never can.

Newton's law of cooling says an object's temperature changes at a rate proportional to the gap between it and the ambient temperature, with a negative constant of proportionality. It is a separable differential equation, and every non-equilibrium solution approaches the ambient temperature without ever reaching it.

A nonremovable discontinuity is a break that cannot be repaired by redefining the function at that one point. Where f is defined on both sides of c, that happens exactly when the two sided limit at c fails to exist; at an endpoint, the relevant one sided limit is what has to exist.

The normal line at a point on a curve is the line through that point perpendicular to the tangent line. Its slope is the negative reciprocal of the derivative there, provided the derivative is not zero.

The nth term test says that if the terms of a series do not approach zero, the series diverges. It can only ever prove divergence, because terms approaching zero says nothing about whether the series converges.

Numerical integration means approximating a definite integral with sums of areas, using left, right, midpoint, or trapezoid estimates. You need it when the integrand has no elementary antiderivative, or when the function is given only as a table of values.

The objective function in an optimization problem is the quantity being maximized or minimized. The constraint is a separate equation relating the variables, used to rewrite the objective in terms of a single variable before differentiating.

An odd function satisfies f of negative x equals negative f of x, making its graph symmetric about the origin. Its integral over any interval symmetric about zero is exactly zero, because the two halves cancel.

A one-sided limit is the value a function approaches as the input comes in from one direction only. The left-hand limit uses inputs below the point and the right-hand limit uses inputs above it. The two-sided limit exists exactly when both agree.

A function is one-to-one when different inputs always give different outputs, which is what the horizontal line test checks. Only a one-to-one function has an inverse, and any strictly increasing or strictly decreasing function is automatically one-to-one.

In an optimization problem you are handed a quantity to make as large or as small as possible, usually with a restriction on the variables.

The order of a differential equation is the order of the highest derivative that appears in it. An equation containing only a first derivative is first order, and one containing a second derivative is second order. Every differential equation in AP Calculus is first order.

The orientation of a parametric curve is the direction the curve is traced as the parameter increases. It belongs to the parametrisation rather than to the shape, since two parametrisations can cover the same set of points in opposite directions.

An oscillating discontinuity happens where a function swings between values infinitely often as it approaches a point, so no limit exists. The standard example is sine of one over x at x equals zero, which crosses every value between negative one and one infinitely many times.

The outer function is the one applied last in a composition, wrapping around everything else. The chain rule differentiates it first, evaluated at the inner function rather than at x, and then multiplies by the inner function's derivative.

An approximation of a definite integral overestimates when it exceeds the true value and underestimates when it falls short. Left and right Riemann sums are decided by whether the function is increasing or decreasing; the trapezoidal and midpoint rules are decided by concavity.

A p-series is the sum of the reciprocals of the positive integers raised to a fixed power p. It converges when p is greater than one and diverges when p is less than or equal to one.

A parameter is the independent variable, usually called t, whose values generate the points of a parametric curve. Each value of t produces one point, so the parameter carries the order and the pace of the tracing, information an equation in x and y alone cannot hold.

Parametric arc length is the distance along a curve given by x of t and y of t. Over a parameter interval that traces the curve exactly once it equals the integral of the square root of the sum of the squares of x prime of t and y prime of t, which is the integral of speed.

For a parametric curve the slope dy over dx is the ratio of dy over dt to dx over dt. The second derivative is not the ratio of second derivatives: you differentiate the first derivative with respect to t and then divide by dx over dt again.

Parametric equations define the coordinates of a point separately as functions of a third variable called the parameter, usually time. They describe not just the shape of a curve but the direction and speed with which it is traced.

Partial fraction decomposition rewrites a rational function as a sum of simpler fractions with linear denominators. Each piece then integrates to a natural logarithm, which is what makes otherwise impossible rational integrals routine.

The nth partial sum of a series is the total of its first n terms. The series converges precisely when this sequence of partial sums approaches a finite limit, which is what makes an infinite sum meaningful.

A particular solution is the one member of a differential equation's solution family that passes through a specified point. The initial condition is what determines the arbitrary constant and selects that single curve.

A periodic function repeats its values on a fixed period: some p > 0 satisfies f(x + p) = f(x) for every x in the domain, and the period means the smallest such p.

A piecewise function uses different rules on different intervals of its domain. The interesting points are the boundaries between pieces, where you must check that the one-sided limits agree with each other and with the defined value before calling the function continuous.

The area enclosed by a polar curve is the integral of one half r squared with respect to theta. The region is built from circular sectors rather than rectangles, which is where the one half and the square come from.

Polar coordinates locate a point by its distance from the origin and the angle it makes with the positive horizontal axis. A polar curve expresses that distance as a function of the angle, which makes circles and roses far simpler to describe than in rectangular form.

A polar curve expresses distance from the origin as a function of the angle. Familiar shapes include circles, cardioids, limacons, and rose curves, and a negative r means the point is plotted in the opposite direction from the angle.

Polar symmetry means a curve r equals f of theta repeats across a line or about the pole. Replacing theta by negative theta without changing the equation shows symmetry about the polar axis, replacing theta by pi minus theta tests the vertical line, and replacing r by negative r tests the pole.

Polar to Cartesian conversion rewrites a polar point or curve in rectangular form using x equals r cos theta and y equals r sin theta. Going the other way, r squared equals x squared plus y squared. For a curve r equals f of theta, it turns theta into a parameter.

A polynomial function is a finite sum of terms, each a constant times x to a nonnegative integer power, such as 3x cubed minus 5x plus 2. Every polynomial is continuous and differentiable at every real number, which is why direct substitution always evaluates its limits.

A position function gives an object's location as a function of time. Its derivative is velocity, and the difference between two of its values is the displacement over that interval, which is why the Fundamental Theorem connects the two descriptions.

A power function is a constant times x raised to a fixed real exponent, such as 3x cubed, the square root of x, or 1 over x squared. The variable is the base and the exponent is a constant, which is exactly the setup the power rule differentiates.

The power rule says that to differentiate a power of the variable, multiply by the exponent and then reduce the exponent by one. It holds for every real exponent, including negative and fractional ones.

A power series is an infinite sum of constant multiples of powers of the variable minus a fixed centre. It converges for the centre always, and generally on an interval around it whose half-width is the radius of convergence.

A power series representation writes a function as an infinite sum of powers of x minus a centre, valid on an interval around that centre. Most are built by substituting into the geometric series, or by differentiating or integrating a series you already know.

Prime notation writes the derivative of f as f prime of x, the second derivative as f double prime, and so on. It is compact and reads well for named functions, but unlike Leibniz notation it does not state which variable you differentiated with respect to.

The product rule differentiates a product of two functions: multiply the first by the derivative of the second, then add the second times the derivative of the first. The derivative of a product is never the product of the derivatives.

Definite integrals obey a small set of rules: swapping the bounds reverses the sign, equal bounds give zero, constants factor out, sums split apart, and an integral can be broken at any interior point.

The Pythagorean identity says that sine squared plus cosine squared equals one, with two more forms found by dividing that equation through by cosine squared or by sine squared.

The quotient rule differentiates a fraction: the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the denominator squared. The order matters because of the subtraction.

Radian measure sizes an angle by the arc it cuts off divided by the radius, so a full circle is two pi radians. Every calculus formula for trigonometric functions assumes radians. In degrees the derivative of sine would be pi over 180 times cosine instead of cosine.

Every radius function measures the distance from the axis of revolution to one boundary of the region. In a disk or washer integral that distance is the quantity that gets squared; in a shell integral the same distance is multiplied by the height instead. A washer needs two at once, an outer R and an inner r.

The radius of convergence is the distance from the centre of a power series to the edge of the region where it converges. It is found by applying the ratio test and solving the resulting inequality for the variable.

The range of a function is the set of all output values it actually attains. Calculus finds it by locating the absolute extrema and checking end behavior, rather than by guessing from the formula.

Rate in minus rate out is the standard accumulation setup: the net rate is the rate entering minus the rate leaving. The amount at time t is the starting amount plus the integral of that difference, and it can only turn around where the rates cross, so the maximum is such a crossing or an endpoint.

A rational function is a ratio of two polynomials, continuous wherever the denominator is nonzero. Each zero of the denominator gives a hole if that factor cancels completely out of the denominator, and a vertical asymptote if it appears more times in the denominator than in the numerator.

Rectilinear motion is movement along a straight line. Position differentiates to velocity, velocity differentiates to acceleration, and integrating runs the chain backwards, which is why one motion problem can test both halves of the course.

A recursive sequence is given by one or more starting values together with a rule that produces each new term from the terms before it, instead of an explicit formula in n. Unless you can find a closed form, and arithmetic and geometric recursions always have one, you reach a distant term by iterating.

A region bounded by curves is the set of points enclosed between two graphs over an interval. Its area is the integral of upper minus lower in x, or right minus left in y. When the curves alone close the region, their intersection points give the limits; otherwise the named boundary lines do.

A related rate is the rate of change of one quantity expressed in terms of the rate of change of another that it depends on. These problems link the quantities with an equation, then differentiate both sides with respect to time.

A relative extremum is a point that is highest or lowest compared only with nearby points. It occurs at a critical point where the first derivative changes sign, and it need not be the largest or smallest value the function takes overall.

A relative maximum is a value that is larger than every nearby value of the function. It occurs at a critical number where the derivative changes from positive to negative, meaning the graph stops rising and starts falling. It is a local peak and need not be the largest value overall.

A relative minimum is a value that is smaller than every nearby value of the function. It occurs at a critical number where the derivative changes from negative to positive, meaning the graph stops falling and starts rising. It is a local valley and need not be the smallest value overall.

The relative rate of change of a positive quantity is its derivative divided by its current value, so growth is measured as a fraction of size rather than in absolute units. It is usually reported as a percent per unit time, and it equals the derivative of the natural logarithm of the quantity.

A removable discontinuity is a point where the limit exists but the function either is not defined there or is defined to be a different value. It appears as a hole in the graph, and redefining the single point would make the function continuous.

Which way the slice points decides the whole setup. A representative rectangle is the thin slice you draw inside a region: perpendicular to the axis of revolution it sweeps a disk or washer, parallel to the axis it sweeps a shell. Its thickness is dx or dy, and that choice fixes the variable of integration.

The reverse power rule integrates a power of x by adding one to the exponent and dividing by that new exponent, then adding a constant. It works for every exponent except negative one. That one excluded case, one over x, integrates instead to the natural log of the absolute value of x.

A Riemann sum approximates a definite integral by slicing the interval into subintervals and adding up the area of a rectangle on each one. As the number of subintervals grows without bound, the sum approaches the exact value of the integral.

A right Riemann sum uses the right endpoint of each subinterval as the rectangle height. For an increasing function every rectangle rises above the curve, so the sum overestimates the integral; for a decreasing function it underestimates.

The right-hand limit is the value a function approaches as the input moves toward a point through values above it. It is written with a plus superscript on the target. The two-sided limit exists only when the right-hand and left-hand limits are equal.

Rolle's Theorem says that if a function is continuous on a closed interval, differentiable inside it, and takes the same value at both endpoints, then there is an interior point where the derivative is zero. It is the Mean Value Theorem with equal endpoint values.

A root of a function is an input where the output equals zero. The same number is called a zero of the function, and on a graph it is an x-intercept. Roots matter because zeros of a derivative locate critical points, and the Intermediate Value Theorem proves a root exists without finding it.

The root test takes the limit of the nth root of the absolute value of the terms. A limit less than one means the series converges absolutely, greater than one means it diverges, and exactly one means the test tells you nothing.

A sample point is the input inside a subinterval where a Riemann rectangle takes its height. Left, right, and midpoint sums are the three standard choices, but any point in the subinterval is allowed, and for a continuous function every choice gives the same limit.

A secant line is a straight line through two points on a curve. Its slope equals the average rate of change between those points, and as the two points slide together the secant line approaches the tangent line.

The second derivative is what you get by differentiating the derivative. It measures how the rate of change is itself changing, it determines concavity, and in motion problems it is acceleration.

The second derivative of a parametric curve is the derivative of dy over dx taken again with respect to the parameter t, then divided by dx over dt. It is not the second derivative of y over the second derivative of x. Its sign gives the concavity of the curve.

The second derivative test classifies a critical point by evaluating the second derivative there. A negative value means a local maximum, a positive value means a local minimum, and a value of zero makes the test inconclusive.

Separation of variables solves a differential equation by moving all the y terms with dy to one side and all the x terms with dx to the other, then integrating both sides. It only works when the derivative factors into a function of x times a function of y.

A sequence is an ordered list of numbers indexed by the positive integers. It converges when its terms approach a single finite limit, and the tools for deciding that are limit techniques applied with n in place of x.

A series is the sum of the terms of a sequence, often infinitely many. It converges when the sequence of its partial sums approaches a finite limit, and that limit is defined to be the sum of the series.

The shell method computes a volume of revolution by slicing the region parallel to the axis of rotation and treating each slice as a thin cylindrical shell. Its integrand is two pi times the radius times the height.

Sigma notation writes a sum compactly using the Greek capital sigma, with an index variable, a starting value below and a stopping value above. It is the standard way to express a Riemann sum and every infinite series in the course.

A sign chart is a number line recording where a function or one of its derivatives is positive or negative. You mark the zeros and undefined points, test one value in each interval, and read off the signs to locate increasing and decreasing behavior and concavity.

A slant asymptote is a non-horizontal line that a curve approaches as x goes to positive or negative infinity. A rational function has one exactly when the degree of the numerator is one more than the degree of the denominator, and you find it by dividing.

A slope field is a grid of short line segments, each drawn with the slope that a differential equation assigns to that point. It shows the shape of every solution curve at once without solving the equation.

The slope of a polar curve is dy over dx, not dr over d theta. Using x equals r cos theta and y equals r sin theta, dy over dx equals dr over d theta times sin theta plus r cos theta, all divided by dr over d theta times cos theta minus r sin theta.

A smooth curve is one that is continuously differentiable with no corners or cusps. For a parametric curve x of t, y of t, this means x prime and y prime are continuous and never both zero, so the curve has a tangent direction everywhere. Arc length only needs x prime and y prime to be continuous.

A solid of revolution is the three-dimensional shape swept out when a plane region is rotated about a line. Its volume is computed by integrating cross-sectional areas, using the disk method, the washer method, or the shell method.

The special trigonometric limits are sine of x over x approaching 1 and one minus cosine of x over x approaching 0, both as x approaches zero. They hold only when x is measured in radians, and they are what let you differentiate sine and cosine from the definition.

Speed is the magnitude of velocity. Along a line that is the absolute value of the velocity function, and for a curve given parametrically it is the magnitude of the velocity vector, so speed is never negative in either case.

A particle is speeding up when its velocity and acceleration have the same sign, and slowing down when they have opposite signs. Speeding up means the speed is increasing, not that the motion is in the positive direction, so a particle can speed up while moving backward.

The squeeze theorem says that if a function is trapped between two others near a point, and those two share the same limit there, the trapped function must have that limit too. It is the standard way to evaluate limits that resist algebra, such as x squared times sine of one over x.

The derivative of a sum is the sum of the derivatives, and the same holds for a difference. This is what makes differentiation term by term legal, and it is why a polynomial can be differentiated one piece at a time.

For a geometric series whose common ratio has absolute value less than one, the sum equals the first term present divided by one minus the ratio. This is one of the very few infinite series with a clean closed-form total, and it powers many comparison and power-series arguments.

The tangent line to a curve at a point is the straight line through that point whose slope equals the derivative there. It is the best linear approximation of the curve near that point.

A Taylor polynomial is the finite piece of a Taylor series obtained by stopping after a chosen degree. It approximates the function near the centre, and higher degree generally means a better approximation over a wider range.

The Taylor remainder is the exact difference between a function and its Taylor polynomial of a given degree, so the function equals the polynomial plus the remainder. You rarely compute it exactly. You bound its size instead, which turns an approximation into a result with guaranteed accuracy.

A Taylor series represents a function as an infinite polynomial whose coefficients come from the function's derivatives at a single centre point. Each coefficient is a derivative at the centre divided by the factorial of its order.

A telescoping series is one where consecutive terms cancel in the partial sum, leaving only a few surviving pieces at the ends. It is one of the rare series whose exact sum you can compute, by taking the limit of that simplified partial sum.

Term-by-term differentiation means differentiating a power series one term at a time, as if it were a long polynomial. Inside the interval of convergence the result is the derivative of the sum, and it keeps the same radius of convergence, though behavior at the endpoints can change.

Term-by-term integration means integrating a power series one term at a time. Inside the interval of convergence the result is an antiderivative of the sum, with the same radius of convergence. It is the standard way to derive the series for arctangent and for the natural log of one plus x.

Every piece of a region counts positively in total area, so the interval is split at each sign change and the pieces where the function is negative are negated before being added. In symbols it is the integral of the absolute value of f, and between two curves the integral of the absolute value of f minus g.

Total distance traveled is the integral of the absolute value of velocity, which is the integral of speed. Unlike displacement it counts motion in both directions positively, so it is always at least as large as the size of the displacement.

The trapezoidal rule approximates a definite integral by joining consecutive points with straight segments and summing the resulting trapezoids. It equals the average of the left and right Riemann sums and is usually more accurate than either.

A turning point is a point on a graph where the function changes from increasing to decreasing or from decreasing to increasing. The derivative must change sign there, so every turning point is a critical point, but not every critical point is a turning point.

The unit circle is the circle of radius 1 centred at the origin, where travelling a signed arc length t from the point (1, 0), counterclockwise when t is positive and clockwise when t is negative, lands you at the point (cos t, sin t). It supplies the exact trig values calculus expects, with t measured in radians.

The units of a derivative are the units of the output divided by the units of the input. If a tank holds W gallons after t minutes, W prime is measured in gallons per minute, and a second derivative divides by the input unit once more, giving gallons per minute per minute.

The variable of integration is the letter named by the differential, and it tells you what you are integrating with respect to and which other letters count as constants. In a definite integral it is a dummy name, so renaming it leaves the value unchanged.

The magnitude of a vector is its length, computed as the square root of the sum of the squares of its components. The magnitude of a velocity vector is the speed of the particle, so it is a single non-negative number, and integrating it over a time interval gives total distance traveled.

A vector-valued function assigns a vector to each value of a parameter, most often describing the position of a particle over time. Differentiating each component separately gives the velocity vector, and differentiating again gives acceleration.

Velocity is the derivative of position with respect to time. It is a signed quantity, so its sign tells you the direction of motion, and its absolute value is the speed.

A vertical asymptote is a vertical line that a graph approaches without ever reaching, occurring wherever at least one one-sided limit is infinite. For a rational function, these sit at the zeros of the denominator that remain after all common factors cancel.

A vertical tangent occurs at a point where the derivative grows without bound with the same sign from both sides, so the tangent line is vertical. The function remains continuous there but is not differentiable, since infinite slope is not a number.

Volume by cross sections integrates the area of each slice along an axis. The slices are a stated shape such as a square, a semicircle, or an equilateral triangle built on a base region, and no rotation is involved at all.

The washer method computes the volume of a solid of revolution with a hole through it by subtracting the inner circle's area from the outer circle's area on every slice. It is used whenever the rotated region does not touch the axis of rotation.