Which one do I use?

Differentiability is the stronger condition: every differentiable function is continuous, but plenty of continuous functions are not differentiable. Continuity means no break in the graph; differentiability additionally means no corner, cusp, or vertical tangent.

A removable discontinuity is a hole where the limit exists but the value is missing or wrong. A jump has two different one-sided limits. An infinite discontinuity has at least one infinite one-sided limit. Only the removable kind can be repaired by redefining a single point.

The Intermediate Value Theorem guarantees that a continuous function attains some output value, which is how you prove a root exists. The Mean Value Theorem guarantees that a differentiable function attains some slope, namely the average rate of change across the interval.

A vertical asymptote occurs where a one-sided limit is infinite, and a graph can never cross one. A horizontal asymptote comes from a finite limit as the input grows without bound, describes long-run behaviour only, and can be crossed any number of times.

A limit existing is the weaker condition. The limit only asks what the function approaches from both sides, so it can exist at a hole where the function is undefined. Continuity asks for more: the limit must exist, f(a) must be defined, and those two numbers must be equal.

These are the same fact in two forms. The limit at infinity is the computation you carry out, and the horizontal asymptote is the line that limit certifies: the graph has the asymptote y equals L exactly when the limit is the finite number L. The computation gives a number, the asymptote is a line.

Ask what the two-sided limit does. If it exists as a finite number, the discontinuity is removable: the graph has a hole, and redefining that one point repairs it. If the function grows without bound near the point, the discontinuity is infinite: the graph has a vertical asymptote, and no redefinition can fix it.

A two-sided limit exists exactly when both one-sided limits exist and are equal, and its value is their common value. If the two sides disagree the two-sided limit does not exist, even though both one-sided limits do. At the endpoint of a domain only one side is available, and that is normal.

Substitution is always the first move on a limit at a finite point, and what it returns tells you what to do next. A real number is the answer whenever f is continuous at that point. A nonzero number over zero is undefined and signals a vertical asymptote, so the limit is infinite or does not exist.

L'Hopital's rule applies only to a differentiable quotient in the form 0 over 0 or infinity over infinity. The squeeze theorem needs only upper and lower bounds that share a limit, so it handles oscillating limits like x squared times sine of 1 over x, where L'Hopital does not apply.

A vertical tangent sits at a point where the function is defined and continuous but its derivative runs off to the same infinity from both sides; a vertical asymptote sits where at least one one sided limit of the function itself is plus or minus infinity.

Calculus uses radians, always. The derivative of sin x is cos x only in radians; in degrees it is pi over 180 times cos x. The reason is the special limit sin x over x, which equals 1 in radians and pi over 180 in degrees, and every trig derivative is built on it.

A function being undefined at a point says nothing about its limit there, because the limit only depends on nearby values and never on the point itself. A removable discontinuity is exactly the case where the function is undefined but the limit exists perfectly well.

Average rate of change measures over an interval and equals the slope of the secant line joining the endpoints. Instantaneous rate of change measures at a single point and equals the slope of the tangent line, which is the derivative.

Use the product rule when two functions are multiplied side by side, and the chain rule when one function sits inside another. The test is to ask what you would compute last if you evaluated by hand: a multiplication means product rule, an outer function means chain rule.

A secant line passes through two points on a curve and its slope is the average rate of change. A tangent line touches at a single point and its slope is the derivative there. Shrinking the gap between the two points turns the secant into the tangent.

The tangent line at a point has slope equal to the derivative there. The normal line is perpendicular to it, so its slope is the negative reciprocal of the derivative. Both pass through the same point on the curve.

Use the power rule when the variable is the base and the exponent is a constant, as in x cubed. Use the exponential rule when the variable is the exponent and the base is a constant, as in two to the x. When the variable is in both places, neither rule works and you need logarithmic differentiation.

The derivative is a rate: a function whose value at x is the instantaneous slope. The differential is an amount: the estimated change in y caused by a small step dx, found by multiplying the derivative by that step. Use the derivative when the question asks how fast, the differential when it asks how much.

They are inverse operations. Differentiating a function produces exactly one answer, its rate of change. Antidifferentiating produces a whole family, because any constant vanishes when you differentiate, which is why the plus C is not optional. The Fundamental Theorem of Calculus is the bridge between them.

Integration by parts is the product rule integrated and then rearranged, which is exactly where its minus sign comes from. The product rule differentiates a product outright and finishes.

Every quotient can be written as a product with a negative power, so the quotient rule is optional and never required. Use it when the denominator is messy enough that you would not want to expand it, and rewrite as a product when the denominator is a single power.

The one-sided slopes separate them: a corner has two different finite slopes, while a cusp has slopes running to positive infinity on one side and negative infinity on the other. Both keep the graph continuous, and at both the derivative fails to exist, so both are critical numbers.

Differentiate explicitly when the equation is already solved for the output variable. Use implicit differentiation when the variables are mixed together and isolating one would be messy or impossible, as with a circle.

Implicit differentiation is not a separate rule; it is the chain rule applied to y, treated as an unknown function of x. The derivative of y that appears whenever you differentiate a term containing y is exactly the chain rule factor. Reach for it when the equation mixes x and y instead of naming the inside function.

They are the same rule read in opposite directions: the chain rule multiplies by the derivative of the inner function, and u-substitution divides that factor back out. In this reverse-chain-rule form, substitution works only when the inner derivative is already sitting in the integrand as a factor.

Logarithmic differentiation is implicit differentiation applied to the equation ln y equals ln f of x, so it is a special case rather than a rival. Reach for it when the exponent contains x, since neither the power rule nor the exponential rule applies there.

Related rates is implicit differentiation performed with respect to time. Differentiating with respect to x gives dy/dx and leaves the x terms alone; differentiating with respect to t gives both dx/dt and dy/dt, so every changing letter picks up a chain factor instead of just one.

Related rates problems link two quantities changing over time and differentiate with respect to time. Optimization problems find a largest or smallest value and set a derivative equal to zero. Both start from a geometric relationship, but only optimization uses a constraint to eliminate a variable.

Velocity is signed, so it tells you both how fast and in which direction. Speed is the absolute value of velocity and is never negative. Speed increases exactly when velocity and acceleration have the same sign, which is not the same as acceleration being positive.

Velocity is the first derivative of position and acceleration is the second, so acceleration measures how the velocity is changing, not how fast the particle is going. When velocity and acceleration share a sign the particle is speeding up; when their signs differ it is slowing down.

Position is where the particle is at a single instant. Displacement is the change in position across an interval, found by integrating velocity, so it can be zero even though the particle moved the whole time. Add the displacement to the starting position to get the final position.

The tangent line approximation is the first-degree Taylor polynomial, so these are not rival methods: one is the opening case of the other. Each extra degree adds a term built from a higher derivative, and the first of those corrects for curvature, which is exactly what a straight line cannot do.

Positive acceleration does not mean speeding up. Acceleration is the rate of change of velocity, and it can be positive while a particle slows down. What decides speeding up is whether velocity and acceleration have the SAME sign, not whether acceleration is positive.

Both use the same tangent line. Linearization estimates the VALUE of the function near a point, giving f of a plus f prime of a times the change in x. The differential estimates the CHANGE in the function, which is just f prime of a times dx. One answers how much, the other how much more.

Average velocity is displacement divided by elapsed time, so it can be zero or negative. Average speed is total distance divided by elapsed time, so it is never negative. They are equal only when the particle never changes direction.

A local extremum is the largest or smallest value compared only to nearby points. An absolute extremum is the largest or smallest across the entire interval. Local extrema occur only at critical points; absolute extrema occur at a critical point or at an endpoint.

The first derivative test classifies a critical point by whether the derivative changes sign there, and it never fails. The second derivative test checks the sign of the second derivative at the point, which is faster but says nothing when that value is zero.

A critical point is where the first derivative is zero or undefined, and it is where extrema can occur. An inflection point is where the second derivative changes sign, and it is where concavity flips. They answer different questions and often sit at different places.

Rolle's Theorem is the special case of the Mean Value Theorem where the two endpoint values are equal. When they match, the average rate of change is zero, so the guaranteed slope is zero and the tangent line is horizontal.

Increasing means the first derivative is positive, so the function is rising. Concave up means the second derivative is positive, so the slope itself is rising. The two are completely independent: a function can be increasing while concave down.

Critical points are candidates; extrema are verdicts. Every interior extremum sits at a critical point, but x cubed has one at the origin and no extremum. A first derivative sign change promotes a candidate to a local extremum; comparing values at the candidates and the endpoints promotes one to an absolute extremum.

The Extreme Value Theorem needs only continuity on a closed interval and hands you an absolute maximum and an absolute minimum. The Mean Value Theorem also needs differentiability on the open interval, and hands you a point where the instantaneous rate equals the average rate.

The candidates test compares function values at critical points and endpoints to find the absolute maximum and minimum on a closed interval. The first derivative test reads sign changes of the derivative to label relative extrema, and it never looks at endpoints.

The objective function is the quantity you maximise or minimise, and the constraint is the equation you solve in order to eliminate a variable from it. In the substitution method the constraint is solved and substituted, and only the objective is differentiated.

A definite integral has bounds and evaluates to a number. An indefinite integral has no bounds and evaluates to a family of functions plus a constant of integration. One is a value, the other is a function, and that single difference explains when the constant is needed.

Part 1 differentiates an accumulation function and returns the integrand, which is the statement that differentiation undoes integration. Part 2 evaluates a definite integral as an antiderivative at the top bound minus the same antiderivative at the bottom.

Use substitution when the integrand contains a function and something close to its derivative, since substitution reverses the chain rule. Use integration by parts when the integrand is a product of two unlike types, such as a polynomial times an exponential, since parts reverses the product rule.

Left and right Riemann sums differ only in whether each rectangle takes its height from the left or right endpoint of its subinterval. For an increasing function the left sum underestimates and the right sum overestimates, and for a decreasing function the roles swap.

The trapezoidal rule joins consecutive points with straight chords, so on a concave up curve it overestimates. The midpoint rule takes each rectangle's height from the centre of its subinterval and errs in the opposite direction, underestimating on a concave up curve.

A definite integral computes net signed area, so region below the horizontal axis counts as negative and can cancel region above it. True geometric area integrates the absolute value of the function, so every piece contributes positively.

A Riemann sum is a finite approximation: it adds the areas of n rectangles across n subintervals. The definite integral is the exact value those sums approach as n goes to infinity and the subinterval width goes to zero. One is an estimate you can compute by hand, the other is the limit it converges to.

Average value is a number: divide the integral of the function over an interval by the length of that interval. The Mean Value Theorem for Integrals is the guarantee that a continuous function actually takes on that number at some point c inside the interval. One computes, the other certifies.

An integral is improper when a limit of integration is infinite or the integrand blows up somewhere on the interval. You replace the bad endpoint with a variable, evaluate an ordinary definite integral, and take a limit. If that limit is finite the integral converges; otherwise it diverges and has no value.

Use substitution when the integrand holds an inner function together with its own derivative as a factor. Use partial fractions when the integrand is a rational function whose denominator factors and no inner derivative appears, so you split it into simpler fractions and integrate each as a logarithm.

Check the degrees first: if the numerator degree is at least the denominator degree, long division runs first and partial fractions applies only to the proper remainder. Partial fractions on an improper fraction produces a contradiction, because constant numerators cannot rebuild a numerator of that degree.

Type one has an infinite limit of integration; type two has a finite interval but an integrand that blows up at an endpoint or somewhere inside it.

A definite integral with constant limits is a single number, so its derivative is zero. An accumulation function puts the variable in the upper limit, so it assigns an output to every input x, which is what makes it a function, and differentiating it hands back the integrand.

An improper integral extends a definite integral to an unbounded interval or integrand, defined as a limit; a series adds discrete terms. The integral test links them: when the terms come from a positive, continuous, eventually decreasing function, the two share a verdict but almost never a value.

Left and right sums are decided by whether the function increases or decreases; trapezoidal and midpoint sums are decided by concavity instead. On a concave up curve the trapezoidal rule overestimates and the midpoint rule underestimates, whichever direction the function happens to be heading.

An antiderivative is any single function whose derivative is f. The indefinite integral is the entire family of them, which is why it carries a plus C. So x squared over 2 is an antiderivative of x, while the indefinite integral of x is x squared over 2 plus C.

A Riemann sum approximates with rectangles, using one function value per subinterval, and whether it over or underestimates depends on whether the function is increasing or decreasing. The trapezoidal rule uses both endpoints to form a trapezoid, and its error direction depends on concavity instead.

The general solution of a differential equation is the whole family of functions that satisfy it, written with an arbitrary constant. A particular solution is the single member of that family passing through a given initial condition.

Exponential growth has a rate proportional to the current amount and increases without any ceiling. Logistic growth multiplies that by a braking factor, so growth slows as the quantity nears a carrying capacity and is fastest at exactly half of it.

A slope field is a picture: short segments showing the slope a differential equation assigns at each point, sketching every solution at once. Euler's method is a numerical procedure that starts from an initial condition and steps along tangent lines to produce actual approximate values.

Use Euler's method when the question hands you a step size, or when the equation will not separate. Solve exactly by separating variables whenever the equation allows it. Euler underestimates a solution that stays concave up and overestimates one that stays concave down, because every step follows a tangent line.

A slope field draws the slope the differential equation assigns at each point, so it shows every solution at once. A solution curve is one function satisfying the equation, picked out of that family by an initial condition. Sketching a solution curve means following the segments through the given point.

Displacement is the net change in position, found by integrating velocity, and it can be zero even after a long trip. Total distance travelled integrates the absolute value of velocity, so movement in both directions adds up.

Average value integrates a function over an interval and divides by the interval length, answering what constant height would give the same area. Average rate of change divides the change in output by the change in input, answering how fast the function moved on average.

Use the disk method when the region being rotated touches the axis of rotation, so each slice is a solid circle. Use the washer method when a gap separates the region from the axis, because that gap becomes a hole and each slice becomes a ring.

Washers slice perpendicular to the axis of rotation and integrate in the axis variable. Shells slice parallel to it and integrate in the other variable. Choose whichever lets you avoid rewriting the equation in terms of a variable it is not already solved for.

Both methods integrate cross-sectional area along an axis. A solid of revolution always has circular or ring-shaped slices because it is swept by rotation. A solid with known cross-sections can have squares, semicircles, or triangles instead, and the problem tells you which.

Slice perpendicular to the axis of rotation and you get disks; slice parallel to it and you get shells. Both return the same volume, so the choice is practical: take the direction whose slices need no rewriting of the curves, because the thickness variable is the one your functions are already solved for.

Arc length measures how long a curve is, integrating with respect to whatever variable traces it. Distance travelled integrates speed over a time interval, so it counts every retrace. For a particle that never doubles back the two agree, but only distance travelled counts a stretch of path covered twice.

Use polar area when the region is swept out by an angle and the boundary is r as a function of theta, because the natural slice is a circular sector and that is where the factor of one half comes from. Use Cartesian area when the boundary is y as a function of x and the slice is a vertical strip of width dx.

Parametric equations define the coordinates separately as functions of a parameter, usually time. Polar equations define distance from the origin as a function of angle. Polar is really a special case of parametric, which is why the slope formulas match.

The parametric second derivative is the t derivative of dy/dx, divided again by dx/dt. It is not the second t derivative of y over the second t derivative of x. In Cartesian form you differentiate twice in x and stop; in parametric form every step down costs another division by dx/dt.

A sequence is an ordered list of numbers; a series is the sum of that list. A sequence converges when its terms approach a limit, while a series converges when its partial sums approach a limit, which is a much stronger requirement.

The ratio test compares consecutive terms and is the right choice when factorials or products of consecutive integers appear. The root test takes the nth root and is the right choice when the whole term is raised to the nth power. Both are inconclusive when the limit equals one.

Direct comparison requires you to prove a term by term inequality against a known series. Limit comparison only requires the two series to grow at comparable rates, judged by the limit of their ratio, which makes it easier to apply when the inequality is awkward.

A series converges absolutely when the series of absolute values also converges. It converges conditionally when it converges as written but the absolute values diverge, meaning the convergence depends entirely on cancellation between positive and negative terms.

A geometric series has a constant ratio between consecutive terms, with the index in the exponent, and converges when the absolute ratio is below one. A p-series has the index in the base raised to a fixed power and converges when that power exceeds one.

A Maclaurin series is a Taylor series centred at zero. They are not different objects: the Maclaurin case is simply the most common centre, which is why the standard series for the exponential, sine, and cosine functions are all Maclaurin series.

A Taylor polynomial is a finite truncation, so it approximates the function and carries an error you can bound. A Taylor series continues forever and, inside its interval of convergence, equals the function exactly.

The alternating series error bound is the absolute value of the first omitted term, and it applies only when the series alternates and passes the alternating series test. The Lagrange error bound works for any Taylor polynomial but requires bounding the next derivative on the interval.

The radius of convergence is a single number, the distance from the centre to where convergence stops, found with the ratio test. The interval of convergence is the full set of inputs where the series converges, which requires testing each endpoint separately.

Every Taylor series is a power series. A power series is any sum of coefficients times powers of x minus c, with the coefficients free to be anything, and the only question you can ask is where it converges. A Taylor series is the one whose coefficients come from the derivatives of a named function at the centre.

The alternating series test proves that the alternating series itself converges, and says nothing about the series of absolute values. Absolute convergence is the stronger conclusion, and you reach it by running a separate test on the absolute values, usually a comparison, a p-series, or the ratio test.

The p-series result is the integral test already carried out for terms of the form one over n to the power p: it converges when p is greater than 1 and diverges otherwise. Quote the p-series rule when the terms match that form exactly, and run the integral test yourself when they do not.

The nth term test can only prove divergence: if the terms fail to approach zero the series diverges, and if they do approach zero the test tells you nothing at all. The ratio test can prove either outcome, and it is the tool for factorials, constants raised to the nth power, and power series.

A series converges when its partial sums approach a finite limit, and that limit is the sum. It diverges in every other case, which covers partial sums that grow without bound and partial sums that oscillate forever without settling on any one value.

Every Maclaurin series is built from the derivatives of a function at zero. The binomial series is that construction already carried out for one plus x raised to a power k, so you quote its coefficient pattern instead of differentiating, and it holds for x strictly between negative 1 and 1.

The actual error is the real difference between an approximation and the true value, which you can only compute if you already know the answer. An error bound is a guarantee that the error is no bigger than a stated number, computable without knowing the answer. Bounds are what exams ask for.