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.

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.

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.

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.

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.

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.

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.

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.

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.

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.