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AP Calculus AB review: key concepts

AP Calculus AB review cards: limits, derivative rules, related rates, optimization, integrals, the FTC and differential equations, each with a worked example.

42 cards5 topicsAbout 15 minutes a roundAP Calculus AB style course content (one semester of university differential and integral calculus)

Independent study material written by StudyBrix. AP and SAT are trademarks of College Board and ACT is a trademark of ACT, Inc.; neither organisation is involved with or has reviewed this deck.

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About this deck

This deck is for high school students preparing for an AP Calculus AB exam, and for anyone reviewing single-variable calculus before a first university course. It follows the usual order of the course: limits and continuity, the definition and rules of differentiation, implicit and inverse derivatives, applications such as related rates, motion and optimization, then Riemann sums, the Fundamental Theorem of Calculus, accumulation, area and volume, and finally separable differential equations.

You should already be fluent with algebra, function notation, exponent and logarithm rules, and the unit circle in radians. If any of those feel shaky, fix them first, because most lost marks in calculus are algebra or trigonometry slips rather than calculus errors.

Work each example on paper before you flip the card, then check every step. Free-response questions reward written justification, so practise saying why a theorem applies (for example, that a function is continuous on a closed interval before you quote the Extreme Value Theorem or the Mean Value Theorem). Mix the tags once you know them, since an exam question rarely announces which tool to use.

Common mistakes: forgetting the chain rule on the inside function, such as writing cos(x^2) as the derivative of sin(x^2); dropping the constant of integration or the units in a context answer; and confusing displacement (the integral of velocity) with total distance (the integral of speed). Another frequent slip is treating a critical point as an extremum without checking a sign change or the endpoints.

Before you start: Strong precalculus: algebra, functions and their graphs, exponents and logarithms, and trigonometry in radians.

All 42 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Limits and continuity

Limit of f(x) as x approaches a: The single value that f(x) gets arbitrarily close to as x gets close to a from both sides. It can exist even when f(a) is undefined.Example: (x^2 - 9)/(x - 3) = x + 3 for x not equal to 3, so the limit as x approaches 3 is 6.
Continuity at x = a: Three conditions must all hold: f(a) is defined, the limit as x approaches a exists, and that limit equals f(a).Example: f(x) = (x^2 - 1)/(x - 1) fails at x = 1 because f(1) is undefined, even though the limit there is 2.
Removable discontinuity: A hole in a graph: the limit exists at the point but the function is undefined there or has a different value. Redefining one point repairs it.Example: Defining g(1) = 2 for g(x) = (x^2 - 1)/(x - 1) makes g continuous at x = 1.
Limits at infinity for rational functions: Compare degrees: if top and bottom have equal degree the limit is the ratio of leading coefficients; if the bottom degree is larger the limit is 0. This gives a horizontal asymptote.Example: As x grows without bound, (3x^2 + 1)/(2x^2 - 5) approaches 3/2.
Squeeze theorem: If g(x) <= f(x) <= h(x) near a, and g and h have the same limit L at a, then f also has limit L at a.Example: -x^2 <= x^2 sin(1/x) <= x^2 and both bounds go to 0, so x^2 sin(1/x) approaches 0 as x approaches 0.
Intermediate Value Theorem: If f is continuous on [a, b] and N is between f(a) and f(b), then f(c) = N for at least one c in (a, b).Example: f(x) = x^3 + x - 1 has f(0) = -1 and f(1) = 1, so it has a zero between 0 and 1.
sin(x)/x as x approaches 0: This standard trigonometric limit equals 1 when x is in radians; rewrite other forms to match it.Example: sin(5x)/x = 5 * sin(5x)/(5x), so the limit as x approaches 0 is 5.

Derivatives

Derivative as a limit: f'(x) = lim as h approaches 0 of (f(x + h) - f(x))/h: the instantaneous rate of change and the slope of the tangent line.Example: For f(x) = x^2: ((x + h)^2 - x^2)/h = 2x + h, which approaches 2x.
Power rule: The derivative of x^n is n x^(n - 1) for any real constant n.Example: d/dx x^7 = 7x^6 and d/dx x^(-2) = -2x^(-3).
Product rule: (fg)' = f'g + fg': differentiate one factor at a time and add the results.Example: d/dx [x^2 sin x] = 2x sin x + x^2 cos x.
Quotient rule: (f/g)' = (f'g - fg')/g^2, valid where g is not zero. Order matters in the numerator.Example: d/dx [x/(x + 1)] = ((x + 1) - x)/(x + 1)^2 = 1/(x + 1)^2.
Chain rule: For a composite f(g(x)), the derivative is f'(g(x)) * g'(x): outside derivative evaluated at the inside, times the inside derivative.Example: d/dx (3x + 1)^5 = 5(3x + 1)^4 * 3 = 15(3x + 1)^4.
Derivatives of sin, cos and tan: d/dx sin x = cos x, d/dx cos x = -sin x, d/dx tan x = sec^2 x (x in radians).Example: d/dx sin(2x) = 2cos(2x) by the chain rule.
Derivatives of e^x and ln x: e^x is its own derivative; ln x has derivative 1/x for x > 0. Combine with the chain rule for composite forms.Example: d/dx ln(x^2 + 1) = 2x/(x^2 + 1).
Implicit differentiation: Differentiate both sides of an equation in x and y with respect to x, treating y as a function of x (so y^2 becomes 2y dy/dx), then solve for dy/dx.Example: x^2 + y^2 = 25 gives 2x + 2y y' = 0, so y' = -x/y; at (3, 4) the slope is -3/4.
Derivative of an inverse function: If g is the inverse of f and f(a) = b, then g'(b) = 1/f'(a), provided f'(a) is not zero.Example: f(x) = x^3 + x: f(1) = 2 and f'(1) = 4, so the inverse has slope 1/4 at x = 2.
Differentiability and continuity: A function that has a derivative at a point must be continuous there, but a continuous function can fail to be differentiable at a corner, cusp or vertical tangent.Example: |x| is continuous at 0 but has a corner there, so it has no derivative at 0.
L'Hopital's rule: For a limit of the form 0/0 or infinity/infinity, the limit of f/g equals the limit of f'/g' if that second limit exists. Check the form first.Example: (e^x - 1)/x is 0/0 at x = 0; the ratio of derivatives e^x/1 gives a limit of 1.

Applications of derivatives

Local linear approximation: Near x = a, f(x) is approximately f(a) + f'(a)(x - a), the tangent line value.Example: For sqrt(x) at a = 4: L(x) = 2 + (x - 4)/4, so sqrt(4.1) is about 2.025.
Related rates: Write an equation linking the quantities, differentiate both sides with respect to time, then substitute the known values at the instant asked about.Example: A = pi r^2, dr/dt = 2 cm/s, r = 5 cm: dA/dt = 2 pi r dr/dt = 20 pi cm^2/s.
Velocity and acceleration from position: Velocity is the first derivative of position and acceleration is the second derivative; both carry units.Example: s(t) = t^3 - 6t^2 + 9t: v(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3), a(t) = 6t - 12.
Speeding up or slowing down: An object speeds up when velocity and acceleration have the same sign and slows down when the signs differ.Example: With v(t) = 3t^2 - 12t + 9 at t = 0.5: v = 3.75 and a = -9, so the object is slowing down.
Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has f'(c) equal to the average rate of change (f(b) - f(a))/(b - a).Example: f(x) = x^2 on [1, 3]: average slope (9 - 1)/2 = 4, and f'(c) = 2c = 4 at c = 2.
Critical point: An interior x value where f'(x) is zero or undefined; the only interior places a local maximum or minimum can occur.Example: f(x) = x^3 - 3x: f'(x) = 3x^2 - 3 = 0 at x = -1 and x = 1.
First derivative test: If f' changes from positive to negative at a critical point, f has a local maximum there; negative to positive gives a local minimum; no sign change gives neither.Example: For x^3 - 3x, f' goes from + to - at x = -1, so f(-1) = 2 is a local maximum.
Second derivative test: At a critical point c with f'(c) = 0: f''(c) > 0 means a local minimum, f''(c) < 0 means a local maximum, and f''(c) = 0 gives no conclusion.Example: For x^3 - 3x, f''(x) = 6x and f''(1) = 6 > 0, so f(1) = -2 is a local minimum.
Concavity and inflection points: f is concave up where f'' > 0 and concave down where f'' < 0. An inflection point is where concavity changes on a continuous graph.Example: f(x) = x^3 has f''(x) = 6x, which changes sign at 0, so (0, 0) is an inflection point.
Candidates test for absolute extrema: On a closed interval, evaluate f at every critical point inside the interval and at both endpoints; the largest value is the absolute maximum and the smallest is the absolute minimum.Example: x^3 - 3x on [0, 3]: f(0) = 0, f(1) = -2, f(3) = 18, so the maximum is 18 and the minimum is -2.

Integrals and accumulation

Left and right Riemann sums: Approximate an integral by adding rectangle areas whose heights are taken at the left or right end of each subinterval.Example: x^2 on [0, 2] with two subintervals of width 1: left sum 0 + 1 = 1, right sum 1 + 4 = 5.
Trapezoidal sum: Approximate an integral with trapezoids; for equal widths it equals the average of the left and right Riemann sums.Example: For x^2 on [0, 2] with width 1: (1 + 5)/2 = 3, while the exact value is 8/3.
Definite integral as net change: The integral of a rate of change over [a, b] gives the total change in the quantity over that interval; area below the axis counts as negative.Example: If water flows in at r(t) litres per minute, the integral of r from 0 to 10 is the net litres added in 10 minutes.
Fundamental Theorem of Calculus, part 1: If F(x) is the integral of f(t) from a to x and f is continuous, then F'(x) = f(x). With an upper limit u(x), multiply by u'(x).Example: d/dx of the integral of cos t from 0 to x^2 is cos(x^2) * 2x.
Fundamental Theorem of Calculus, part 2: If F is any antiderivative of a continuous f, the integral of f from a to b equals F(b) - F(a).Example: The integral of 2x from 1 to 3 is 3^2 - 1^2 = 8.
Antiderivative of x^n: For n not equal to -1, the antiderivative is x^(n + 1)/(n + 1) + C; for n = -1 it is ln|x| + C.Example: The integral of x^3 dx is x^4/4 + C.
u-substitution: Undo the chain rule: choose u as an inside function whose derivative also appears, rewrite the integral in u, integrate, then substitute back (or change the limits).Example: With u = x^2 + 1, the integral of 2x(x^2 + 1)^4 dx is (x^2 + 1)^5/5 + C.
Average value of a function: On [a, b], it is 1/(b - a) times the integral of f from a to b.Example: For x^2 on [0, 3]: (1/3)(27/3) = 3.
Displacement versus total distance: Displacement is the integral of velocity; total distance travelled is the integral of |velocity|, so split at the times where velocity changes sign.Example: v(t) = t - 2 on [0, 4]: displacement 0 m, total distance 2 + 2 = 4 m.
Area between two curves: Integrate (top function minus bottom function) between the intersection points or given bounds.Example: Between y = x and y = x^2 on [0, 1]: integral of (x - x^2) = 1/2 - 1/3 = 1/6.
Volume by the disc method: Revolving the region under y = f(x) about the x-axis gives volume pi times the integral of f(x)^2 over the interval; washers subtract an inner radius squared.Example: y = sqrt(x) on [0, 4] about the x-axis: pi times the integral of x from 0 to 4 = 8 pi.

Differential equations

Slope field: A grid of short segments whose slopes show dy/dx at each point; solution curves follow the segments.Example: For dy/dx = x + y, the segment at (1, 0) has slope 1 and the one at (0, 0) is flat.
Separation of variables: Move all y terms to one side with dy and all x terms to the other with dx, integrate both sides, add the constant, then use any initial condition to find it.Example: dy/dx = xy gives dy/y = x dx, so ln|y| = x^2/2 + C and y = A e^(x^2/2).
Exponential growth model: When the rate of change is proportional to the amount, dy/dt = ky, the solution is y = y0 e^(kt).Example: y0 = 200, k = 0.05, t = 10: y = 200e^0.5, which is about 329.7.