AP Calculus AB flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Limit definition

The statement lim(x→a) f(x) = L means values of f(x) become arbitrarily close to L as x approaches a, without requiring x to equal a.

Two-sided limit condition

A two-sided limit exists exactly when the left-hand limit and right-hand limit both exist and are equal to the same value L.

Continuity at a point

f is continuous at x = a when f(a) is defined, lim(x→a) f(x) exists, and the limit equals f(a). All three conditions must hold.

Removable discontinuity

A discontinuity where the two-sided limit exists but does not equal f(a), or f(a) is undefined; the gap can be removed by redefining a single point.

Squeeze theorem

If g(x) ≤ f(x) ≤ h(x) near a and both g and h have limit L at a, then f also has limit L at a.

Intermediate Value Theorem

If f is continuous on [a, b] and k lies between f(a) and f(b), then some c in (a, b) satisfies f(c) = k.

Derivative limit definition

f′(a) = lim(h→0) [f(a+h) − f(a)]/h, the instantaneous rate of change of f at x = a, equal to the tangent line slope there.

Differentiability implies continuity

If f is differentiable at a point, then f is continuous there. The converse fails: |x| is continuous but not differentiable at x = 0.

Power rule

For any real constant n, the derivative of xⁿ is n·xⁿ⁻¹. The derivative of any constant is zero.

Product and quotient rules

(fg)′ = f′g + fg′ and (f/g)′ = (f′g − fg′)/g², valid where the derivatives and denominators exist.

Tangent line equation

The tangent line to y = f(x) at x = a is y = f(a) + f′(a)(x − a), using the point and the derivative as slope.

Average rate of change

Over an interval [a, b], the average rate of change is [f(b) − f(a)]/(b − a), the slope of the secant line through the endpoints.

Chain rule

The derivative of f(g(x)) equals f′(g(x))·g′(x): differentiate the outer function evaluated at the inner, then multiply by the inner derivative.

Implicit differentiation

Differentiate both sides of a relation in x and y with respect to x, applying the chain rule to y-terms, then solve algebraically for dy/dx.

Inverse function derivative

If g is the inverse of f, then g′(y) = 1/f′(g(y)), provided f′(g(y)) is nonzero at that point.

Logarithmic differentiation

Take the natural log of both sides, apply log properties to simplify, then differentiate implicitly. Useful for products, quotients, and variable exponents.

Derivative of arctangent

d/dx arctan x = 1/(1 + x²). Related inverse trig derivatives include d/dx arcsin x = 1/√(1 − x²) and d/dx arccos x = −1/√(1 − x²).

Velocity and acceleration

For position s(t), velocity is v(t) = s′(t) and acceleration is a(t) = v′(t) = s″(t). Speed is the absolute value of velocity.

Related rates procedure

Write an equation relating the quantities, differentiate both sides with respect to time t, then substitute known values and rates to solve for the unknown rate.

Linear approximation

Near x = a, f(x) ≈ f(a) + f′(a)(x − a). This uses the tangent line to estimate function values close to the point of tangency.

L'Hôpital's rule

For a 0/0 or ∞/∞ indeterminate form, lim f/g = lim f′/g′, provided the limit of the quotient of derivatives exists.

Marginal analysis

Marginal cost, revenue, or profit is the derivative of the total function, approximating the change in the total from producing one additional unit.

Critical point

A point c in the domain of f where f′(c) = 0 or f′(c) is undefined. Critical points are candidates for local extrema.

First derivative test

At a critical point, f′ changing from positive to negative gives a local maximum; changing from negative to positive gives a local minimum.

Second derivative test

If f′(c) = 0 and f″(c) > 0, f has a local minimum at c; if f″(c) < 0, f has a local maximum there.

Concavity and inflection

f is concave up where f″ > 0 and concave down where f″ < 0. An inflection point is where concavity changes sign.

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) satisfies f′(c) = [f(b) − f(a)]/(b − a).

Fundamental Theorem of Calculus

Part 1: d/dx ∫ₐˣ f(t) dt = f(x). Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a) for any antiderivative F of f.

Net change theorem

The definite integral of a rate of change over [a, b] equals the total net change in the quantity: ∫ₐᵇ F′(x) dx = F(b) − F(a).

Separable differential equation

An equation writable as g(y) dy = h(x) dx. Integrate both sides and solve for y, using an initial condition to find the constant.

Exponential growth and decay

The equation dy/dt = ky has solution y = y₀e^{kt}. Positive k gives growth, negative k gives decay, and y₀ is the initial value.

Volume by washers

Rotating a region about an axis gives V = π∫ₐᵇ ([R(x)]² − [r(x)]²) dx, where R is the outer radius and r the inner radius.