AP Calculus BC flashcards

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

Limit definition

lim_{x→a} f(x) = L means f(x) can be made arbitrarily close to L by choosing x sufficiently close to a, but not equal to a.

Continuity at a point

f is continuous at a exactly when f(a) is defined, lim_{x→a} f(x) exists, and the limit equals f(a).

Squeeze theorem

If g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L as x approaches 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 as a limit

f'(x) = lim_{h→0} [f(x+h) − f(x)]/h, the instantaneous rate of change of f at x.

Differentiability implies continuity

If f is differentiable at a, then f is continuous at a; the converse fails, as |x| shows at x = 0.

Product rule

The derivative of a product is (fg)' = f'g + fg'; each factor is differentiated once while the other is held fixed.

Quotient rule

For f/g, the derivative is (f'g − fg')/g², valid wherever g(x) is nonzero.

Chain rule

For a composition, d/dx f(g(x)) = f'(g(x))·g'(x); differentiate the outer function and multiply by the inner derivative.

Implicit differentiation

Differentiate both sides with respect to x, attaching dy/dx to every derivative of y, then solve algebraically for dy/dx.

Inverse function derivative

If g is the inverse of f, then g'(x) = 1/f'(g(x)), provided f'(g(x)) is nonzero.

Derivative of arctangent

d/dx arctan x = 1/(1 + x²) for all real x; this follows from inverting the restricted tangent function.

Related rates procedure

Write an equation linking the quantities, differentiate both sides with respect to time t, then substitute the known values and rates.

Linear approximation

Near x = a, f(x) ≈ f(a) + f'(a)(x − a); the tangent line supplies the best first-degree estimate of f.

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.

Particle at rest

A particle with position s(t) is at rest when its velocity v(t) = s'(t) equals zero; acceleration is v'(t).

Critical point

A point c in the domain of f where f'(c) = 0 or f'(c) is undefined; candidates for local extrema.

Second derivative test

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

Concavity and inflection

f is concave up where f'' > 0 and concave down where f'' < 0; inflection points are 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, Part 1

If F(x) = ∫_a^x f(t) dt with f continuous, then F'(x) = f(x); differentiation undoes accumulation.

u-substitution

Reversing the chain rule: let u = g(x), so du = g'(x) dx, converting ∫ f(g(x))g'(x) dx into ∫ f(u) du.

Integration by parts

∫ u dv = uv − ∫ v du; choose u so its derivative simplifies, often following the LIATE ordering.

Net change theorem

The integral of a rate of change over [a,b] gives the net change: ∫_a^b F'(x) dx = F(b) − F(a).

Separable differential equation

An equation dy/dx = f(x)g(y) solved by writing (1/g(y)) dy = f(x) dx and integrating both sides, then applying initial conditions.

Logistic growth solution

Solutions of dy/dt = ky(1 − y/L) have the form y = L/(1 + Ae^{−kt}), approaching carrying capacity L as t grows.

Euler's method

A numerical scheme using y_{n+1} = y_n + h·f(x_n, y_n) with fixed step size h to approximate solutions of dy/dx = f(x,y).

Washer method

Volume about an axis is V = π∫_a^b ([R(x)]² − [r(x)]²) dx, subtracting the inner radius squared from the outer.

Average value of a function

The average value of f on [a,b] equals (1/(b − a))∫_a^b f(x) dx.

Parametric slope

For x = x(t), y = y(t), the slope is dy/dx = (dy/dt)/(dx/dt), valid where dx/dt is nonzero.

Polar area

The area swept by r = f(θ) from θ = α to θ = β is A = (1/2)∫_α^β r² dθ.

Ratio test

With L = lim |a_{n+1}/a_n|, the series converges if L < 1, diverges if L > 1, and is inconclusive if L = 1.