Calculus Memorization Quiz

These flashcards cover vocabulary and key concepts related to limits, derivatives, continuity, and integral calculus as outlined in the memorization quiz.

Limit Definition of a Derivative

The limit as h approaches 0 of (f(x+h) - f(x))/h.

Equation of a Tangent Line

y - f(a) = f'(a)(x - a).

Product Rule

(f(x) * g(x))' = f'(x) * g(x) + f(x) * g'(x).

Quotient Rule

(f(x)/g(x))' = (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2.

Chain Rule

(f(g(x)))' = f'(g(x)) * g'(x).

Derivative of x

1.

Derivative of sin(x)

cos(x).

Derivative of cos(x)

-sin(x).

Derivative of tan(x)

sec^2(x).

Derivative of csc(x)

-csc(x)cot(x).

Derivative of sec(x)

sec(x)tan(x).

Derivative of cot(x)

-csc^2(x).

Derivative of sin^(-1)(x)

1/√(1-x^2).

Derivative of cos^(-1)(x)

-1/√(1-x^2).

Derivative of tan^(-1)(x)

1/(1+x^2).

Derivatives of ln(x)

1/x.

Derivative of e^x

e^x.

Derivative of a^x

a^x ln(a).

Derivative of x + y

1.

Continuity conditions

1. f(a) is defined, 2. lim x->a f(x) exists, 3. lim x->a f(x) = f(a).

Differentiability conditions

1. f(a) is defined, 2. lim x->a (f(x) - f(a))/(x - a) exists.

Extreme Value Theorem (EVT)

If f(x) is continuous on [a, b], then there exists a maximum and minimum on that interval.

Candidate Test

Use critical points and endpoints to find absolute extrema.

First Derivative Test

Use sign changes of f'(x) to find relative extrema or to determine where a function is increasing or decreasing.

Intermediate Value Theorem (IVT)

If f(x) is continuous on [a, b] and N is between f(a) and f(b), then there exists c in (a, b) such that f(c) = N.

Mean Value Theorem (MVT)

If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) where f'(c) = (f(b) - f(a)) / (b - a).

Rolle's Theorem

If f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

f '(x) > 0

f(x) is increasing.

f '(x) < 0

f(x) is decreasing.

f '(x) = 0 or DNE

f(x) could be a local extremum.

f ''(x) > 0

f(x) is concave up.

f ''(x) < 0

f(x) is concave down.

f ''(x) = 0 or DNE

f(x) could be an inflection point.

Speeding up

v(x) and a(x) must be both positive or both negative.

Slowing down

v(x) and a(x) must be of opposite signs.