AP Calculus: Applications of Derivatives flashcards

Question 1

What is an absolute maximum of f at c?

Accepted Answer

f(c) is at least as large as f(x) for every x in the domain of f.

Question 2

What is an absolute minimum of f at c?

Accepted Answer

f(c) is no larger than f(x) for every x in the domain of f.

Question 3

What is a local maximum of f at c?

Accepted Answer

An interval I exists where f(c) >= f(x) for every x in I.

Question 4

What is a local minimum of f at c?

Accepted Answer

An interval I exists where f(c) <= f(x) for every x in I.

Question 5

What is a local extremum?

Accepted Answer

A point where f has either a local maximum or a local minimum.

Question 6

When is c a critical number of f?

Accepted Answer

When f'(c) = 0 or f'(c) is undefined.

Question 7

What does Fermat's theorem say about a local extremum at c?

Accepted Answer

If f has a local extremum at c, then c is a critical point of f.

Question 8

Extreme Value Theorem: what is guaranteed for f continuous over a finite closed interval?

Accepted Answer

f attains an absolute maximum and also an absolute minimum on it.

Question 9

Where can the absolute extrema of a continuous function on a closed, bounded interval occur?

Accepted Answer

At a critical point or at an endpoint of the interval.

Question 10

Must f have a local extremum at each of its critical points?

Accepted Answer

No. A local extremum must occur at a critical point, but a critical point need not give one.

Question 11

State Rolle's theorem.

Accepted Answer

If f is continuous over [a, b], differentiable over (a, b), and f(a) = f(b), then f'(c) = 0 for some c in (a, b).

Question 12

State the Mean Value Theorem.

Accepted Answer

If f is continuous over [a, b] and differentiable over (a, b), then some c in (a, b) has f'(c) = (f(b) - f(a)) / (b - a).

Question 13

If f'(x) = 0 over an interval I, what follows about f?

Accepted Answer

f is constant over I.

Question 14

Two differentiable functions f and g satisfy f'(x) = g'(x) over I. How are f and g related?

Accepted Answer

They differ by a constant: f(x) = g(x) + C.

Question 15

What does the concavity test say when f'' > 0 over an interval I, with f twice differentiable?

Accepted Answer

f is concave up over I.

Question 16

What does the concavity test say when f'' < 0 over an interval I, with f twice differentiable?

Accepted Answer

f is concave down over I.

Question 17

A critical point c has f' > 0 just left of c and f' < 0 just right of c. What does f have at c?

Accepted Answer

A local maximum at c.

Question 18

A critical point c has f' < 0 just left of c and f' > 0 just right of c. What does f have at c?

Accepted Answer

A local minimum at c.

Question 19

Second derivative test: f'(c) = 0 and f''(c) > 0. What does f have at c?

Accepted Answer

A local minimum at c.

Question 20

Second derivative test: f'(c) = 0 and f''(c) < 0. What does f have at c?

Accepted Answer

A local maximum at c.

Question 21

What is an inflection point of f at (c, f(c))?

Accepted Answer

f is continuous at c and f changes concavity at c.

Question 22

When is y = L a horizontal asymptote of f?

Accepted Answer

When the limit of f(x) is L as x -> infinity or as x -> -infinity.

Question 23

What is an oblique asymptote of f?

Accepted Answer

A line y = mx + b that f(x) approaches as x -> infinity or x -> -infinity.

Question 24

Rational function p(x)/q(x) with deg p < deg q: what is the horizontal asymptote?

Accepted Answer

y = 0.

Question 25

Rational function p(x)/q(x) with deg p = deg q: what is the horizontal asymptote?

Accepted Answer

y = a_n / b_n, the ratio of the leading coefficients of p and q.

Question 26

Which indeterminate forms does L'Hopital's rule treat directly for a quotient?

Accepted Answer

0/0 and infinity/infinity.

Question 27

Name the indeterminate forms of a limit.

Accepted Answer

0/0, infinity/infinity, 0*infinity, infinity - infinity, 0^0, infinity^0, 1^infinity.

Question 28

Which grows faster as x grows: e^x or x^p for p > 0?

Accepted Answer

e^x grows faster than any power x^p with p > 0.

Question 29

What is the linear approximation of f at x = a?

Accepted Answer

L(x) = f(a) + f'(a)(x - a).

Question 30

How is the differential dy defined?

Accepted Answer

dy = f'(x) dx.

Question 31

What is the Newton's method update for approximating a root of f(x) = 0?

Accepted Answer

Start from a guess x_0; then x_n = x_(n-1) - f(x_(n-1)) / f'(x_(n-1)).

Question 32

What is an antiderivative of f?

Accepted Answer

A function F with F'(x) = f(x) for all x in the domain of f.

Question 33

If F is one antiderivative of f, what form does every antiderivative of f take?

Accepted Answer

F(x) + C for some constant C.

Question 34

What are related rates?

Accepted Answer

Rates of change of two or more related quantities that are all changing over time.

Question 35

In a related rates problem, when may a variable quantity be replaced by a number?

Accepted Answer

Only after an equation relating the rates has been found by differentiating.

AP Calculus: Applications of Derivatives flashcards

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