Theorems

Question 1

Mean Value Theorem (MVT): If f(x) is ____ on [a, b] and ___ on (a, b), then there exists a number c in (a, b) such that ____

Answer 1

continuous, differentiable, f’(c) = f(b) - f(a) / b - a

Application: This gives you the secant line, or the average slope of the entire function with the intervals determining the points for it

Question 2

Extreme Value Theorem (EVT): If f is _____ on [a, b], then f has both a _____ and a _____.

Answer 2

continuous, absolute maximum, absolute minimum

Question 3

Intermediate Value Theorem (IVT): If f is ____ on [a, b], and f(a) ___ f(b), then there is a value c in (a, b) such that f(c) ____.

Answer 3

continuous, does not equal, = k

Question 4

2nd Derivative Test: if f’(c) = ___, and f’‘(x) > ___, then (c, f(c)) is a _______. If f’(c) = ____, and f’‘(x) < ____, then (c, f(c)) is a _____.

Answer 4

0, 0, relative minimum, 0, 0, relative maximum

Question 5

What are Critical Numbers (two things) about f’(x):

Answer 5

f’(x) is undefined or equals 0

Question 6

Rolle’s Theorem (Special case of MVT): if f is ____ on [a, b] and ___ on (a, b), and f(a) ___ f(b) there is at least one number c in which ______.

Answer 6

continuous, differentiable, =, f’(c)=0

Question 7

Concave Upward: f’‘(x) > ____ and f’(x) is _____

Answer 7

0, increasing

Question 8

Concave Downward: f’‘(x) < ____ and f’(x) is ____

Answer 8

0, decreasing

Question 9

Particle Position: Moving right when v(t) ____, moving left when v(t) ____.

Answer 9

> 0, < 0

Question 10

Particle Speed: Speeding up when v(t) ____ and a(t) ____. Slowing down when v(t) ____ and a(t) ___ OR VICE VERSA.

Answer 10

Speeding Up: > 0, > 0, OR < 0 < 0. (They must be the same)

Slowing Down: >0, < 0. (They must be opposite)

Question 11

Candidates Test: Find ___, then set equal to __ OR ___. Plug in found ____ AND ___ into ___. Then determine ___ based on y values.

Answer 11

f’(x), 0, find where it’s undefined, x values, endpoints, f(x), extrema

Question 12

First Derivative Test for Relative Minimum (on interval [a, b]): if f’(c) = ___ OR ____, f’(x) ___ from (a, f’(c)), and f’(x) ___ from (f’(c), b).

Answer 12

0, is undefined, < 0, > 0,

Question 13

First Derivative Test for Relative Maximum (on interval [a, b]): if f’(c) = ___ OR ____, f’(x) ___ from (a, f’(c)), and f’(x) ___ from (f’(c), b).

Answer 13

0, is undefined, > 0, < 0,