Theorems#

Intermediate Value Theorem

Let $f(x)$ be continuous on the interval $[a,b]$ . There exists a c between a and b such that

$f(a) \leq f(c) \leq f(b)$

Mean Value Theorem

Let $f(x)$ be continuous on the interval $[a,b]$ . There exists a c between a and b such that

$f^{\prime} (c) = \frac{f(b) - f(a)}{b-a}$

Extreme Value Theorem

Let $f(x)$ be continuous on the interal $[a,b]$ . Then $f(x)$ has at least one maximum value and a minimum value on the interval $[a,b]$ .

2007, Free Response, #3

$x$	$f(x)$	$f^{\prime}(x)$	$g(x)$	$g^{\prime}(x)$
1	6	4	2	5
2	9	2	3	1
3	10	-4	4	2
4	-1	3	6	7

The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table above gives values of the functions and their first derivatives at selected values of x. The function h is given by $h ( x ) = f ( g ( x ) ) - 6$ .

Explain why there must be a value $r$ for $1 < r < 3$ such that $h ( r ) = - 5$ .
Explain why there must be a value $c$ for $1 < c < 3$ such that $h^{\prime} ( c ) = - 5$ .

2018, Free Response, #4

t (years)	2	3	4	5	6
$H(t)$ (meters)	1.5	2	6	11	15

The hieght of a tree at time $t$ is given by a twice-differentiable function $H$ , where $H(t)$ is measured in meters and $t$ is measured in years. Selected values of $H(t)$ are given in the table above.

Use the data in the table to estimate $H^{\prime}(6)$ . Using correct units, interpret the meaning of $H^{\prime}(6)$ in the context of the problem.
Explain why there must be at least one time $t$ , for $2 < t < 10$ , such that $H^{\prime}(t) = 2$ .

2014, Free Response, #4

TODO

2014, Free Response, #5

TODO