Theorems

Theorems#

  1. 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.

  1. Explain why there must be a value r for 1 < r < 3 such that h ( r ) = - 5.

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

  1. 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.

  1. 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.

  2. Explain why there must be at least one time t, for 2 < t < 10, such that H^{\prime}(t) = 2.

  1. 2014, Free Response, #4

TODO

  1. 2014, Free Response, #5

TODO