Honors AP Calculus / Mr. Hansen

(9 pts.) Name: _______________________________

10/22/2009

 

 

Test #3 (100 points): No calculator permitted

 

 

Part I: Definitions (6 pts. each)
Give a mathematically complete and correct definition of each of the following. Do not simply describe the notation (for example, in #2, you cannot simply say, “derivative of function f, evaluated at point c”).

 

 

1.

antiderivative

 

 

 

 

2.

 

 

 

 

3.

Function g is continuous at x = 3.

 

 

 

 

 

 

 

 

 

 

4.

Average rate of change. (An example is helpful but not required.)

 

 

 

 

 

 

 

 

 

Part II: Free Response (points as marked in parentheses)
Show adequate work to justify your answers. Circle or box all final answers.

 

 

5.
(12 pts.)

If  and f (−1) = 2, find f (2).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.  (10)

Compute  where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.  (10)

Compute  where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8.  (6)

State the chain rule (any valid version).

 

 

 

 

 

 

 

 

 

 

 

 

9.  (10)

A circle’s radius grows at a rate of 2 cm/sec. Find the area’s instantaneous rate of growth when r = 3 cm. Give units!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10.(15)

Sketch (and label) position, velocity, and acceleration graphs for a particle whose position is 3 when time t = 0, given that v(t) = −4t.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11.  (4)

Would it surprise you to know that Benoit Mandelbrot coauthored a 2005 article predicting a collapse of risk assessment models based on conventional Gaussian (“normal” bell curve) theories? ___

What sort of investment pricing model do you speculate that Mandelbrot advocated in his article? ____________

 

 

12.  (4)

(Note: This problem appeared only on an alternate version of the test.)

Give an example of a function f that is continuous everywhere, differentiable everywhere except at a single point, and for which a piecewise quadratic antiderivative can be defined.