AP Calculus AB / Mr. Hansen
10/12/2005

Name: _________________________

Sample Test

Time limit: 32 minutes (48 minutes for extended time).
Calculator is permitted for all problems.
Do not spend too much time on any single problem.
Clues to some problems that you cannot solve initially may be found elsewhere in the test.
All problems are free-response.
Use appropriate mathematical notation. Legibility counts.

 

1.

Sketch a function f that is not defined at x = 3 and has a step discontinuity there.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

Explain clearly why, even if #1 is modified so that the function is defined at x = 3, the function cannot possibly be differentiable there. Try to consider at least two different cases, i.e., different ways in which the function can be defined at x = 3. A formal proof is not expected.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.

State the IVT. Standard abbreviations are encouraged.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.

Let

 

 

 

Explain clearly in what way(s) the given function violates the hypotheses of the IVT for any closed interval that includes 0.

 

 

 

 

 

 

 

 

5.

Prove, rigorously, that even though function f in #4 violates the hypotheses of the IVT for [−2.356, 0.786], a closed interval that includes 0, the IVT conclusion is nevertheless true. You may find a sketch to be helpful, but a sketch is not required.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.

Sketch a function that violates both the hypothesis and the conclusion of the IVT.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.

Sketch two functions f and g that are continuous on  and have the property that f (a) > g(a), while f (b) < g(b), where a < b. Let h(x) = f (x) − g(x), and take it as a given that the difference of continuous functions is continuous. Prove that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8.

Let h(x) = x3 – 10x2 + 3x + 31.

(a)

Suppose that someone doubts that this polynomial function is continuous. (We already know that all polynomials are continuous, but assume that there is someone who did not get the memo.) Use the definition of continuity (all 3 parts) to prove that h is continuous on [2, 3]. Limits can be proved using limit properties; no need for epsilons and deltas here.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Compute the sign (+ or –) of h(2) and of h(3). (Exact values of h(2) and h(3) are not required.)

 

 

 

 

 

 

(c)

What does IVT allow you to conclude from (a) and (b) regarding roots?

 

 

 

 

 

 

 

 

 

 

 

 

9.(a)

Sketch a continuous function f that has a cusp at x = 1.5, such that f is differentiable on

 

 

 

 

 

 

 

 

 

 

 

 

(b, c)

What role does the line x = 1.5 play for f ¢ ? for f itself?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10.

Find  if .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11.

Solve the following differential equation for y subject to the initial condition (2, −1):



The term “initial condition” refers to the fact that the unknown curve for y must pass through (2, −1).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12.

In #11, compute y when x = 2.3. (Note that this question can be answered even if you were unable to answer #11 itself.)