Honors AP Calculus / Mr. Hansen
2/3/2009

Name: _________________________
Bonus (for Mr. Hansen’s use only): ________

Are you taking the AMC (American Mathematics Contest) after school on Tuesday, 2/10/2009? ____

Test through Chapter 9 (Calculator Required)

 

Rules
  • You may not write calculator notation anywhere unless you cross it out. For example, fnInt(X^2,X,1,2) is not allowed; write  instead.
  • Adequate justification is required for free-response questions.
  • All final answers in free-response portions should be circled or boxed.
  • Decimal approximations must be correct to at least 3 places after the decimal point.
  • Questions in Part I are worth 3 points each. All other questions are worth 8 points each.

 

 

Part I
(9 pts.)

Unlike the scoring on the AP exam, there is no penalty in this section for wrong guesses (other than the loss of the points you could have earned, of course). In the small blank provided, write a capital letter A if the statement is always true, S if it is sometimes true, or N if it is never true. There is no partial credit.

 

 

___ 1.

The centroid of a solid of revolution is a point of the solid itself. (Note: The question is not asking about the centroid of a cross section. It is asking about the centroid of the solid.)

 

 

___ 2.

A closed polar curve is traced exactly once as the parameter, , varies from 0 to .

 

 

___ 3.

A differentiable curve y = f (x) has a uniquely defined arc length over any closed interval.

 

 

Part II
(8 pts. each)

Show adequate work. Circle your final answer. Final answers must be accurate to 3 decimal places in order to earn full credit. Answers without clear setup will not earn full credit.

 

 

4.

Let f (x) = x2 – 3, g(x) = ln (x + 2). Let R be the region in Quadrants I and IV bounded by the graph of f, the graph of g, and the y-axis. Compute the volume formed when R is revolved about the line y = 4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.

Repeat #4 for a different method.

 

6.

(a) State (do not prove) the parametric arc length formula for rectangular coordinates.



(b) State the equations that convert from polar coordinates to rectangular coordinates. You may purchase these for 4 points if necessary, since they are required in #7.

 

 

 

 

 

 

 

 

7.

Use #6 to prove that the arc length of a differentiable polar function  equals .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8.

Plot and lightly shade (do not compute) the area outside the circle r = 2 but inside the graph of for a single lobe of the second curve. (The second curve has four lobes.)

 

9.

Now compute the area that you shaded in #8.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10, 11.

Minimize and maximize the function y = x3 + 7x2 – 22 on the interval [–4, 2].

 

12.

For the function given in #10 and #11, find all points of inflection on , and prove that these are indeed points of inflection by using the definition given in class.