Honors AP Calculus / Mr. Hansen
2/13/2013

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

Test through §9-5 (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.
  • Anything that is given on this test may be used elsewhere without rejustification.

 

 

1.

Use integration by parts to prove the reduction formula

 

2.

Let region R be defined as the equilateral triangular region in the xy-plane having vertices at (–2, 0), (2, 0), and

 

 

(a)

Find the area of R by calculus. (There are no additional points for checking against the common geometry area formula, but you are free to do that on your own if you wish.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Compute, by plane slicing, the volume of the solid created when R is rotated about the axis x = –3.5. Use planes perpendicular to the y-axis. A sketch is required.

 

(c)

Compute the same volume requested in part (b), but this time, use cylindrical shells. A sketch is optional.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)

Compute the same volume requested in part (b), but this time, use radial slicing. A sketch is optional.

 

3.

Consider the portion of the polar curve r = 2 cos  – 7 that lies within Quadrants IV and I. Linear units for this problem are meters.

 

 

(a)

Sketch the requested portion of the curve, labeling the values of  at the endpoints.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Compute the arc length of the requested portion of the curve.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)

Compute the area in Quadrants IV and I that lies between the given curve and the curve defined by r = 9. A sketch is required.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.

Compute each of the following. Show at least enough work to indicate that you know what you are doing (and did not simply get lucky). Different people may require different amounts of work.

 

 

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)


 

(c)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.

Consider the function y = f (x) = x2ex on the closed interval [–4, 1].

 

 

(a)

Find the coordinates (ordered pairs) of all points of inflection on [–4, 1]. Justify your answer. If there are no points of inflection on the given interval, state that and provide justification.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Find the coordinates (ordered pairs) of the function’s absolute maximum and minimum on the given interval. Justify your answers. You may continue on the reverse side if necessary.