Honors AP Calculus / Mr. Hansen
1/31/2011

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

Test through Chapter 8 (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.

 

 

1.

At Bisneyland, there are already 200 people (mostly employees) within the park gates at 9:00 a.m. when the gates open. The function F(t) = 230 + 640 sin(0.9t + 0.8) is a good model of the net flow rate into the park, in people per hour, where t denotes elapsed time (in hours) after 9:00 a.m.

 

 

(a)

Find a time t for which F(t) is negative.

 

 

(b)

State the clock hour (and include a.m. or p.m., please) corresponding to your answer in part (a).

 

 

(c)

Interpret what F(t) < 0 means, in the context of this problem, for the value of t you gave in part (a). Write a complete, grammatically correct sentence.

 

 

 

 

 

 

 

 

 

 

(d)

Let P(t) denote the population of people within the park gates at time t, where again t is measured in hours and t = 0 corresponds to 9:00 a.m. Find the first point of inflection for P(t) on the interval [0, 7]. Justify your answer.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(e)

Find the times between 9:00 a.m. and 4:00 p.m., inclusive, to the nearest clock hour and minute, at which the population P of people in the park is maximized and minimized. Justify your answers. Please use reverse side of paper.

 

 

2.

A lozenge, believe it or not, is defined to be a special type of rhombus. Let L denote the lozenge in the xy-plane defined by the four vertices (0, 0),  (2, 0), and

 

 

(a)

Use plane slicing to compute the area of L. Be sure to indicate whether your slices are dx or dy slices. Since the area is so easy to check with geometry (Arhombus = 0.5 times the product of the diagonals’ lengths), a wrong answer is not really acceptable here. Use plane slicing, though. A sketch is required for full credit.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Use cylindrical shells to compute the volume created when L is revolved about the line  A sketch is optional but highly recommended. In fact, if you show a sketch and raise your hand, you will be told whether your sketch is correct or not.

 

(c)

Explain why, in part (b), the integral with limits 2 to 3 cannot simply be doubled in order to obtain the final answer. A single sentence of explanation will suffice.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)

Use plane slicing (planes perpendicular to the y-axis) to cross-check your answer to part (b).

 

(e)

Use radial slicing to triple-check your answer to part (b). If you are running low on time, be sure to at least show the setup.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.(a)

What sort of figure is the polar graph r = 2 cos  for ?

 

 

 

 

(b, c)

Compute the area of the figure in part (a) using simple geometry and also using the polar area formula. Explain any discrepancy you encounter.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.

Compute the arc length (in Quadrant I only) of the parabola y = 9 − x2. You have enough room here, but if you need more room, write OVER.