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Honors AP Calculus / Mr. Hansen |
Name: _________________________________ |
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Elapsed time (to nearest 15 minutes): _________ |
Take-home test on Chapters 10 and 11
Due by end of class Monday, 4/12/99
General Instructions (please read carefully):
i) The following symbol should be a less-than-or-equal sign: £ . If it is not, please contact me immediately to receive an official hard copy of this test sheet.
ii) Use unlined (i.e., blank) paper. See me if you need some.
iii) Use pen. Yes, you read correctly--use pen, not pencil.
iv) Show all work on your own paper. Scratch paper (not to be turned in) is optional.
v) Neatness counts: 15% of your grade will be based on clarity (neatness, handwriting, and logical ordering of arguments). Crossouts, using an "X" through sections that you wish to be ignored, will not count against you. You may also make rubouts, minor scratchouts, and minor corrections (fixing sign errors or wrong constant values, for example) without penalty.
vi) The suggested time for this test is approximately 90 minutes. Record your elapsed time in the blank provided above. This is for informational purposes only and will not affect your score.
vii) Each problem is worth 20 points.
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1.(a) |
Graph the function y = x + 2 sin x using the domain 0 £ x £ 10. |
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(b) |
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(c) |
Demonstrate that your answer to part (b) is not the average of the values of the function at x=2 and x=7. |
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(d) |
Show on the graph at least one geometrical meaning of the average value you computed in part (b). |
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2.(a) |
At a certain instant, the length of a rectangle is decreasing at 5 cm/sec and the width is increasing at 3 cm/sec. Compute the rate at which the area is changing if the dimensions at this instant are 100 cm long by 50 cm wide. (Include proper units.) |
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(b) |
Someone who had not studied calculus might be tempted to claim that since the length is decreasing more rapidly than the width is increasing, the area must clearly be decreasing. Is this true or false if the dimensions are 100 cm long by 50 cm wide? |
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(c) |
Does your answer to part (b) change if the dimensions at the instant in question are 10 cm long by 4 cm wide? Explain. |
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(d) |
What can you say in general about the truth or falsity of the claim in part (b)? |
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3.(a) |
At time t=0 seconds, a moving object has a velocity of 6 ft/sec. The table below shows the acceleration of the object at 5-second increments. Calculate the approximate net displacement of the object in the 20-second interval: |
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t |
acceleration (ft/sec2) |
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0 |
3 |
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(b) |
Calculate the object’s approximate total displacement in the 20-second interval. |
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4.(a) |
An above-ground cylindrical gasoline storage tank 10 feet long and 6 feet in diameter has its axis 8 feet above the ground. (Note: The tank is oriented horizontally, not vertically.) Sketch the tank with a suitable coordinate system, and use a definite integral to calculate the volume of the tank. |
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(b) |
A pump at ground level pumps gasoline with weight density of 44 lbs./ft3 into the tank. The output of the pump hose is positioned so that it is always at the top of the level of gasoline in the tank. How much work does the pump do in filling the tank half full? |
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(c) |
Compute the total weight of gasoline in the tank if the tank is filled to the 9-foot level (i.e., 9 ft. above ground level). |
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5.(a) |
An object whose position vector is given by r(t) = (7 sin 0.5t) i + 3(1 + cos t) j, where t is in seconds, i denotes the unit vector in the x direction, and j denotes the unit vector in the y direction, moves back and forth along a parabolic path in the xy-plane. Sketch this path. |
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(b) |
Compute the velocity vector when t = 4.5 seconds. |
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(c) |
Compute the acceleration vector when t = 4.5 seconds. |
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(d) |
Plot the vectors from parts (b) and (c) on the same axes you used in part (a). |
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(e) |
Compute and plot the tangential and normal components of the acceleration vector. |
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(f) |
Provide both an algebraic and a geometric explanation of why the object must be speeding up when t = 4.5 seconds. |