Honors AP Calculus / Mr. Hansen 12/13/2007

Name: _________________________

Test #6 Through Chapter 7: 100 pts., Calculator Required

• Illegible or cryptic responses may not earn full credit, even if your answers are “correct.”
• Give all final numeric approximations correct to at least 3 decimal places (AP standard).
• Circle or box your final answer if applicable (AP standard).
• Do not erase large sections. If you wish for a section to be ignored during grading, simply mark it through with a single large “X” and proceed. This is also an AP standard.
• Check here to indicate that you have read these instructions and understand them: o

Part I: Short Answer (4 pts. each)

Warning: These are very tricky, and there is essentially no partial credit. Think carefully! Do not write DNE if there is a more descriptive conclusion you can make. “Nothing” may be a correct answer.

1.	If , then what can we conclude about
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2.	If , and the quotient is of the indeterminate form as , then what can we conclude about
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3.	If does not exist (and is also neither +¥ nor –¥), then what can we conclude about
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Part II: Differential Equations and Slope Fields (10 pts. each)

For problems 4 through 10, consider the equation .

4.	How would you classify the equation? You should give a 3-part description that addresses order, separability, and category. For example, if you think it is a third-order separable partial differential equation, you would write, “3rd order separable PDE.”
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5.	On the reverse side, sketch the slope field on the lattice points in . Warning: Show only those 25 lattice points. Do not show all of the slope field that a graphing calculator utility may give you.
6.	On your slope field diagram, or on a separate set of axes if you prefer, sketch the particular solution that passes through the point (–1.5, 0).
7.	Use your superior analytic powers to find the exact particular solution that passes through (–1.5, 0). Do not round anything; keep all equations and constant(s) in exact unrounded form. Show all work below.

8.	Use Euler’s Method with a step size of 0.75 to estimate the y-intercept of the particular solution that passes through (–1.5, 0). Show your work, including the calculation of dy and the means by which each new y-value is obtained. Round only at the end. (You may show intermediate results with . . . to indicate the insignificant decimal places.)
9.	Prove that the original equation given on page 1 forces all solutions, regardless of the values of x and y, and regardless of the initial condition specified, to be concave upward.

10.	Explain briefly how #9 would imply that a correct answer to #8 would always be less than the “true” y-intercept computed from a correct answer to #7. Note that you can answer this question even if your answers to problems 7, 8, and 9 are incorrect.

Part III: Growth Problem (20 pts.)

Justify your steps. Justification can be rather brief, but it is required.

11.

A bacterial colony is growing at an instantaneous rate proportional to the number of bacteria present at time t. If there are 10 million bacteria at time t = 1 hour and 20 million bacteria at time t = 1.5 hours, compute (a) the instantaneous growth rate at time t = 2 hours and (b) the estimated size of the colony at time t = 2 hours.