MATH 222 Test 1 Spring 2001 Name__________________

Instructions: Answer all the questions on the paper provided and show all your work so that partial credit may be given. You may use a calculator on all questions.

I. Find each of the following limits. In each case in which L”Hopital’s Rule is required explicitly show when it is used. (8 points each)

1. = 1.__________

2. 2.__________

3. 3.__________

II. Find the (x,y) coordinates of all points on the graph of at which the tangent lines are horizontal. (12)

III. Evaluate the following two indefinite integrals. In one case a substitution is required while the other uses integration by parts. In each case say which method needs to be used and explicitly show the steps of the evaluation. (24)

1.

2.

IV. The following indefinite integral requires a trigonometric substitution. Perform the substitution and find the resulting integral of a product of trigonometric functions. Do not evaluate this integral. (10)

V. Let R be the region bounded the graph of , the x-axis, the y-axis, and the vertical line that crosses the graph of at a height of y = 4. Find the area of R. (8)

VI. The bass population in a lake undergoes logistic growth. The initial population is 100 fish. After one year the population has grown to 150. It is estimated that the carrying capacity of the lake is 2000 fish. (22)

1. Find an expression for the size of the bass population at time t in years.

2. What is the population (according to the model) at the end of three years?

3. How long does it take the population to reach half the carrying capacity?

4. Sketch the graph of the population.