Calculus Basic Skills Pre-Test



Test #7 – Areas and Integrals - Practice Problems Name ____________________

Advanced Placement Calculus

Mr. Honigs

Multiple Choice Section (Weight = 50%)

Circle the letter of the most appropriate choice for each multiple choice item. The most appropriate choice will be written in exact simplified form.

1. [pic].

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

2. [pic]

a. [pic]

b. [pic]

c. [pic]

d. 0

e. [pic]

3. [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

4. [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

5. [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

6. [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

7. [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

For #8 & #9: compute the following definite integrals using what you know about area formulas.

8. [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

9. [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

10. A smooth curve with equation [pic] is such that its slope at each x equals [pic]. If the curve goes through the point (-1, 2), then its equation is

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. none of these

Free Response Section (Weight = 50%)

Show all your work for problems in this section. Justify answers with full sentences. Use complete and correct notation on each and every item you work.

1. Let K represent the piece of the curve [pic] that lies in the first quadrant. Let R be the region bounded by K and the coordinate axes.

a. Find the slope of the line tangent to K at [pic].

b. Estimate the area of R using a rectangular midpoint approximation with [pic].

c. Find the exact area of R.

d. (DO NOT ATTEMPT) Find the volume generated when R is rotated around the x-axis.

2. a. Sketch a graph of the following integral. Shade the area underneath the curve

represented by the integral [pic].

b. Compute the value of the definite integral [pic].

c. Find the total area under the curve from [pic] to [pic].

[pic]

3. Let f be a continuous function defined on [pic] whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let g be the function given by [pic].

a. Find the values of [pic] and [pic].

b. For each of [pic] and [pic], find the value or state it does not exist.

c. Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of these points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers.

d. For [pic], find all values of x for which the graph of g has a point of inflection. Explain your reasoning.

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