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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