Mathematics

1.

Evaluate . (5 points)

2.

Use geometry to evaluate where . (5 points)

3.

Find F ‘(x) for . (5 points)

4.

Find the antiderivative of . (5 points)

5.

Evaluate . (5 points)

6.

Use your calculator to evaluate . (5 points)

7.

R is the 4th quadrant region enclosed by the x-axis and the curve y = x^{2} – 2kx, where k > 0. Find the value of k so that the area of the region R is 36 square units. (5 points)

8.

The base of a solid is bounded by the curve , the x-axis and the line x = 1. The cross sections, taken perpendicular to the x-axis, are squares. Find the volume of the solid. (5 points)

9.

Find the average value of f(x) = cos(x) on the interval [0, ]. (5 points)

10.

Find the domain for the particular solution to the differential equation , with initial condition y(2) = 2. (5 points)

11.

The slope field for a differential equation is shown in the figure. Determine the general solution of this equation.

(5 points)

12.

The function f is continuous on the interval [4, 15], with some of its values given in the table above. Estimate the average value of the function with a Trapezoidal Approximation, using the 4 intervals between those given points. (5 points)

x | 4 | 9 | 11 | 14 | 15 |
---|---|---|---|---|---|

f(x) | -6 | -11 | -18 | -21 | -25 |

13.

If the graph of f ‘(x) has an x-intercept at x = c, which of the following must be true? (5 points)

14.

Which of the following shows that f(x) grows faster than g(x)? (5 points)

15.

Find F ‘(x) for . (5 points)

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