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1.

Write and then solve for the differential equation for the statement: “The rate of change of y with respect to x is inversely proportional to y^{2}.” (10 points)

2.

Solve the differential equation with the initial condition y(0) = 1. (10 points)

3.

a. Solve the differential equation

b. Explain why the initial value problem with y(0) = 4 does not have a solution.

4.

The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of . Give 3 decimal places for your answer. (10 points)

x | 1 | 1.1 | 1.2 | 1.5 | 1.7 | 1.9 | 2.0 |
---|---|---|---|---|---|---|---|

f(x) | 1 | 2 | 4 | 6 | 7 | 9 | 10 |

5.

Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for . (10 points)

1.

The figure below shows the graph of f ‘, the derivative of the function f, on the closed interval from x = -2 to x = 6. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4.

Find the x-value where f attains its absolute maximum value on the closed interval from x = -2 to x = 6. Justify your answer. (10 points)

2.

A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec^{2} is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10.

What is the velocity of the car when t = 6? You must show your work and include units in your answer.

(10 points)

3.

Show that f(x) = 2000x^{4} and g(x) = 200x^{4} grow at the same rate. (10 points)

4.

A radar gun was used to record the speed of a runner (in meters per second) during selected times in the first 2 seconds of a race. Use a trapezoidal sum with 4 intervals to estimate the distance the runner covered during those 2 seconds. Give a 2 decimal place answer and include units. (10 points)

t | 0 | 0.5 | 1.2 | 1.5 | 2 |
---|---|---|---|---|---|

v(t) | 0 | 4.5 | 7.8 | 8.3 | 9.0 |

5.

Water flows into a tank according to the rate , and at the same time empties out at the rate , with both F(t) and E(t) measured in gallons per minute. How much water, to the nearest gallon, is in the tank at time t = 10 minutes. You must show your setup but can use your calculator for all evaluations. (10 points)