[SOLVED] 10 Short Answer Calculus
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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 y2.” (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/sec2 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) = 2000x4 and g(x) = 200x4 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)
