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 dy dx equals the quotient of y times the cosine of x and the quantity 1 plus y squared with the initial condition y(0) = 1. (10 points)


3.

a. Solve the differential equation y prime equals the product of 4 times x and the square root of the quantity 1 minus y squared
b. Explain why the initial value problem y prime equals the product of 4 times x and the square root of the quantity 1 minus y squared 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 the integral from 1 to 2 of f of x, dx . 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 the integral from 0 to 8 of x squared, dx . (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.

Graph of a function on the domain negative 2, 6 and range negative 3, 4. Graph increases from negative 2, 3 until 2, 0 and from 4, negative 2.5 to 6, 4. Graph decreases from 2, 0 to 4, negative 2.5. There are x intercepts at 2, 0 and 4, negative 2.5.

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.

Linear graph from x equals 0 to x equals 30 with a negative slope. y intercept is y equals 10 and x intercept is x equals 10. (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 F of t equals the quotient of t plus 6 and the quantity 1 plus t , and at the same time empties out at the rate E of t equals the quotient of the natural log of the quantity t plus 2 and the quantity t plus 1 , 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)

 
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