10 Short Answer Calculus
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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)
Solve the differential equation with the initial condition y(0) = 1. (10 points)
a. Solve the differential equation
b. Explain why the initial value problem with y(0) = 4 does not have a solution.
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)
Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for . (10 points)
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)
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.
Show that f(x) = 2000x4 and g(x) = 200x4 grow at the same rate. (10 points)
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)
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)