MATHEMATICS H161--AU03 MIDTERM THREE
Do all problems. Show all steps for your work. (Even if you get an answer just from a calculator, you need to explain why it is correct.) Formulas for your use are on the back of the last page.
1. Find the following indefinite integrals.
(a) (5 points) The integrand is x ln x
(b) (5 points) The integrand is x[(x + 1)1/2]
2. Find the following indefinite integrals.
(a) (5 points) The integrand is x2 /(1 - x2)1/2
(b) (5 points) The integrand is (x2 - x)/[(x + 1)( x2 + 1)]
3. (10 points) A particle moves along the x-axis according to the formula x(t) = sin (5t + 2).
(a) Calculate the velocity function v(t).
(b) If speed is the absolute value of velocity, what is the maximum speed attained
by the particle?
(c) Calculate the acceleration function a(t).
(d) Show that x = x(t) satisfies the differential equation 25x + x'' = 0.
4. (10 points) A ball is thrown upward from the ground. Eight seconds later it is moving downward at 80 feet per second. How many seconds after it is thrown upward does it hit the ground?
5. (10 points) Find the area of the surface obtained by revolving about the x-axis the graph of
y = x3 from x = 0 to x = 1.
6. (10 points) A person standing on the ground and weighing 150 pounds climbs 30 feet up a vertical ladder. A chain of length 100 feet is coiled on the ground next to the man, and one end of it is attached to the man's left ankle before he starts to climb. The chain weighs 2 pounds per foot. How much work does the man do in climbing up?
7. (10 points) A rectangular gate 10 feet high and 5 feet wide is set in the face of a dam. The water level behind the dam is at the top of the gate and the water is 100 feet deep. The other side of the gate faces the open air. What is the hydrostatic force on the gate? (Just use the symbol w for the density of water in pounds per cubic foot.)
sin2 x = (1 Ð cos 2x)/2
cos2 x = (1 + cos 2x)/2
sin (x + y) = sin x cos y + cos x sin y
cos (x + y) = cos x cos y - sin x sin y
The area of the surface obtained by revolving the graph of y = f(x) about the x- axis from x = a to x = b is [formula given here].