MATHEMATICS H161--AU03 MIDTERM TWO
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. Calculate the following integrals.
(a) (5 points) indefinite integral. The integrand is (tan3 x)/(sin2 x)
(b) (5 points) definite integral from 0 to pi/4 of the integrand in problem 1,
(c) (5 points) indefinite integral. The integrand is 1/(9x2 + 4)
2. (5 points) Evaluate the definite integral from 2 to 7 of the integrand
square root of (- x2 + 4x + 21)
3. (10 points) Find the area between the curve y = cosh x and the x-axis from x = 0 to x = 1.
4. (10 points) Find the length of the curve y3/2 from x = 0 to x = 1.
5. (10 points) Use the approximate value ln 2 = 0.7 (not the value you would get from your calculator) to find the area between the graph of y = 1/x and the x-axis from x = 2 to x = 8.
6. (10 points) At 2:00 pm a count of a bacteria culture was 50 per square centimeter. At
3:00 pm the count was 70 per square centimeter. If the culture grows at the same rate until midnight of the same day, at what time (to the nearest 10 minutes) will the count be 300 per square centimeter?
7. (a) (5 points) Evaluate the definite integral from 3 to 4 of the integrand 1/(x Ð 2).
(b) (5 points) Evaluate the definite integral from - 4 to - 3 of the integrand 1/(x Ð 2).
8. (10 points) State carefully the fundamental theorem of calculus.
9. (extra credit 10 points, but you can't get more than 100% = 85 points on the exam) Prove the fundamental theorem of calculus.
ln 3 = 1.09861
ln 5 = 1.60944
ln 6 = 1.79176
ln 7 = 1.94591
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