MAT136 - Term Test 2
Part A: Multiple Choice Questions (12 marks)
1. Calculate the integral 4x ln x dx.
(a) 1 − e2
(b) 3e2 − 1
(c) e2 + 1
(d) 4e
(e) 2x2 − e2 + 1
2. Calculate the integral cos2
(x)sin3
(x) dx.
(a) 3/1 cos3
(x) − 5/1 cos5
(x) + C
(b) 5/1 cos5
(x) − 3/1 cos3
(x) + C
(c) 4/1 sin4
(x) − 6/1 sin6
(x) + C
(d) 6/1 sin6
(x) − 4/1 sin4
(x) + C
(e) 12/1 cos3
(x)sin4
(x) + C
3. What is the form. of the partial fractions decomposition of
4. Let f be a positive, continuous function with domain [1, ∞). Suppose we know that f(x) dx diverges.
Which of the following integrals could possibly converge? (i.e., Which of the following integrals does not necessarily diverge?)
5. Find the limit of the sequence an = ln(5n
2 + n) − ln(15n
2 + 2n + 4), if it converges.
(a) − ln(3)
(b) 0
(c) 1
(d) ln(3)
(e) The sequence diverges
6. Calculate the value of the series if it converges.
(a) 0
(b) 5/16
(c) 5
(d) 20
(e) The series diverges.
Part B: Long-answer problems (28 points)
1. (6 points, 2 points per part) Determine whether the following statements are true or false.
Circle your choice of “true” or “false”. If you believe a statement is true, briefly explain why. If you believe a statement is false, give an example to show it’s false. A correct true/false answer without justification will receive 0 points (no points for guessing).
(a) Let f be a continuous function with domain R. If f(x) = 0, then f(x) dx converges.
(b) If {an} is an increasing sequence with negative terms (i.e., an < 0 for all n), then {an} converges.
(c) Let f be a continuous function with domain R. Consider the sequence an = f(n) for all n ∈ N.
If an = 0, then f(x) = 0 as well.
2. (5 points) Consider the region R in the first quadrant bounded by the axes and the curves y =
√x − 1 and y = 7−x. R is the shaded region in the diagram to the right.
For all parts of this question, you do not have to evaluate any integrals.
(a) (1 point) Find a definite integral (or sum of definite integrals) that equals the area of R. No justification required.
(b) (2 points) Find a definite integral (or sum of definite integrals) that equals the volume of the solid of revolution obtained by rotating R around the y-axis. Put your answer on the line, and show your work/explain in the space below.
(c) (2 points) Find a definite integral (or sum of definite integrals) that equals the volume of the solid of revolution obtained by rotating R around the line y = −136. Put your answer on the line, and show your work/explain in the space below.
3. (6 points) The alien spaceship has been hovering over the McLennan Physical Laboratories at the St. George campus for the past month. Recall that it was difficult to see the precise shape of the ship. A group of chemistry students believe that this was caused by the unusual properties of a metal that coats the outside of the ship. They name the previously unknown element Estravium.
(a) (3 points) Estravium decays according to an exponential decay model E(t) = E0 e
−kt, where t is measured in months. During the one month students have observed the ship, 5% of the initial amount of Estravium coating the ship has decayed. If the Estravium continues to decay in this way, how long (in months) will it take until only 25% of the original amount remains? Give an exact answer in units of months, which may involve logarithms, on the line. Show your work in the space below.
(b) (3 points) An alien scientist is designing a way of regenerating the Estravium coating over time, and needs help from UTM students to solve an integral involving the function E mentioned above.
Calculate k
2
t E(t) dt.
(The k in the integral is the same constant k as in part (a). You may leave your answer in terms of E0 and k.)
4. (6 points) Let
(a) (4 points) Calculate the indefinite integral f(x) dx.
(b) (2 points) Does the improper integral f(x) dx converge or diverge? Circle your choice, and justify your answer below. If it converges, also calculate its value.
5. (5 points) Consider the four sequences whose general terms are given below.
(a) (2 points) Which of the four sequences above are bounded? Which of them are monotone? List your answer(s) below. No justification required.
Now consider two more sequences, about which you only know the following properties.
(b) (1 point) Is either of {En} or {Fn} necessarily bounded or necessarily unbounded? Briefly explain your answer.
(c) (1 point) Neither of {En} or {Fn} is necessarily monotone. Briefly explain why.
(d) (1 point) Exactly one of {En} or {Fn} is necessarily convergent. State which one converges on the line, and show it must converge in the space below. Be sure to clearly state any theorems you use in your explanation.