MAT136H5S - Integral Calculus
APRIL 2022 FINAL EXAMINATION
Part A: Multiple Choice Questions (40 marks)
For Part A, fill in your answers in the multiple choice answer table on the last page of the exam. (You may use a pen or pencil.) There may be a penalty for not indicating answers on the last page.
There is only ONE correct answer for each question. Each question is worth 4 marks.
No marks are given for incorrect answers, or for filling in more than one circle in the same row. There may be a penalty for answers indicated outside of the multiple choice answer table.
1. Evaluate the integral dx.
2. The expression is a Riemann sum that is equal to an integral. Which one?
3. Let Then what is g
′
(x)?
4. Which one of the following integrals represents the volume obtained by rotating the region bounded by y = sin(x2), the x-axis and around y-axis?
5. A group of doctors are conducting a clinical trial for an experimental cancer drug treatment plan. The graph shows the probability of patient survival as a function of time, t (measured in months). The efficacy of the drug can be defined by
The drug is considered beneficial at time τ if E(τ ) > 0.
Which of the following statements are TRUE?
I. Patients in the treatment group are more likely to survive for at least 12 months than those in the placebo group are.
II. The efficacy of the drug at τ = 30 is 0.
III. The drug is beneficial at 36 months.
(a) I and II are true.
(b) I and III are true.
(c) Only I is true.
(d) Only II is true.
(e) All are true.
6. For which values of p is the series convergent?
(a) Converges for p > 4.
(b) Converges for p < 1.
(c) Converges for p ≥ 3.
(d) Converges for p > 1.
(e) Converges for p > 3.
7. Which one of the following sequences is convergent? (There is only one correct answer.)
8. Suppose that the sequence of the partial sums of the series is given by Which ONE of the following statements is TRUE?
(a) is convergent but we do not have enough information to evaluate its sum.
(b) The sequence converges and the series diverges.
(c) = 0.
(d) The sequence converges to 0 and the series converges to 2/5.
(e) Both the sequence and the series an are divergent.
9. Suppose you wanted to use a Taylor or Maclaurin series to approximate the value of cos(1.01) without a calculator. Which of the following is the best choice for the centre, c, of your series?
10. Which one of the following power series is convergent only at one point?
Part B: Written Answer Questions (60 marks)
In Part B, you must fully justify all your answers. Provide full solutions and show all your work. Part marks may be given. A correct answer with false reasoning or little to no justification will not receive any marks. Simplify your answers as much as possible, unless stated otherwise. Points are indicated by each question.
1. (8 marks) Evaluate the integral dx.
2. (10 marks - 2 marks each) Suppose that the complete graph of the function f consists of straight lines and a quarter circle, as shown. Answer the questions and briefly justify your answers.
(a) Evaluate the definite integral f(t) dt.
(b) Suppose g(x) = f(t) dt where f is given in the graph. What is g
′
(6)?
(c) For what value(s) of x does g have an absolute maximum?
(d) Now suppose that f gives the outdoor temperature at a certain location at time t (hours).
i. Find the average temperature over time from t = 1 to t = 5.
ii. Find f
′
(t) dt and explain what it represents in terms of temperatures.
3. (12 marks - 2 marks each) Which of the following statements are TRUE? Justify your answers BRIEFLY (in a sentence or at most two). Also circle your final answer for each part.
(a) A Riemann sum of a positive function using right endpoints, will result in an overestimate of the area under the graph.
TRUE / FALSE
(b) The partial fractions decomposition of is of the form.
TRUE / FALSE
(c) If 0 < f(x) ≤ 1 for all x and f(x) dx converges, then (f(x))3
dx converges too.
TRUE / FALSE
(d) If a monotone sequence of positive terms is divergent, then it must have an infinite number of terms larger than a million.
TRUE / FALSE
(e) For an infinite series an, if then an = 0.
TRUE / FALSE
(f) If the power series converges for x = 1, then it must converge for x = 8.
TRUE / FALSE
4. (a) (5 marks) Evaluate the integral
(b) (5 marks) Is the improper integral convergent or divergent? If it is convergent, evaluate it.
5. (a) (8 marks) Does the series converge absolutely, converge conditionally, or diverge? Show your work. (Use any test for convergence that you think is appropriate.)
(b) (2 marks) Suppose that you wanted to estimate the sum of the series How many terms would be needed to ensure you get an estimate that is accurate to within 0.001?
Note:
6. You have an oyster farm that starts with 5000 oysters. By the end of every month the population of oysters doubles, then a certain percentage c of the oysters die (a percentage of the number of oysters at the beginning of the month die), and then you harvest 100 live oysters to sell.
Let Pn be the population of Oysters at the beginning of month n.
(a) (2 marks) Write Pn+1 recursively in terms of Pn and c.
(b) (2 marks) If c is 0%, then describe what happens to the oyster population after a long time.
(c) (2 marks) If c is 100%, then describe what happens to the oyster population after a long time.
(d) (4 marks) Find a rate c so that the oyster population tends to 5000 over a long time.