MAT136 - Term Test 1
Part A: Multiple Choice Questions (12 marks)
1. Which of the following expressions equals
(a) (100)(101) + (90)(5)
(b) + (90)(5)
(c) (100)(101) − (10)(11) + (90)(5)
(d) (100)(101) − (11)(12) + (90)(5)
(e) (100)(101) − (10)(11) + 5
2. Which of the following integrals does this limit of Riemann sums equal?
3. A particle moves in a straight line. Its velocity at time t is 4 − 2t m/s. Find the total distance traveled by the particle over the time interval [0, 3].
(a) 2 m
(b) 1 m
(c) 3 m
(d) 5 m
(e) −3 m
4. Let f and g be continuous functions with domain R which satisfy the following three equations.
Find the value of
(a) −1
(b) 0
(c) 7
(d) 3
(e) 1
5. Let g be a differentiable function with domain R. Let Find a formula for F
′
(x).
(a) F
′
(x) = −g
′
(sin x) · cos x
(b) F
′
(x) = g(7) − g(sin x)
(c) F
′
(x) = −g(sin x)
(d) F
′
(x) = −g(sin x) · cos x
(e) F
′
(x) = g(sin x) · cos x
6. Suppose that f is a continuous function such that What is the value of f(u) du?
(a) 2
(b) 1
(c) 6
(d) 1.5
(e) 3
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, in at most two or three sentences, explain why. If you believe a statement is false, explain why or give an example to show it’s false. A correct true/false answer without justification will receive 0 points (no points for guessing).
(a) equals its right-endpoint Riemann sum for some integer n.
Note: You can explain your answer to this with a picture.
(b) If f is a continuous positive function with domain R, then the definite integral f(x) dx is positive for any real number a.
(c) The average value of f(x) = 2x + 7 on [0, 2] equals f(1).
2. (6 points) Consider the graph below of a function f with domain [0, 10], consisting of straight lines and a semicircle.
Also, let F(x) = f(t) dt.
(a) (1 point) Find the value of F(10). No justification required.
(b) (1 point) Write down an expression involving an integral that equals the average value of f on [4, 10]. You do not have to evaluate any integral. No justification required.
(c) (2 points) On which interval(s) is F decreasing? Put your answer on the line, and briefly explain how you got it in the space provided.
(d) (1 point) At which points in [0, 10] does F have its absolute minimum and maximum? No justification required.
(e) (1 point) Let G(x) = f(t) dt. Find the value of G′
(3.5). No justification required.
3. (6 points) Astronomy students working overnight at the observatory in the McLennan physics building at St. George campus observe a strange object above the Earth, moving down along a straight path directly toward them.
The altitude of the object at time t (measured in minutes) is modeled by the function s(t). At t = 0, its altitude is s(0) = 80 km.
The rate of change of the altitude of the object s
′
(t) is continuous and increasing over the time interval [0, 8]. s
′
(t) is measured in kilometres per minute.
The table below shows selected values of s
′
(t). A plot of these data is also given to the right.
(a) (2 points) Is positive or negative? What does this quantity represent?
(b) (1 point) Explain why the quantity represents the altitude of the object at t = 8 minutes. Answer in at most one or two sentences.
(c) (1 point) The students want to use a Riemann sum to help approximate and underestimate the altitude of the object at t = 8 minutes.
Circle the correct choices to complete the sentences below. No justification required.
• To underestimate the altitude, they need an estimate of s
′
(t) dt that is than its actual value.
• To do this, the students should do a endpoint Riemann sum for s
′
on [0, 8].
(d) (2 points) Use the type of Riemann sum you chose in part (c), with four intervals, to calculate an underestimate of the altitude of the object at t = 8 minutes.
4. (6 points) There is no connection between the two parts of this question.
(a) (3 points) Evaluate the definite integral Put your final answer on the line, and your supporting work in the space below.
(b) (3 points) Find the indefinite integral
Put your final answer on the line, and your supporting work in the space below.
5. (4 points) In this problem, you will compute using Riemann sums.
(a) (1 point) Write two sums, in sigma notation, representing a left-endpoint Riemann sum Ln and a right-endpoint Riemann sum Rn approximating this integral, for a general n. No justification required.
(b) (2 points) Simplify your expression for the right-endpoint Riemann sum until you get a formula for Rn in terms of n only, with no ∑. Put your answer on the line, and show your work in the space below.
(c) (1 point) Evaluate to find the value of the definite integral.
(You will get no credit for using FTC2 here, though you can use it to check your answer on scrap paper.)