# MAT136F Assignment 1 due Friday, May 14 at 11:59 pm Instructions: • This assignment consists of 7 questions for a total of 37 points. • Your solutions must be submitted on Crowdmark by Friday, May 14...

MAT136F Assignment 1
due Friday, May 14 at 11:59 pm
Instructions:
• This assignment consists of 7 questions for a total of 37 points.
• Your solutions must be submitted on Crowdmark by Friday, May 14 at 11:59 pm. You should complete
this assignment well before 11:59 pm to give yourself time to submit everything.
• You may print out this document and write your solutions in the space provided or you may write
• Show all of your work. Unsupported answers will not earn credit.
• Have fun!
Marking Scheme:
Question XXXXXXXXXX
Points XXXXXXXXXX
MAT136F Assignment 1 Page 1 of 7
1. Let f(x) be a continuous function. Suppose the following three facts are known:
• f(0) = 3
• A left endpoint Riemann sum for f(x) on [0, 8] using 4 subintervals yields a value of 30.
• A right endpoint Riemann sum for f(x) on [0, 8] using 4 subintervals yields a value of 48.
(a) (2 points) What is the value of
4∑
k=1
(b) (3 points) What is f(8)? Show your work.
MAT136F Assignment 1 Page 2 of 7
2. Let f(x) = 3x2 + 4x− 1.
(a) (3 points) By taking the limit of a Riemann sum, evaluate
∫ 2
0
Hint: you will need to use the following facts:
n∑
k=1
1 = n,
n∑
k=1
k =
1
2
n(n+ 1), and
n∑
k=1
k2 =
1
6
n(n+ 1)(2n+ 1)
(b) (2 points) By using the Fundamental Theorem of Calculus, evaluate
∫ 2
0
f(x) dx. Show your
work.
MAT136F Assignment 1 Page 3 of 7
3. Find the total area of the following regions. Show your work.
(a) (2 points) The region in the first quadrant below the function y = 5− |x− 3|.
(b) (2 points) The region above f(x) and below the x-axis, where f(x) =
{
x2 − 1, x ≤ 0
ex − 2, x > 0
.
(c) (2 points) The region below y =

9− (x− 4)2 and above the x-axis.
MAT136F Assignment 1 Page 4 of 7
4. For each of the functions below, find the signed area of the region bounded by the function on the
(a) (2 points) f(x) =
x sin(x2)√
cos(x2)
on [0,

π
4
]
(b) (2 points) g(x) = sin(x) sin(2x) on [−π, π]
(c) (2 points) h(x) = x sin(x) sin(2x) on [−π
4
, π
4
]
MAT136F Assignment 1 Page 5 of 7
5. Let f(t) be a continuous function and define F (x) =
∫ 1+x3
1−x
f(t) dt.
(a) (2 points) If possible, find a number c ∈ R such that F (c) = 0. If not possible, explain why not.
(b) (2 points) Alice claims the average value of f(t) on [0, 2] is 1
2
F (1). Is she correct? Explain your
(c) (3 points) For this part only, suppose f(t) = ln(t XXXXXXXXXXWhat is F ′(x)? Show your work.
MAT136F Assignment 1 Page 6 of 7
6. Suppose that at time t = 0 water starts flowing out of a large tank. The water pours out at a rate of
r(t) =
100t+ 50
t2 + t+ 1
litres/min with t measured in minutes.
(a) (1 point) What is the physical interpretation of the statement r(1) = 50?
(b) (2 points) How much water flows out of the tank in the first 5 minutes? Show your work.
(c) (2 points) How long does it take for 300 litres of water to flow out of the tank? Round your
MAT136F Assignment 1 Page 7 of 7
7. (3 points) Let a and b be positive real numbers. Show that the following inequality holds:
0 ≤
∫ b
a
| sin t|
t
dt ≤ ln b− ln a

## Solution

Rajeswari R answered on May 10 2021

Qno.3
a) y = 5-|x-3|
Let us sketch the region first
Only in the I quadrant is given
Y = 5-x+3 if x >=3 and y =5+x-3 , if x <3
i.e. y = 8-x,x≥3,
y = x+2, x<3
Area...

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