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Math 1010U Calculus I
Midterm II: November 2022
First Name
Last Name
Student Number
Signature of Student
Which Program are
you in (circle one)
Science FESNS FEAS
(Energy System and Nuclear Science) (Engineering)
Name of TA
(circle correct one)
David B., Parikshit B., Shaimaa S., Georges K.,
Marian K., Saadat S., Saba.M
FIRST 4 LETTERS IN YOUR LAST NAME:
Instructions:
- There is a 1 mark penalty if the correct TA name is not circled above
- The test is printed double-sided; there are questions on the back of some pages!
- Use a pen to fill in the front page. All midterm answers should be written in ink. If a midterm is
written in pencil, or white-out is used, you may not be allowed to re-submit questions for
remarking.
- A non-programmable, non-graphing calculator is allowed
- There are 6 questions on this test and you will have 70 minutes. Unless otherwise indicated,
YOU MUST SHOW ALL OF YOUR WORK! Answer the questions in the space provided.
- We will NOT be answering any mathematical questions during the test.
Question # Points Available Points Earned
Total 45
1. [ 6 marks] An outdoor sensor light is at the top of 4m gate. Mary is 1.6m tall, and walks away from
the gate in a straight path with a speed of o.3m/s. How fast is the tip of her shadow moving when she is
9m from the gate?
3. Given the function ๐
๐ฅ
3
โ
1
2
๐ฅ
2
(a) [4 marks] Find all of the critical numbers of ๐(๐ฅ).
ANSWER: _________________________________________________________
(b) [4 marks] For the function from part (a) use closed interval method to find the absolute maximum
and the absolute minimum on the interval [โ 1 ,
1
3
]. [If your answer is in decimal form, you
should round to 3 decimal places]
ANSWER: _________________________________________________________________
(c) [3 marks] Find lim
๐ฅโ 0
1 โ๐
โ 3 ๐ฅ
1 โ๐
โ 2 ๐ฅ
ANSWER: _____________________________________________________
4. Let ๐
2
2
. Answer the following questions, knowing that
โฒ
2
โฒโฒ
2
(a) [1 mark] ๐ is increasing on: ____________________________________
(b) [1 mark] ๐ is decreasing on: ____________________________________
(c) [1 mark] ๐ has local max(s) at ๐ฅ =________ and local min(s) at ๐ฅ =_____________
(d) [1 mark] ๐ is concave up on: ______________________________________________
(e) [1 mark] ๐ is concave down on: ____________________________________________
(f) [1 mark] ๐ has inflection point(s) at ๐ฅ =______________________________________
(g) [1 mark] Sketch a graph of the function based on the information above:
6. [2 marks each; 6 marks total] For each of the following questions, select ALL of the
correct answers by clearly shading in the appropriate boxes. Each question is worth 2
marks, but there may be anywhere from 0 to 4 correct answers. You will lose 1 mark for
each mistake (i.e. selecting something thatโs wrong, or missing something thatโs correct,
up to a maximum of 2 marks deduction per question (i.e. no negative marks โบ)
a) Given the function ๐(๐ฅ) = ๐ฅ
2
โ ๐ฅ as ๐ฅ changes from 1 to 1. 2 and considering differentials:
b) The graph of some function
f ( x )
is shown below. Which of the following statements are true?
๓ ฎ f ( x ) has a critical number at ๐ฅ = 3
f ' ( x )๏ผ 0 for ๐ฅ < โ 3
๓ ฎ f ' '( 4 )๏ผ 0
The Mean Value Theorem applies to f ( x )on the interval [- 2 , 0].
c) Let f ( x )be a function such that f ' ( x )๏ฃ 3 for all
x
in the interval [2, 6]. Which the following
is guaranteed by the Mean Value Theorem?
If f ( 2 )= 10 , then f ( 6 )๏ฃ 2 If f ( 6 )= 2 , then f ( 2 )๏ณโ 10
If f ( 2 )= 10 , then f ( 6 )๏ณโ 2 If f ( 6 )= 2 , then f ( 2 )๏ฃ 10
