# Math 1013 M

Midterm 2

Version 22

**Instructions:**

- Please write your responses as neatly as possible.
- If you are uploading photos, make sure you can read the images before you submit them.
- If we can’t read your work, we can’t give you points.

**How to submit: **This is exactly like the crowdmark practice test. You can either:

- Print this test out and complete as normal. When done, scan/photo each question/page and upload to crowdmark using the link they sent you by email.
- Solve each question using your own blank paper. When done, scan/photo each question/page and upload to crowdmark using the link they sent you by email.
- Use a tablet with a stylus to answer questions directly on the pdf file emailed to you by crowdmark. When finished, upload the file to crowdmark using the link they sent you by email.

**Code of honour: **I am trusting that you are all not cheating for this exam. You are more than welcome to use the notes from class to help you answer the questions (in fact, I encourage it!) But please don’t talk to one another nor should you use the internet to ask for solutions to the problems. I trust in you all.

## Exercise 1 (10 points) Find the following limits

## Exercise 2 (10 points) Find the following limits

## Exercise 3 (10 points) Find the following limits

**Exercise 4 **Differentiate the following:

**Exercise 5 (10 points) **Suppose that s is a function of r. Find d/dr s of the following function:

r^{4}s^{2 }+ sin(rs)^{2 }+ csc(r + s) = 5

**(Bonus - 5 points) **Suppose further that r is a function of q. Find _{dq}__ ^{d }__s

^{2}.

## Exercise 6 (25 points)

A 2 meter tall bear is walking in the woods and sees honey flavoured chips strapped to a flashlight hanging from a tree 3 meters above the ground. The bear, smelling honey, decides to raid the chips package and starts walking towards the honey at 3 meters per second. As the bear walks, the flashlight is casting a shadow behind the bear which gets shorter and shorter.

(1) When the bear is 3 meters away from the honey, how fast is the length of his shadow decreasing?

(Assume that the honey chips and the flashlight are at exactly the same point.)

**Exercise 7 (5 points) **Find a good approximation for

f(x) = arccot(x) at x = 8/9

**Exercise 8 (20 points) **Using the techniques in class, sketch the curve of the following function.

f(x) = x^{3 }− 18x^{2 }+ 81x

Things to include:

- Where does the function cross the x axis?
- What are the critical points? What are the inflection points?
- Where is the graph increasing/decreasing?
- Where is the graph concave up/down?

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