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Math 1013 M Exercise

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Midterm 2

Version 22


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

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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

Math 1013 M exercise 1 Find the following limits

Exercise 2 (10 points) Find the following limits

Math 1013 M exercise 2 Find the following limits

Exercise 3 (10 points) Find the following limits

Math 1013 M exercise 3 Find the following limits

Exercise 4 Differentiate the following:

Math 1013 M 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:

r4s2 + sin(rs)2 + csc(r + s) = 5

(Bonus - 5 points) Suppose further that r is a function of q. Find dqd s2.

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) = x3 − 18x2 + 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?
Math 1013 M exercise 8 Answer

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