School of Mathematics and Statistics
Carleton University
Math. 69.104
TEST 1
SOLUTIONS

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Student Number: PART I: Multiple Choice Questions
(Choose and CIRCLE only ONE answer)

1. [2 marks] Let . Then is equal to:

   , (b) , (c) , (d) .

2. [2 marks] Let . Which of the following expressions represents the value of ?

   (a) , , (c) , (d) .

3. [2 marks] Evaluate the limit:

   , (b) , (c) The limit does not exist, (d) .

4. [2 marks] Let y be given implicitly as a differentiable function of x by . Then the slope of the tangent line of the curve y = y(x) at the point (x, y) where x=0, y=1 is equal to:

   (a) , (b) , (c) , .

5. [2 marks] Answer TRUE or FALSE:

   The function f defined by f(x) = |x| is differentiable at x=0.

   (a) TRUE,

PART II: Show all work here.
No additional pages will be accepted

1. [5+5 marks] Find the required limits:

   a) .

   b) Solution: a)

   

   b) This is a limit involving an indeterminate form of the type 0/0: So, we can use L'Hospital's Rule.

2. [5+5 marks] Evaluate the required derivative of each of the following functions:

   a) . Find .

   b) . Find Solution: a) Let . Then

   

3. [10 marks] Use Newton's method to approximate the positive root of the function . Use the starting value and use your calculator to find .

   Given that we know that Newton's Method yields

   

   Next, we use THIS estimate for in the definition of . In this way we find

   

   Finally, using this estimate for in the definition of we find

   

   You should compare this value with the square root of 2 as found using your calculator.

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