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School of Mathematics and Statistics
Carleton University
Math. 69.107
TEST 1 SOLUTIONS

Print Name: 

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

1. [2 marks] Let  . Which of the following expressions represents the value of  ?

(a)  ,   , (c)  , (d)  .

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

(a)  , (b)  , (c)  ,   .

3. [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=1, y=0 is equal to:

  , (b)  , (c)  , (d)  .

4. [2 marks] Evaluate the limit: 

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

5. [2 marks] Answer TRUE or FALSE:

If f is continuous at a point x=a in its domain, then f is differentiable at x=a.

(a) TRUE,  FALSE

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

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

a)  .

b) 

Solution a)  or use L'Hospital's Rule directly after the second equation. So



Solution b) The form of the limit is 0/0 so L'Hospital's Rule can be used. Thus, 

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

a)  . Find  .

b)  . Find 

Solution a) 

At x=0,  and  , so  .

Solution b)  .

Next, we use the Product Rule:



3. [10 marks] Find the equation of the tangent line to the curve  at the point x=2.

Solution  . At x=2 the value is  . The equation of the tangent line is given by  where  and  . In other words, the equation looks like y - 1 = 24(x-2) or y = 24x -47.

Total: /40
 

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Mon Oct 26 11:01:40 EST 1998