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SOLUTION OF EXAM 1 MATH 131 Fall 2002

1. a) (10) Evaluate the limit

by an algebraic procedure. Show all your algebraic steps.

b) (5) Support your result in part a) with a calculator. Describe your procedure and show as much data as possible.

 x y 1.9 .377868 1.99 .375310 1.9999 .375003 1.999999 .375000 2.1 .371661 2.01 .374685 2.0001 .374997 2.000001 .374999

As the values of approach 2 either from the right or from the left (but not equal to 2) the values of approach .

2. Given the function

where , are constants.

a) (5) Find and .

b) (5) What conditions do and have to satisfy, so that is continuous at .

For to be continuous at we need the limit exist and equal . Since and exists if and only if we see that the condition on and is .

c) (5) Find and , such that is everywhere differentiable.

First must be continuous everywhere, thus from part b). Notice that is differentiable at each . The derivative at is the limit . It exists if and only if the left hand and the right hand derivative limits are equal. We have

Therefore and .

3. a) (10) Find all horizontal and vertical asymptotes for the function

Explain your reasons.

The vertical asymptotes are the lines such that the denominator is zero at , but the numerator is not. We have , i.e. or . Since and the vertical asymptotes are and .

The horizontal asymptotes are the lines where is the limit of as approaches positive or negative infinity. We have

Therefore the only horizontal asymptote is .

b) (5) Graph the function in part a) indicating all horizontal and vertical asymptotes.

4. (10) Determine the derivative of the given function using the definition of the derivative as a limit

One can first write . Then

Another way is

5. A ball is dropped from a tower of height feet. Its height after seconds, in feet above the ground, is given by the formula

a) (5) Find the velocity of the ball after seconds.

The velocity function is the derivative of the height function with respect to the time . Thus .

b) (5) It is observed, that the ball hits the ground with a speed of feet per second. How long was the ball in the air? Hint: The speed is the absolute value of the velocity.

At the moment when the ball hits the ground the velocity is ft/s (the velocity vector points downward). We get , i.e. , which means that the ball was 4 seconds in the air.

c) (5) Use the information in part b) to determine the height of the tower.

When the ball hits the ground the height of the ball is zero. On the other hand, it is the height of the ball after 4 seconds, thus we get , i.e. and .

6. a) (8) Suppose is a number, such that

Find an equation for the tangent line to the graph of at the point .

The slope of the tangent line to the graph of at the point is equal to the derivative of at . By definition the derivative of at is the limit . We are given that this limit is equal to 17, so the slope of the tangent is 17. Since the tangent passes through the point the equation of the tangent is , i.e. .

b) (7) Find a formula for the derivative of the following function

Therefore,

7. a) (10) Find a formula for the derivative of the function

By the quotient rule

b) (5) Find the equation of the tangent line to the graph of at the point .

The slope of the tangent line to the graph of at is . By part a)

Since the line passes through we have

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Eyal Markman 2002-10-22