Math 1A – Chapter 2 Test II Solutions – Fall ’04    

 

1.      The graph of f is given.

a.       Find each limit, or explain why it doesn’t exist:

i.	ii.	iii.          doesn’t exist.
iv.	v.	vi.

b.      Where does the function have removable discontinuities?
ANS:  There are removable discontinuities where x = –2 and x = 3.

c.       Where does the function have a jump discontinuity?
ANS:  The function has a jump discontinuity where x = –4.

2.      Find the limit

a.	b.
c.	d.

 

3.       Use the intermediate value theorem to prove that   has a solution in (-1,0).
SOLN:  Consider .  Since f is a sum of continuous functions, it is continuous.  Also 0 is between  and  so by the Intermediate Value Theorem, there is  such that f(c) = 0, which is a solution to given equation.

4.      If the tangent to y = f(x) at (5,4) passes through the point (1,2), find f(5) and .
SOLN:  f(5) = 4 is given. The slope of the tangent line is

5.      Find the derivative of  using the definition of the derivative.
SOLN:  One way is to plug directly into the formula for the limit of a difference quotient:
Alternatively, it’s a bit simpler to do division to write  and then plug into the difference quotient:

6.      Suppose that we don’t have a formula for g(x) but we know that g(2) = -4 and  for all x.

a.       Use a linear approximation to estimate g(1.95) and g(2.05).
SOLN:  Using the line tangent to g(x) at x = 2, that is,  we have  and

b.      Are your estimates in part (a) too large or too small?  Explain.
SOLN:   Since  is increasing, g(x) is concave up and so the tangent line lies beneath the curve.  Therefore both estimates are underestimates.

 

7.      Is there a number a such that   exists?  If not, why not?  If so, find the value of a and the value of the limit.
SOLN:    can exist only if the numerator goes to zero at -2.  That is .  If a = 8 then

8.      Consider .

a.       What theorem is essential to evaluating this limit.  Why are the conditions of the theorem met?
SOLN:  The relevant theorem says that if   (i.e. the limit exists) and if f is continuous at b (i.e.  ) then .  Since  is a composition of continuous functions, it is also continuous, thus the conditions of theorem are met.

b.      Use the theorem to evaluate the limit.
SOLN:

9.      For the function  whose derivative function  is graphed below,

a.        is increasing on

b.       is concave up on (–2,1)

c.        has a local maximum where x = -3 and where x = 1.

d.       is positive on (–2,1)

e.        where x = –2 and where x = –1.