Calculus 1  Hour Exam 1   -   Math 220     Fall, 2011

 

 

 

 

 

 

 

1.

 

 

Circle true if the given statement is always true.  Otherwise circle false.

 

a.        Given a function g, if  | g(x) | ≤ x4  for all  x  then  lim g(x)  =  0.

                                                                                            x→0

b.       If the point (-4, ¼) is on the graph of an odd function  g then (-1/4, 4) is another

point on the graph of g.

c.       Given a function g, if  lim  [g(x) – g(4)]/(x – 4) exists then g  is continuous at 4.

                                               x→4

d.      If a function  g  is continuous at 0 then  lim g(x) = 0.

                                                               x→0

e.       A function which is continuous at a point  a  must also be differentiable at a.

f.       If a function is one-to-one then   g(1) = 1.

 

Answers:     a. T        b. F        c. T        d. F        e. F        f. F

 

 

2.

 

Given the graph  y  =  f(x)  and six possible choices for graphs of  f ’(x)  then circle the

correct graph  of  f ‘(x)  for the given graph  y = f(x).

 

 

 

3.

 

 

Let  f(x)  =  4x3 + 2.  Use the definition of a derivative as a limit to prove that

f ‘(x) = 12 x2.  Show each step in your calculation and be sure to use proper terminology

in each step of your proof.

 

 

 

 

 

4.

 

 

Suppose that  f  and  g  are one-to-one functions which take on the following values.

 

     f(-2) = 2,           f(-1) = ½,         f(0) – ½,          f(1) = -2,           f(2) = -4

 

     g(-2) = -4,         g(-1) =- -2,        g(0) – ½,          g(1) = ½ ,          g(2) = 2

 

What is the value of  f-1 ( g-1(-4) ) ?                                   Answer:  1

 

 

 

 

 

 

 

5.

 

 

State the domain of each function.

 

a.                               f(x)  =  cos-1 (x)

b.                              g(x) =  (8 – x) / ln(x – 4)

c.                               h(x) = √ (x2 + 9)

 

Answers:       a.  [ -1, 1 ]         b.  ( 4, 5 ) U ( 5, ∞ )        c.  ( -∞ , ∞ )

 

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