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MATH 105 PRACTICE EXAM 1 Key

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true, explain why.

A)     (7 Points)  If f(x)  and g(x)  are differentiable functions with

           f '(5)  =  10    and    g'(5)  =  4

then if

                         f(x)
     h(x)   =           -  3g(x)
                      2 

        then

            h'(5) = -7

Solution

True:  

        h'(x) = (f(x)/2 - 3g(x))' 

        = f '(x)/2 - 3g '(x)

So that 

        h'(5) = f '(5)/2 - 3g '(5) = 10/2 - 3(4) = -7

B)     (7 Points)  Let f(x)  and g(x)  be continuous functions
     f(1)  >  g(1)       and      f(2) < g(2)
then if
     h(x) = f(x) - g(x)
h(x)  has a root for some value of x between 1 and 2.

Solution

       True:

                    h(1) =  f(1) - g(1) > 0

                    h(2) =  f(2) - g(2) < 0

        Hence by the Intermediate Value Theorem, there is a c with h(c) = 0.

C)     (7 Points)  Let f(x)  and g(x)  be continuous functions such that

           \( \lim\limits_{x \to 2} g(x) = 0 \)   and    \( h(x) = \frac{f(x)}{g(x)} \) 

            Then h(x) has a vertical asymptote at x = 2.

Solution

                  False, Let f(x) = x - 2        and g(x) = x - 2

PROBLEM 2 Find the following limits if they exist:

A) (8 Points) \( \lim\limits_{x \to \frac{3}{4}} \frac{6x^2 +19x - 36}{3x^2 - 7x + 4} \)

Solution

B) (8 Points)  \( \lim\limits_{x \to 0} \frac{x}{tan(x)} \)

Solution

C)    (8 Points)  \( \lim\limits_{x \to 2} \frac{\sqrt{x-2} + 2}{x+2} \) 

Solution

PROBLEM 3

A)   (8 Points)  Find the following limits if they exist

                   i)  \( \lim\limits_{x \to -3^{+}} f(x) \)  ii) \( \lim\limits_{x \to -1} f(x) \)     iii) \( \lim\limits_{x \to 0} f(x) \)     iv) \( \lim\limits_{x \to 1} f(x) \)      v) \( \lim\limits_{x \to 3} f(x) \) 

Solution

     i)  0    ii)  2    iii)  1    iv)  Does Not Exist    v)  Does Not Exist

B)    (8 Points)  At which values is f(x) not continuous?

Solution

  -1, 1, and 3

C)   (8 Points)  At which values is f(x) not differentiable?

Solution

-3, -1, 1, and 3

PROBLEM 4   (20 Points)  Below is the function y = f(x).  Sketch a graph of the derivative y = f ’(x).

Solution

PROBLEM 5   Find f ' (x) for the following

A)    (10 Points)  \(  f(x) = x^2 sec(x) - \frac{2x^3}{cot(x)} \) 

Solution

B)     (11 Points)  \(  f(x) =  \frac{2}{x^5} - (x - cos(x))^2 + \frac{x^3}{\sqrt{x}} + \pi^2 \) 

Solution

       -10x^-6 - 2x + 2cosx - 2xsinx + 2cosxsinx + 5/2 x^3/2

PROBLEM 6 Let \(  f(x) = \sqrt{x-1} \) 

A)    (10 Points)  Use the limit definition of the derivative to find f ’(x).

Solution

B)     (10 Points)  Prove using the \(\epsilon - \delta \)  definition of the limit that \( \lim\limits_{x \to 2} 6 - 2x = 2 \)

Solution

     Let e > 0 , choose d = e/2.

                  Then  

                        |x - 2| < d

                   implies that 

                        |x - 2| < e/2

                    so that

                        |2x - 4| < e

                    or

                        |4 - 2x| < e

                    adding and subtracting two gives 

                        |4 + 2 - 2x - 2| < e

                        |6 - 2x - 2| < e

                   Hence

                        |f(x) - 2| < e

                     So that that the limit exists.