Exam2

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MATH 105 PRACTICE MIDTERM II

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.

  Printable Key

PROBLEM 1  (7 Points each)  Please answer the following true or false.  If false, explain why or provide a counter example.  If true, explain why.

1. Let f(x) be differentiable on [0,1], then there is a positive c less than 1 such that
       f(1) = f(0) + f '(c)
   Solution

2. Let f(x) be a differentiable function such that f '(c) = f ''(c) = 0 , then f has neither a relative maximum nor a relative minimum at x = c .
   Solution

3.  If f ''(x) < 0 for all x between 0 and 1 and f(0) = 0  then f(1) < 0.
   Solution

4. If f(x) is a periodic differentiable function, that is f(x + p) = f(x)  for some p and all x, then f '(x) has an infinite number of roots.
   Solution

PROBLEM 2  Consider the function (22 Points)

       \(  y = x^2 + \frac{1}{x}  \) 

Without the use of the graphing capabilities of your graphing calculator, for the following two functions

1. Determine the relative extrema if any.
   Solution

2. Determine where the function is increasing and decreasing.
   Solution

3. Determine the inflection points if any.
   Solution

4. Determine where the function is concave up and concave down.
   Solution

5. Find any horizontal asymptotes.
   Solution

6. Find any vertical asymptotes.
   Solution

7. Use the above to graph the function, labeling all important points and asymptotes.
   Solution

PROBLEM 3  (20 Points) 

As you are standing outside on a beautiful sunny late afternoon, you notice that your shadow is growing in length at a rate of 2 feet per hour.  If you are 6 feet tall, how fast is the angle of elevation of the sun decreasing when your shadow is 8 feet long?

          Solution

PROBLEM 4   (20 Points Each)

1. Suppose that

               x² + y + sin(xy) = 2

    Find the equation of the tangent line to this curve at the point (0,2).
   Solution

2. Let
   \(  f(x) =  \frac{cos(x^2 + 1)}{1 - x}  \) 
   
   Find
   \(  \frac{df}{dx}  \) 
   
   Solution

PROBLEM 5    (20 Points)

Sketch the graph of a function with the following properties:

• \(  \lim\limits_{x \to -\infty} f(x) = 1  \) 
• \(  \lim\limits_{x \to \infty} f(x) = -1  \) 
• \(  \lim\limits_{x \to -1^{-}} f(x) = -\infty  \) 
•  \(  \lim\limits_{x \to -1^{+}} f(x) = \infty  \) 
• f ''(x) > 0 only for     x < -3,     -1 < x < 2,     and     3 < x < 5
• f '(x) > 0 only for      x < -2 and   x > 4
• f '(-2) = f '(2) = f '(4) = 0
• f ''(-3) = f ''(2) = f ''(3) = 0

  Solution

PROBLEM 6   (20 Points)

You plan to build a rectangular office building of minimal cost with a with a wall down the middle to subdivide the building into two offices as shown in the picture.  The outer walls cost $500 per foot to construct and the divider wall costs $100 per foot to construct.  If the combined square footage of the two offices is to be 1,000 square feet, what dimensions should you make the building?

Solution