Math 161 Practice Midterm13 September 2000

You must show your work or explain how you found your answers. No calculators.

1. Consider this graph of y = f (x). Find as closely as possible each of the values in parts a)-d):

   

   1. f (4) =$\framebox [2cm]{\Huge\phantom{gH}}$
   2. f'(4) =$\framebox [2cm]{\Huge\phantom{gH}}$
   3. The slope of the tangent line at x = 4 is $\framebox [2cm]{\Huge\phantom{gH}}$.
   4. $$\lim_{h\rightarrow 0}^{}$$$${\frac{f(4+h) - f(4)}{h}}$$=$\framebox [2cm]{\Huge\phantom{gH}}$
   5. For what values of x is f''(x) > 0?
   6. Plot, on the same axes, the graph of y = f'(x).
2. If g(x) = 3x² - 1, use the definition of the derivative to calculate g'(2).
3. Let f (x) = 2sin(x) + $\pi$.

   

   1. On the axes above, draw the graph of f.
   2. On the same axes, sketch the tangent line to the graph of f at x = 2$\pi$.
   3. Find an equation for the tangent line to f at x = 2$\pi$: y =$\framebox [5.3cm]{\Huge\phantom{gH}}$

4. Find the derivative of each of the following functions:

   
   

   1. f (x) = 3x³ - cos x + 6         f'(x) =$\framebox [4cm]{\Huge\phantom{gH}}$

      
      

   2. g(x) = e²ln(x³)         g'(x) =$\framebox [4cm]{\Huge\phantom{gH}}$

      
      

   3. h(x) = $${\frac{\sin \pi}{e^x}}$$         h'(x) =$\framebox [4cm]{\Huge\phantom{gH}}$

      
      

   4. j(x) = 7x$\sqrt{x}$         j'(x) =$\framebox [4cm]{\Huge\phantom{gH}}$
5. $\parbox{3in}{Here is a graph of $y=h'(x)$. Note that this is a graph... ...ive} of $h$. The graph of $h$\ is not shown. On the interval shown, determine}$    

   

   1. all the values of x for which h is increasing.$\framebox [4cm]{\Huge\phantom{gH}}$
   2. all the values of x for which there is a minimum for h.$\framebox [4cm]{\Huge\phantom{gH}}$
   3. all the values of x for which h' is increasing.$\framebox [4cm]{\Huge\phantom{gH}}$
   4. all the values of x for which h is concave down.$\framebox [4cm]{\Huge\phantom{gH}}$
   5. all the values of x for which there is an inflection point for h.$\framebox [4cm]{\Huge\phantom{gH}}$

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