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AP CALCULUS PROBLEM SET Derivative

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									                    AP CALCULUS PROBLEM SET #5                           CURVE ANALYSIS II
(92BC-4)
                                                 
                                                     2 x  x 2 for x  1
1.       Let f be a function defined by f ( x)  
                                                  x  kx  p for x  1
                                                    2
                                                 

(a) For what values of k and p will f be continuous and differentiable at x = 1 ?

(b) For the values of k and p found in part (a), on what interval or intervals is f increasing?

(c) Using the values of k and p found in part (a), find all points of inflection of the graph of f.
    Support your conclusion.




(89-5)
2.




The figure above shows the graph of f ', the derivative of a function f. The domain of f is the set of all
real numbers x such that 10  x  10 .

(a) For what values of x does the graph of f have a horizontal tangent?

(b) For what values of x in the interval (10, 10) does f have a relative maximum? Justify your answer.

(c) For what values of x is the graph of f concave downward?




(99-4)
3. Suppose that the function f has a continuous second derivative for all x, and that f (0) = 2, f ' (0) = 3,
                                                                           2 x
and f "(0) = 0. Let g be a function whose derivative is given by g '(x) = e (3 f (x) + 2f ' (x)) for all x.

(a) Write an equation of the line tangent to the graph of f at the point where x = 0.

(b) Is there sufficient information to determine whether or not the graph of f has a point of inflection when
    x = 0 ? Explain your answer.

(c) Given that g (0) = 4, write an equation of the line tangent to the graph of g at the point where x = 0.

                             2 x
(d) Show that g"(x) = e             (6 f (x)  f ' (x) + 2 f " (x)). Does g have a local maximum at x = 0 ?
    Justify your answer.
(2002-6)
4.
                                x                          
                              f (x)                               
                              f '(x)                                  

Let f be a function that is differentiable for all real numbers. The table above gives the values of f and its
derivative f ‘ for selected points x in the closed interval 1.5  x  1.5 . The second derivative of f has
the property that f " (x) > 0 for 1.5  x  1.5 .

               1.5
(a) Evaluate    (3 f ( x)  4)dx .   Show the work that leads to your answer.
               0


(b) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this line to
    approximate the value of f (1.2). Is this approximation greater than or less than the actual value
    of f (1.2) ? Give a reason for your answer.

(c) Find a positive real number r having the property that there must exist a value c with 0 < c < 0.5 and
    f "(c) = r. Give a reason for your answer.

                                          2 x 2  x  7 for x  0 
                                                                  
(d) Let g be the function given by g(x) =  2                      
                                          2 x  x  7 for x  0 
                                                                  
    The graph of g passes through each of the points (x, f (x)) given in the table above. Is it possible that
    f and g are the same function? Give a reason for your answer.


(2000-3)
5.




The figure above shows the graph of f ' , the derivative of the function f , for  7  x  7 . The graph of
f ' has horizontal tangent lines at x = 3, x = 2, and x = 5, and a vertical tangent line at x = 3.

(a) Find all values of x, for 7 < x < 7, at which f attains a relative minimum. Justify your answer.

(b) Find all values of x, for 7 < x < 7, at which f attains a relative maximum. Justify your answer.

(c) Find all values of x, for 7 < x < 7, at which f "(x) < 0.

(d) At what value of x, for  7  x  7 , does f attain its absolute maximum? Justify your answer.


(2001-4)
6.   Let h be a function defined for all x  0 such that h(4) = 3 and the derivative h is given by
               x2  2
     h '( x)         for all x  0 .
                 x
(a) Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a
    local maximum, a local minimum, or neither at each of these values. Justify your answers.

(b) On what intervals, if any, is the graph of h concave up? Justify your answer.

(c) Write an equation for the line tangent to the graph of h at x = 4.

(d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for x > 4 ? Why?
?

								
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