AP - AB Calculus

							AP - AB Calculus                                         Name ________________________
Sem #1 - Test #5              Limits/Def of derivative
                                             DUE May 22, 2006 BY 8:00am on my desk.
100 points
This take home test – rules – NO help from any book, notes, person, computer etc. Take over
whatever time period you like. You can only see me for help by the end of the day - May 18th.

PLEDGE _____________________________________________________________________

_____________________________________________________________________________
Do as we have done in class. If no work is shown some points may be deducted.
As the AP says…. “Justify your answer.”




1.     Suppose that f(T) is the cost to heat my house, in dollars per day, when the outside
temperature is T degrees Fahrenheit.
       (a) What does f ’(23) = -0.17 mean?




       (b) If f(23) = 7.54 and f ‘(23) = -0.17, approximately what is the cost to heat my house
when the outside temperature is 21°F?
2.      To study traffic flow along a major road, the city installs a device at the edge of the road
at 4:00 a.m. The device counts the cars driving past, and records the total periodically. The
resulting data is plotted on a graph, with time (in hours) on the horizontal axis and the number of
cars on the vertical axis. The graph is shown below; it is the graph of the function C(t) = Total
number of cars that have passed by after t hours.




(a) When is the traffic flow greatest?




(b) From the graph, estimate C‟(3).




(c) What is the meaning of C‟(3)? What are its units? What does the value of C‟(3) you obtained
in (b) mean in practical terms?
3.      The graph of p(t) in figure 2.1.34 gives the position of a
particle at time t. List the following quantities in order, smallest
to largest:
A, average velocity on 1 ≤ t ≤ 3
B, average velocity on 8 ≤ t ≤ 10
C, instantaneous velocity at t = 1
D, instantaneous velocity at t = 3
E, instantaneous velocity at t = 10

               Answer _____, _____, _____, _____, _____



4.     The graph below represents the rate of change of a function f with respect to x; i.e., it is a
graph of f ‘.




You are told that f(0) = 0. On what intervals is f increasing? On what intervals is it decreasing?
On what intervals is the graph of f concave up? Concave down? Is there any value x = a other
than x = 0 in the interval 0 ≤ x ≤ 2 where f(a) = 0? If not, explain why not, and if so, give the
approximate value of a.
5.     Consider the function y =f(x) graphed below. (Notice that f(x) is defined for -5 < x < 6,
except x = 2




(a) For what values of x (in the domain of f) is f „(x) = 0?



(b) For what values of x (in the domain of f) is f „(x) positive?



(c) For what values of x (in the domain of f) is f „(x) negative?



(d) For what values of x (in the domain of f) is f „(x) undefined?



(e) Based on your answers to the above questions, make a sketch of y = f „(x) on the axes below.
Make your sketch as precise as possible.
6.         Given the following data about a function, f,
                                                x    3 3.5 4 4.5 5 5.5 6
                                                f(x) 10 8  7 4   2 0   -1
     (a)        estimate f’(4.25) and f’(4.75)




     (b)        Estimate the rate of change of f at x = 4.5




     (c)        Find, approximately, an equation of the tangent line at x = 4.5




     (d)           Estimate the derivative of f -1 at 2




7.         Find the derivatives of the given functions
                                1
           a)      y  2 x 31      2 x  4 2
                                2x




                               t 3  4t 2  7
           b)       g (t ) 
                                     t2
8.     Brenna believes that every continuous function is differentiable. While her sister Mary
Cae believes that every differentiable function is continuous. Who is correct? Justify your answer
with diagrams and words.




9.      Sketch a graph of a continuous function f(x)
with the following properties:
(a) f”(x) < 0 for x < 4
(b) f”(x) > 0 for x > 4
(c) f”(4) is undefined




10. Decide whether the following statement is true or false and provide a short explanation or
counterexample:
       The 10th derivative of y = x10 is 0