CALCULUS AB
SECTION I, Part C
Time – 45 minutes
Number of questions – 3
A GRAPHING CALCULATOR IS PERMITTED FOR QUESTION 3 ONLY.
Directions: Solve each of the following problems, using the available space for work.
Mark your answers clearly.
AP Calculus-AB Mid-Term Examination – Form A
Free Response Question 1
Let
by
be a function defined for all
for all
, such that
, and the derivative of
is given
.
(A) Find all values of x for which the graph of
has a horizontal tangent, and determine
whether
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 concave up? Justify your answer.
(C) Write an equation for the line tangent to the graph of at
.
(D) Does the line tangent to the graph of at
lie above or below the graph of for
?
Why?
Free Response Question 2
Consider the curve given by
(A) Find
.
.
(B) Find all points on the curve whose x-coordinate is , and write an equation for the tangent
line at each of these points.
(C) Find the x-coordinate of each point on the curve where the tangent line is vertical.
Free Response Question 3–Calculator permitted
(hours)
people
0
1
3
4
6
7
8
248
357
313
491
263
196
187
Concert tickets went on sale at 8:00 am (
) and were sold out by 4:00 pm. The number of
people waiting for assistance by phone or Internet sales to purchase tickets at time is modeled
by a twice-differentiable function for
. Values of
at various times are shown in
the table above.
(A) Use the data in the table to estimate the rate at which the number of people waiting for
assistance was changing at 12:30 p.m. (
). Show the computations that lead to your
answer. Indicate units of measure.
(B) Use a trapezoidal sum with three subintervals to estimate the average number of people
waiting for assistance during the first four hours that the tickets were on sale.
(C) For
, what is the fewest number of times at which
? Give a reason for
your answer.
(D) The rate at which tickets were sold for the time interval is modeled by
tickets per hour. Based on the model, how many tickets were sold for the concert, to the
nearest whole number?