Using Released AP Items to Inspire Improved Instruction
Inspired by #78 BC 2016 International and #3 AB/BC Free Response 2016 International
Graph of f(x)Graph of g(x)
a) If is a continuous function, such that, find and justify your
answer.
b) If find
c) If write the equation of a line tangent to at
d) If find and
e) If find and
f) Find the value of c such that the area of the region enclosed by the graph of and the x-axis in the
first quadrant is 12.
g) Find the value of c for which the average value of over the interval is 4.
h) Evaluate .
i) If find and .
j) If find .
Can you think of another question to be asked?
Inspired by #28 BC 2016 International (only 22% answered correctly)
A particle travels along the x-axis so that at time its velocity is given by What is the total distance the particle travels from to ?
Inspired by #27 BC 2016 International (41% answered correctly; originally an improper integral)
If R is the region between the graph of and the x-axis between and , then find the area of R.
Inspired by #84 BC 2016 International (only 44% answered correctly; originally an area problem)
The figure above shows the region A, which is bounded by the x- and y- axes, the graph of for , and the vertical line The region A is revolved around the x-axis to form a solid. If k increases at a rate of units per second, how fast is the volume of the solid increasing when ?
Inspired by #90 BC 2016 International (only 44% answered correctly)
A water vessel has the shape of a right circular cone. The height of the vessel is 20 cm, and the radius of the opening is 5 cm. Water is poured into the vessel at a constant rate of 3 What is the rate at which the water level is rising when the depth of the water in the cup is 4 cm? (The volume of a cone of height h and radius r is given by )
Inspired by #22AB 2016 International (only 17% answered correctly)
A particle moves along the x-axis so that at time , the acceleration of the particle is . The position of the particle is 8 when and the position of the particle is 15 when . What is the velocity of the particle at time ?
Inspired by #92AB 2016 International (only 27% answered correctly)
In the xy-plane, the graph of the twice-differentiable function is concave down on the open interval (5,7) and is tangent to the line at x = 6. Which of the following statements must be true about the derivative of f ?
A) on the interval (5.9, 6)
B) on the interval (5.9, 6)
C) on the interval (5.9, 6.1)
D) on the interval (5.9, 6.1)
Inspired by #2 AB/BC Free Response 2016 International and #1 AB/BC Free Response 2016 Operational
t(weeks) / 0 / 2 / 5 / 10 / 12
S(t)
(students per week) / 15 / 30 / 50 / 75 / 90
Seniors at Pville High School are being infected with the dreaded senioritis virus. Mrs. Ross tracks the rate of infection over a 12-week period. The rate at which seniors are being infected is modeled by the differentiable and increasing function S(t), measured in students per week, and t is measured in weeks for Values of S(t) are given in the table above for selected values of t.
Mrs. Ross designs the HARDWORK serum to cure senioritis and begins administering it to seniors five weeks after the epidemic begins. Starting with 25 students per week at time t = 5, the continuous rate at which seniors will be cured will increase at a constant rate of 10 students per week per week.
a) Approximate the value of using the data in the table. Show the computations that lead to your answer. Using correct units, explain the meaning of in the context of the problem.
b) Approximate the value of using right Riemann sum with the four subintervals indicated by the table. Is the approximation greater than or less than the actual value of Justify your answer. Explain the meaning of in the context of the problem.
c) Is there a time in the interval such that Justify your answer.
d) Based on the given models for and is there a time in the interval where the rate that seniors are being infected is equal to the rate that Mrs. Ross is able to cure them through administration of the HARDWORK serum? Justify your answer.
e) Based on the given information and your answer from part b, approximately how many students will still be infected with the senioritis virus at the end of the 12th week?
Inspired by #24AB 2016 International (only 29% answered correctly)
Gravel is deposited into a pile with a circular base. The volume V of the pile is given by , where r is the radius of the base, in feet. The circumference of the base is increasing at a constant rate of feet per hour. When the circumference of the base is feet, what is the rate of change of the volume of the pile, in cubic feet per hour?
Inspired by #12AB 2016 International (only 32% answered correctly)
If , what is the area of the region enclosed by the graphs of y = 0, , x = c and x = 1?
Inspired by #7AB 2016 International (only 34% answered correctly)
For which of the following pairs of functions is infinite?
A) and
B) and
C) and
D) and