CHAPTER 12 Miscellaneous Free-Response Practice
Exercises
These problems provide further practice for both parts of Section II of the examination.
Part A. Directions: A graphing calculator is required for some of these problems.
x 2.5 3.2 3.5 4.0 4.6 5.0f (x) 7.6 5.7 4.2 3.1 2.2 1.5- A function f is continuous, differentiable, and strictly decreasing on the interval [2.5,5]; some
values of f are shown in the table above.
(a) Estimate f ′(4.0) and f ′(4.8).
(b) What does the table suggest may be true of the concavity of f? Explain.
(c) Estimate with a Riemann sum using left endpoints.
(d) Set up (but do not evaluate) a Riemann sum that estimates the volume of the solid formed when
f is rotated around the x-axis. - The equation of the tangent line to the curve x^2 y − x = y^3 − 8 at the point (0,2) is 12y + x = 24.
(a) Given that the point (0.3,y 0 ) is on the curve, find y 0 approximately, using the tangent line.
(b) Find the true value of y 0.
(c) What can you conclude about the curve near x = 0 from your answers to parts (a) and (b)? - A differentiable function f defined on −7 < x < 7 has f (0) = 0 and f ′(x) = 2x sin x − e−x^2 + 1.
(Note: The following questions refer to f not to f ′.)
(a) Describe the symmetry of f.
(b) On what intervals is f decreasing?
(c) For what values of x does f have a relative maximum? Justify your answer.
(d) How many points of inflection does f have? Justify your answer. - Let C represent the piece of the curve that lies in the first quadrant. Let S be the region
bounded by C and the coordinate axes.
(a) Find the slope of the line tangent to C at y = 1.
(b) Find the area of S.
(c) Find the volume generated when S is rotated about the x-axis.