AP Calculus BC 2017 Free-Response Questions

Overview

• The free-response questions are part of the AP Calculus BC exam.
• Section II is divided into Part A and Part B.
• Part A requires a graphing calculator and Part B does not.

Section II, Part A

Question 1

• Scenario: A tank with a height of 10 feet.
  • The area of horizontal cross-section at height ( h ) is given by a function ( A(h) ).
  • Function ( A(h) ) is continuous and decreases as ( h ) increases.
  • Data is provided in a table.
• Tasks:
  • (a) Use a left Riemann sum with three subintervals to approximate the volume of the tank. Indicate units.
  • (b) Determine if the approximation overestimates or underestimates the volume. Explain why.
  • (c) Model the area with function ( f ) and find the tank's volume. Indicate units.
  • (d) Water is pumped into the tank. When water height is 5 feet and increasing at 0.26 ft/min, find the rate of volume change using the model from part (c). Indicate units.

Question 2

• Polar Curves:
  • Two curves are given: ( r = f(\theta) = 1 + \sin(\theta) \cos(2\theta) ) and ( r = \cos(\theta) ).
  • Define region ( R ) and ( S ) in the first quadrant.
• Tasks:
  • (a) Find the area of region ( R ).
  • (b) Write an equation involving integrals to find ( \theta = k ) dividing ( S ) into equal areas.
  • (c) Write an expression for the distance ( w(\theta) ) between points ( (f(\theta), \theta) ) and ( (g(\theta), \theta) ). Find average value ( w_A ).
  • (d) Find ( \theta ) where ( w(\theta) = w_A ) and determine if ( w(\theta) ) is increasing or decreasing.

Section II, Part B

Question 3

• Function: ( f ) is differentiable on ([6, 5]) and ( f'(2) = 7 ).
• Tasks:
  • (a) Find ( f(6) ) and ( f(5) ).
  • (b) Identify intervals where ( f ) is increasing. Justify.
  • (c) Find absolute minimum of ( f ) on ([6, 5]). Justify.
  • (d) Evaluate ( f'(3) ) and determine existence.

Question 4

• Scenario: Potato cooling from initial temperature 91°C.
  • Modeled by ( H(t) ) with differential equation ( \frac{dH}{dt} = -\frac{1}{4}(H - 27) ).
• Tasks:
  • (a) Tangent line at ( t=0 ) and approximate temperature at ( t=3 ).
  • (b) Use ( \frac{d^2H}{dt^2} ) to analyze if approximation is under/overestimate.
  • (c) Alternate model ( G(t) ) with different differential equation. Determine temperature at ( t=3 ).

Question 5

• Function: ( f(x) = \frac{2x^3 - 7x^2 + 5x}{2x^2 - 5x + 1} ).
• Tasks:
  • (a) Find slope of tangent at ( x=3 ).
  • (b) Identify and classify critical points in ((1, 2.5)).
  • (c) Evaluate or determine divergence of given integral.
  • (d) Determine convergence/divergence of series. Explain test used.

Question 6

• Function: ( f ) has derivatives of all orders with given conditions.
• Tasks:
  • (a) Show first four nonzero terms of Maclaurin series and general term.
  • (b) Determine type of convergence at ( x=1 ).
  • (c) Write first four nonzero terms and general term for ( g(x) = \int_0^x f(t) , dt ).
  • (d) Use alternating series error bound for Taylor polynomial evaluation at ( x=0 ).

Conclusion

• End of AP Calculus BC Free-Response Questions.