M AP CALCULUS BC Test Booklet 6.2 Approximating Areas with Riemann Sums - Homework 1. Let f be a continuous function such that f changes from increasing to decreasing, and the graph of f changes from concave up to concave down. Which of the following is true about the midpoint Riemann sum approximation for 3 f f(at2 + 10): using 4 subintervals of equal width? 1 1 f(12+1)+f(l.52+1.5) f(1.53+1.5)+f(222) f(22+2)+f(2.52+2.5) f(2.59+2.5)+f(32+3) E (f + f + f + f) (A) 3 is the midpoint Riemann sum approximation and underestimates / f (:32 + 2:) dz. 1 f(1211) 1f(1.5211.5) f(115211.5) +f(22-2) 1122 12) 131252125) f(2,5212.5) 1313213) %(+ + f + f + f) (B) 3 is the midpoint Riemann sum approximation and overestimates / f(a22 + ac) dm. 1 %(f(1.252 + 1.25) + f(1.752 + 1.75) + f(2.3252 + 2.25) + f(2.752 + 2.75)) is the midpoint C ( ) Riemann sum approximation and underestimates / f(a:2 + 3:)d33. 1 5311.252 + 1.25) + ,f(1.752 + 1.75) + f(2.252 + 2.25) + f(2.752 + 2.75)) is the midpoint (D) Riemann sum approximation. There3 is not enough information to determine whether the approximation underestimates or overestimates / f (m2 -i- m)d:. 1 2' 1m- manna The differentiable function R is increasing, and the graph of R is concave down. Values of R(t) at selected values oft are given inthe table above. The trapezoidal sum (5 1)( 102111) i (8 5)( 17-2510) i (13 8)(17;32) l3 approximates R(t)dt. Which of the following statements is true? 1 13 (A) The trapezoidal sum is an underestimate for R(t)ft because R is increasing. 1 13 (B) The trapezoidal sum is an overestimate for R(t)dt because R is increasing. 1 13 (C) The trapezoidal sum is an underestimate for R(t)ft because the graph of R is concave down. 1 13 (D) The trapezoidal sum is an overestimate for R(t)dt because the graph of R is concave down. 1 AP Calculus Page 1 of 3 AP CollegeBoard Test Booklet 6.2 Approximationg Areas with Riemann Sums - Homework 3. a 2 3 10a 12a f (a) 5 2 The continuous function f is decreasing for all x. Selected values of f are given in the table above, where a is a 12a constant with 1