MATH 2B Study Guide - Midterm Guide: Improper Integral, Trigonometric Substitution, Asymptote

Consider the function f(x)=x²-5x+4 on the interval [0,5]. The graph of f is shown below.

A. Estimate the integral f(x)dx using the approximation L₅ (The approximation using 5 rectangles and left endpoints as sample points).

B. Sketch the approximating rectangles used L5 on the graph above.

(c) Sketch a graph of a continuous function y = f(1) that satisfies the following: • f(-5) = f(0) = 0; • f'(x) < 0 for -5 < <-3 and 1 < x < 5; • f'(x) > 0 for -3 < x < 1; • f'(x) < 0 for -2 < x < 3; • F"(x) > 0 for -5 < x < -2 and 3 < x < 5; Clearly label and local extrema and points of inflection.