Question

Activity: A Journey Through Calculus from A to Z x g'(x) sin(x - 1) x-1 kx2 - 8x +6, * 1 1<x<3 -4 13 h(x) = f'(2) 14e2x-6 – x2 +5, x>3 108 2 3 e -1 Consider f'(x), the derivative of the continuous function f. defined on the closed interval (-6,71 except at x = 5. A portion of f' is given in the graph above and consists of a semi circle and two line segments. The function h(x) is a piecewise defined function given above where k is a constant. The function g(x) and its derivatives are differentiable. Selected values for the decreasing function g'(x), the second derivative of g, are given in the table above. (E) Find all x value(s) on the open interval (-2,5) where f(x) has a point of inflection. Give a reason for your answer. (F) Find the average rate of change of h(x), in terms of k, over the interval (2,5) (G) If f(3) = 5, write an equation of the tangent line to f(x) at x = 3. (H) Use a right Riemann sum with the four subintervals indicated in the table to approximate g'(x)dx. Is this approximation an over or under estimate? Give a reason for your answer.

Please show ALL of your work as if you don't have a calculator. Thanks!

Answer 1

t'(w) = 2-14-22 – 2404201 24-n 25045 = 2 5n<7. i ' f(n) =o. at noo and u as is there is no sign Chang of f'(n) ie t (m) yo on to and film yo nyo so at nao is the point of inflection. • At nou , f (m) Yoney and f (m) 20 suyu So at nou, f has manimum Answer : n=o. © Average rate of change of h (m) in [25] n(5)-h(2) (4 61 - 25+5)-au - 16+6) L 5-2 ue" – 20-ukt 10 s ue – uk-10 3

G (3) 25. f' (3)=1. i Equation of tangens une of the coure Yoffer) 8-5. 1. n-3. y_574-3 ? Hry +2=0 f"( 13 Right Riemann sum of g (m) ²9" (m) dr - 4 a 0 2 m Bxlo + EX 8)+(2xe) + 1X(4) = 30 + 8 +2e-1 = 37+2 e. Since the ewure of gun) is decreasing and sum of the area of the rectangles as is lower than me value of the accual carrea So this is rende under estima