(1 point) Let f(x) = 8 sin(2) a.) If'(2) < b.) By the Mean Value Theorem,...
Question 5 (1 point) S2x4, Let f(2) - <x< 0 5 sin(x), 0 < x < Evaluate the definite integral [ f(x) f(x)dx. 5 O + 10 873 - 10 O 1/25 - 10
1. (a) State and prove the Mean-Value Theorem. You may use Rolle's Theorem provided you state it clearly (b) A fired point of a function g: (a, bR is a point cE (a, b) such that g(c)-c Suppose g (a, b is differentiable and g'(x)< 1 for all x E (a, b Prove that g cannot have more than one fixed point. <「 for (c) Prove, for all 0 < x < 2π, that sin(x) < x.
Let us verify the Mean Value Theorem with the function f(x) = VE on the interval (2,8). Solution. We have f is continuous on (2,8) f is differentiable on (2,8). f'(o) – f(8) – f(2) 8 - 2 We have f'(x) = The only value that satisfies the Mean Value Theorem is
Use the Mean Value Theorem to demonstrate that In(1 + x) < x, given that x > 0.
20. The function f(x)=e satisfies the hypotheses of the Mean Value Theorem on the interval [0, 16] Find all values of c that satisfy the conclusion of the theorem. a. - Sin 2e b. Sin c. -Sin d. Sin 2e2
Theorem 20.8 (The Mean Value Theorem for Integral Calculus). Let f a, bR be continuous, and g a, bR be integrable and nonnegative. Then, there exists acE (a,b) such that (20.3) f(x)g(a)dx - f(c g(x)dr (ii). Apply Theorem 20.8 to show that 1 100 32 Jo (1 +r2)5 Theorem 20.8 (The Mean Value Theorem for Integral Calculus). Let f a, bR be continuous, and g a, bR be integrable and nonnegative. Then, there exists acE (a,b) such that (20.3) f(x)g(a)dx...
1-8 please 1. Find the value c that satisfies Rolle's Theorem for f(x) = cos x on A / B./2 C. D. E. 0 F. None of the above 311/4 2. The function f is graphed below. Give the number of values that satisfy the mean value theorem on the interval (-6,6). A. 0 B. 1 C. 2 D. 3 E. 4 F. None of these Page 1 of 5 1. The graph off) is shown. Find the value(s) where)...
(1 point) Let f(x) = 0 if x < -4 5 if – 4 < x < 0 -3 if 0 < x < 3 0 if x 2 3 and g(x) = Los f(t)dt Determine the value of each of the following: (a) g(-8) = 0 (b) g(-3) = 5 (c) g(1) = (d) g(4) = (e) The absolute maximum of g(x) occurs when x = 0 and is the value It may be helpful to make a graph...
1. (10 points) Let f(x) = S1 (+2) - 1, 2<1 1 - 4 sin(), >1 and g(x) = 5-24 (a) Sketch as much off and g as will fit on the graph below. Draw them in two different (dark) colors or one solid/one dashed, and clearly label the graph. -6 -4 -2 2 4 6 8 10X (b) Find the exact value of the limit below, and explain your reasoning in complete sentences. lim (g(x)) =
Problem 2 Find o satisfying the Mean Value Theorem for f(x) on the interval (0, 1). (a) f(x) = f* (b) f(x) = x2