Homework Sourse13e function f x ex on the interval 0 ln 4 a Show that the M
13.e function f (x) e^x on the interval [0, ln 4] a. Show that the Mean Value Theorem can be used. b. Approximate the value point that is guaranteed by the theorem. c. Make a sketch of the function. Include the secant line that goes through the endpoints and the tangent line that goes through the point found in part b.
Solution
More precisely, if a function f(x) is continuous on the closed interval [a, b], where a < b, and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that [1] This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour. To get at that average speed, the car either has to go at a constant 100 miles per hour during that whole time, or, if it goes slower at one moment, it has to go faster at another moment as well (and vice versa), in order to still end up with an average of 100 miles per hour. Therefore, the Mean Value Theorem tells us that at some point during the journey, the car must have been traveling at exactly 100 miles per hour; that is, it was traveling at its average speed.![13.e function f (x) e^x on the interval [0, ln 4] a. Show that the Mean Value Theorem can be used. b. Approximate the value point that is guaranteed by the theo](/WebImages/37/13e-function-f-x-ex-on-the-interval-0-ln-4-a-show-that-the-m-1112810-1761590431-0.webp "13.e function f (x) e^x on the interval [0, ln 4] a. Show that the Mean Value Theorem can be used. b. Approximate the value point that is guaranteed by the theo")
