Question 2. Our goal is to prove that f(x) = x + e* has exactly one root (i.e. f(x) = 0 for exactly one value of x)." a) First, we really need this function to be both continuous and differentiable. Explain in words how we know this is true. b) Now, let's prove that f(x) has at least one root. What theorem do we need for that? Hint: This is a review problem! We're looking all the way back at section 2.5. c) Use the theorem you stated in part a) to prove that f(x) has at least one root.

Billy is walking from the front door of his house to his bus stop, which is 940 feet away from his front door. As Billy walks out his front door he walks in a straight path toward his bus stop at a constant rate of 7.5 feet per second.a) Define a function f to determine Billy's distance from his bus stop in terms of the number of seconds he has been walking.b) What is the independent quantity and what is the domain of ff (the values the independent quantity can take on)?c) What is the dependent quantity and what is the range of ff (the values the dependent quantity can take on)?d) What do each of the following represent: f(0) and f(60.25)?e) Use function notation to represent the following:  i) Billy's distance from the bus stop after he has walked 23.6 seconds   ii) The change in Billy's distance from the bus stop as the number of seconds since Billy left his front door increases from 25 seconds to 48 seconds f) If t represents the number of seconds since Billy left his front door,…

The first and second derlvatives of a function f are given. Use these to find the Information below. f'x)=(3-x)(x-5) f"(x)=2(x-5) (7-2x) a. On what Interval(s) is fincreasing? Type your answer in interval notation. If there is more thon one interval, separate them with a comma, Use "infinity" (without the quotes) for o. b. For what values(s) of x does fhave a local maximum? Round your answer(s) to 2 decimal places, if necessary. If there is more than one x-value, separate them with o comma. If there is no local maximum, type "none" (without

based on what we know a function to be today, what did Euler mean by an analytic expresstion?