a.) Does this function satisfy the hypothesis of the Mean Value Theorem on the interval [1, 3]? ? v 1)) Does it satisfy the conclusion? ? v c) At what point c is f'((:) : @? If such a point 0 exists, give an exact answer when possible, or give an approximation based on the graph. If no such 6 exists, enter DNE into the blank. 6 : EEE (1 point) Consider the function f graphed below. a) Does this function satisfy the hypothesis of the Mean Value Theorem on the interval [0, 3]? ? v b) Does it satisfy the conclusion? '2 v c) At what point c is f' (c) : w ? If such a point c exists, give an exact answer when possible, or give an approximation based on the graph. If no such 6 exists, enter DNE into the blank. C : === (1 point) Suppose that a car's speedometer reads 23 miles per hour and then 4 minutes later the speedometer reads 2? miles per hour At some point during those 4 minutes, was the acceleration ever exactly 60 miles per hour per hour? ? v (1 point) Our goal is to use the racetrack principle to prove that e'\" 2 1 + a: for all at: Z 0. Below is an outline of a proof of this statement. At each step, select the correct word or phrase which makes this proof valid. First, we know that the functions 8\" and 1 + a: are ? v on [0, co) and ? v on (0, 00). Also, we know that $033) : iii , and %(1 + 11:) : EEE , .Therefore 833) is ? v + :12) on (0,00). 80, by the ? v ,we may conclude that e: Z 1 + at. (1 point) Our goal is to use the racetrack principle to prove that In (ac ) 0 for all :3. Does this make your answer to part (a) an underestimate or an overestimate? O underestimate O overestimate (1 point) Imagine you are trying to compute V91 but don't have access to a calculator. We will walk through the steps of how we can use local linearization to estimate 9.1. a) To use local linearization, we need to choose a function f and a value a near 9.1 at which we want to linearize: f(:1:) : iii and a. : b) Using the function f and value a that you found in part a), nd the equation for the local linearization L(:.c) off at a: : a: L(:c) : c) Plugging m : 9.1 into the equation for the local linearization in part b), estimate 9.1: Estimate: 9.1 m iii d) Is this an overestimate or underestimate? O overestimate O underestimate 8) Finally, check your results with the calculator. (1 point) a ) Find the equation of the tangent line to f ( ) = ek at x = 0. Tangent Line: y = b) How does the tangent line that you found in part a) relate to the local linearization of ek near x = 0? O A. There is no relationship between the local linearization of f at a = 0 and the tangent line to f at x = 0. O B. The local linearization of f at a = 0 is parallel to the tangent line to f at x = 0. O C. The local linearization of f at x = 0 is exactly the tangent line to f at x = 0. O D. The local linearization of f at a = 0 is perpendicular to the tangent line to f at x = 0. c) Based on your answer to part b), what is the equation for the local linearization of f (a) = ek at ac = 0? Local Linearization: L(ac ) = d) Observe that we use the first derivative to find the local linearization of f at x = 0, which allows us to approximate f near x = 0. We do this by finding the linear function L which satisfies L(0) = f(0) and L' (0) = f' (0). It turns out that we can approximate f near x = 0 even better by finding a quadratic function Q which satisfies Q (0) = f(0), Q'(0) = f' (0), and Q"(0) = f"(0). Find a quadratic function which approximates f near a = 0. That is, find a quadratic function Q which satisfies Q (0) = f(0), Q'(0) = f'(0), and Q"(0) = f"(0) Q (a) =(1 point) The value of cos(49 ) is known to be approximately 0.656059. Solve for the tangent line of cos(a ) at x = . Use the tangent line to estimate the value of cos(49). cos(490 ) ~