calculus homework

5.1 The Mean Value Theorem m Write your questions and thoughts here! We use the MVT to justify conclusions about a function over an interval. Mean Value Theorem: If a function f is continuous over the interval and differentiable over the interval then there exists a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval . 1. Use the function f(x) : x2 + 3x + 10 to answer the following. a. On the interval [2, 6], what is the average rate of change? b. On the interval (2, 6), when does the instantaneous rate of change equal the average rate of change? MVT vs IVT Mean Value Theorem Intermediate Value Theorem MVT IVT o The derivative (instantaneous rate of o On a given interval, you will have a 3/- change) must equal the average rate of value at each of the end points of the change somewhere in the interval. interval. Every y-value exists between these two y-values at least once in the interval. t 30 minutes 80 var) 4 feet per minute A hot air balloon is launched into the air with a human pilot. The twicedifferentiable function h models the balloon's height, measured in feet, at time t, measured in minutes. The table above gives values of the Mt) and the vertical velocity 120?) of the balloon at selected times t, a. For 5 S t S 20, must there be a time 1: when the balloon is 50 feet in the air? Justify your answer. b. For 20 S t S 30, must there be a time t when the balloon's velocity is 1.5 feet per second? Justify your answer. 5.1 The Mean Value Theorem m Calculus l. Skater Sully 1s rldrng a skateboard back and forth on a street that runs north/south. The twice-differentiable function S models Sully's position on the street, measured by how many meters north he is from his starting point, at time t, measured in seconds from the start of is ride. The table below gives values of the S (t) and Sully's velocity v(t) at selected times t. _"n seconds 56) "- meters 1105) 3.2 meters er second a. For 0 S t S 20, must there be a time t when Sully is 2 meters south of his starting point? Justify your answer. b. For 30 S t S 60, must there be atime t when Sully's velocity is 1.1 meters per second? Justify your answer. 2. A particle is moving along the x-axis. The twice-differentiable function 5 models the particles distance from the origin, measured in centimeters, at time t, measured in seconds. The table below gives values of the 5(t) and the velocity v(t) of the article at selected times t. t S (t) 5 2 1 0 CH1 em oer second a. For 20 S t S 25, must there be a time t when the particle is at the origin? Justify your answer. b. For 3 S t S 10, must there be a time t when the particle's velocity is 1.5 cm per second? Justify your answer. ,3 a. A hot air balloon is launched into the air with a human pilot, The twice-differentiable rnetion h models the balloon's height, measured in feet, at time 6, measured in minutes. The table below gives values of the h(t) and the vertical velocity v(t) of the balloon at selected times t. . t 0 6 10 40 mlnutes W) 0 46 35 105 feet "(0 0 6 20 1 feet per minute a. For 6 S t S 10, must there be a time t when the balloon is 50 feet in the air? Justify your answer. b. For 10 S t S 40, must there be a time t when the balloon's velocity is 3 feet per second? Justify your answer. Using the Mean Value Theorem, find where the instantaneous rate of change is equivalent to the average rate of chan_e. 4. y=x25x+2 on [4,2] 5. y=sin3xon [0,71] 1 _ 6. y = (5x + 15)? on [1,3] 7- y ex 011 [0'1\" 2] Calculator active problem 8, A particle moves along the x-axis so that its position at any time t 2 0 is given by x(t) = t3 31?2 + t + 1. For What values of t, 0 S t S 2, is the particle's instantaneous velocity the same as its average velocity on the closed interval [0, 2]? N0 calculator on this problem. 9. The table below gives selected values of a function . The function is twice differentiable with f "(x) > 0. Which of the following could be the value of f '(5)? (A) 0.5 (B) 0.7 (C) 0.9 (D) 1.1 (E) 1.3 10. Let g be a continuous function. The graph of the piecewise-linear function g', the derivative of g, is shown above for 4 S x S 4. Find the average rate ofchange ofg'(x) on the interval 4 S x S 4. Does the Mean Value Theorem applied on the interval 4 S x S 4 guarantee a value of c, for 4