4.2 Homework - The Mean Value Theorem (Homework)

\fReview the Explore It, then use it to complete the exercise below. EXPLORE IT X CONCEPT EXPLORE & TEST WHEN WOULD I USE THIS YOU WILL LEARN ABOUT: The Mean Value Theorem for Derivatives speed limit = 50 mph sot secant line, 40- slope = 60 mph 19-14463764 Position (miles) S tangent lines with slope = 60 mph 0.25 0.75 Time (hours) In the diagram above, imagine an officer noting the time in which an individual enters and exits a toll road (via the time stamps on the receipts or stubs). The officer could issue a ticket based on the Mean Value Theorem if the ratio of the distance traveled (f(b)-f(a)) to the travel time (b-a) is greater than the speed limit. The MVT states that at some point between a and b, the slope of the graph (the speed of the car at an instant) must be equal to the slope of the secant line through the endpoints. In other words, there must be some c in the interval (a,b) such that f(c)=f(b) - f(a)b - a. -Select- positive Click here to access the Explore It in a new window. negative increasing Select Scenario 3, which is about exponential functions. Change the function to h(x) = 3ex - 2, and set the interval being considered to [-1, 4]. decreasing constant (a) To the nearest hundredth, the slope of the secant line is . This means the secant line ---Select--- . One of the reasons this makes sense is that the function h(x) = 3ex - 2 is always ---Select-- v (b) To the nearest hundredth, the value of c that satisfies the conclusion of the Mean Value Theorem in this case is c =(a) In the viewing rectangle [3, 3] by [-5, 5]; graph the function for) = x3 2x and its secant line through the points [2, 4) and (2, 4). Y Y O O (b) Find the exact values of the numbers cthat satisfy the conclusion of the Mean Value Theorem for the interval [2, 2]. (Enter your answers as a comma-separated list.) C: 9. [-/1 Points] DETAILS SCALCET8 4.2.036. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER At 2:00 PM a car's speedometer reads 30 mi/h. At 2:20 PM it reads 50 mi/h. Show that at some time between 2:00 and 2:20 the acceleration is exactly 60 mi/h2. Let v(t) be the velocity of the car t hours after 2:00 PM. Then V(1/3) - (0) with v'(c) = Since v'(t) is the 1/3 - 0 By the Mean Value Theorem, there is a number c such that 0