Mean Value Theorem: If f is a continuous function on [(1,1)] and f is differentiable on ((1,6), then there exists at least one number c in (a, 5) such that (5) - f0!) f'(6) = \"it- 50. 1. Looking at the statement of the Mean Value Theorem, explain in words what the difference quotient on the right hand side of the equation above represents in terms of the slope of a line. 2. Looking at the statement of the Mean Value Theorem, explain in words what f' (c) represents in terms of the sIOpe of a ne. 3. Putting together the two pieces above, explain in your own words what the Mean Value Theorem says. You may want to provide a sketch to help with your explanation. 4. Let at) = 3:3 and let's restrict our view to looking at f on [1, 1]. (a) Sketch a graph of f(:L') = 2:3. (b) Find the slope of the secant line connecting the points (0, 0) and (1, 1). Give the equation of the secant line and graph it on the graph of f(:r). (c) Find the number(s) c in (1,1) such that f'(c) has slope equal to the slope of the secant ne you found above and explain why such a number must exists. Give the equation of the tangent ]ine(s) and graph the line(s) on the graph of f (9:). (d) What does the fact that the nes all having the same 310pe mean about their relationship in the plane? 5. You are riding in a bus on the highway. The bus's speed is being monitored by radar at toll stations along the highway. The bus passes under the rst radar at 12pm going 55 mph. At 12:05, 7 miles down the road, the bus passes under the second radar going 55 mph. Did the bus exceed the speed limit of 55 mph on your trip? Explain your answer. 6. Think of another context where the Mean Value Theorem could apply. Write your own word problem ]ike the one above, that uses the Mean Value Theorem to answer a question. Give a complete answer and explanation along with your