Intro to Calculusname

Intro to CalculusName:

Date:

Mean Value Theorem

Let be a function that satisfies the following hypothesis:

1. is continuous on the closed interval [a, b]

2. is differentiable on the open interval (a, b)

Then there is a number c in (a, b) such that

Eq 1:

OR

Eq 2:

What do you recognize in the two equations?? What is the Mean Value Theorem really telling us??

Class Examples:

1a. Show that the function satisfies the hypotheses of the Mean Value Theorem on the interval [0, 2].

b. Find all numbers c in the given interval that satisfy the equation .

2a. Show that the function satisfies the hypotheses of the Mean Value Theorem on the interval [0, 2].

b. Find all numbers c in the given interval that satisfy the equation .

If an object moves in a straight line with position function , then the average velocity between t = a and t = b is and the instantaneous velocity at t = c is . By the Mean Value Theorem there is some time t = c between a and b that the instantaneous velocity is equal to the average velocity. For example, if a car traveled 180 km in 2 hours, then the speedometer must have read 90 km/hour at least once.

3. A car accelerating from zero takes 8 sec. to go 352 ft. What is its average velocity for the 8 second interval in mph? What would the speedometer read at some point during the acceleration?

4. Suppose that and for all values of x. How large can possibly be?

5. Explain why each of the following functions fails to satisfy the conditions of the Mean Value Theorem on the interval [-1, 1].

a. b.

Homework

For #1-4:

a) state whether or not the function satisfies the hypothesis of the Mean Value Theorem.

b) if it does, find each value of c in the interval (a, b) that satisfies the equation:

1.

2.

3. Reminder:

4.

For #5-6: Let A = (a, f(a)) and B = (b, f(b)). Write an equation for:

a) the secant line AB

b) the tangent line to in the interval (a, b) that is parallel to AB (Hint: you must find c)

5.

6.

III. Interpretation Problems

7. A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 miles on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?

8. It took 20 seconds for the temperature to rise from 0F to 212F when thermometer was taken from a freezer and placed in boiling water. Explain why at some moment in that interval the mercury was rising at exactly 10.6F/second.

9. Classical accounts tell us that at 170-aor trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the

trireme’s speed exceeded 7.5 knots (sea miles per hour).

10. A marathoner ran the 26.2-mile New York City Marathon in 2.2 hours. Show that at least twice, the marathoner was running at exactly 11 mph.

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