4.2 Mean Value Theorem

ObjectiveSWBAT apply the Mean Value Theorem and

find the intervals on which a function is increasing or decreasing

• The Mean Value Theorem is one of the most important theorems in all of Calculus, so focus!!!

-The function has to be 1) continuous on a closed interval, 2) differentiable over the entire open interval, and if those conditions hold true, then the Mean Value Theorem guarantees us a number in the middle of the interval somewhere (between the endpoints a and b) such that the derivative at c is equal to the slope of the line between the endpoints a and b (secant line).

Somewhere in the interval there is a point whose derivative is exactly the same as the slope of the secant line (meaning there is a tangent line parallel to the secant line).

• Another way of looking at the MVT is that the average rate of change over the entire interval is equal to the instantaneous rate of change at some point on the interval

• Other things to note:– The MVT is an existence theorem: • it does not tell you how to find c• it does not tell you the value of c• It does not tell you how many c’s exist

– c is an x-value

The Mean Value Theorem states that the value of c is guaranteed in the open interval from (a, b). Therefore, since 1 is an endpoint, it is not in that interval. So the only value of c that is guaranteed by the MVT is -1/3.

Therefore, the function is not differentiable at -1 or 1. However, it doesn’t need to be! The function only needs to be differentiable between the endpoints, and it is here. Therefore the MVT applies.

Set the derivative to 0 to find the value(s) of the ordered pair(s) that contain tangent line(s) through them that are parallel to the secant line.

So the tangent line we are looking for has a slope of 0 and goes through the ordered pair .

• For this to occur, the instantaneous velocity (the derivative of the position function) would need to equal the average velocity.

• From the previous slide, we know the average velocity is -17.84657 m/s. So this is what the derivative would need to equal.

The object’s velocity equals its average velocity at 1.82108 seconds.

• One of the applications of the MVT is that it tells us when a function is increasing (the graph is going up) or decreasing (the graph is going down). [note: include endpoints when referencing locations for the function decreasing and increasing]

Example 4: What interval is the function decreasing? Increasing? Constant?

Example 5: What is the value of the derivative when the function is decreasing? Increasing? Constant?

Antiderivatives• As you may have guessed using your powers of

intuition, an antiderivative is the opposite of a derivative.

• Finding the function from the derivative is a process called antidifferentiation, or finding the antiderivative.