The Mean Value Theorem and Taylor’s Theorem

Unit 14.4 The Mean Value Theorem and Taylor’s Theorem

Take a moment to recall the Mean Value Theorem ([cross-reference to target(s) "MVT" missing or not unique]), and how we might use it.

🔗

Example 14.17.

Let \(f(x)\) be the position (mile marker) of a PA Turnpike driver at time \(x\text{.}\) Suppose the driver entered the Turnpike at Mile 75 (New Stanton) at 4pm and exited at Mile 328 (Valley Forge) at 7pm. What does the Mean Value Theorem tell you in this case? The average slope of \(f\) over interval \([4pm,7pm]\) is the difference quotient \((f(7) - f(4))/(7-4) = (325 - 75) / 3 = 83 \frac{1}{3}\text{.}\) Thus there is some time \(c\) between 4pm and 7pm that \(f'(c) = 83 \frac{1}{3}\) MPH, in other words, that this driver was traveling at a speed of \(83 \frac{1}{3}\) MPH.

🔗

Checkpoint 198.

In Example 14.17, the speed in question is above the posted speed limit along the Turnpike, 70 MPH.

🔗

Should this driver receive a speeding ticket? Give a mathematical argument for each side -- don’t rely on mercy, or ``everybody does it", or things like that.

🔗

Example 14.18.

Let \(f(x) := 1/x\) and let \(a < b\) be positive real numbers. What, explicitly in terms of \(a\) and \(b\text{,}\) is the number \(c\) guaranteed by the Mean value theorem?

🔗

Actually, Example 14.18 is a bit beside the point. The Mean Value Theorem only asserts that there is some number \(c\); it says nothing about what \(c\) actually is.

🔗

Why is the Mean Value Theorem in this chapter? We can rewrite (5.2) to read:

\begin{equation} f(b)=f(a)+f'(c)(b-a)\tag{14.1} \end{equation}

which looks an awful lot like the linear approximation to \(f\) at \(x=a\text{:}\)

\begin{equation*} L(x)=f(a)+f'(a)(x-a) \end{equation*}

except we’ve plugged in \(x=b\) and the place we’re evaluating \(f'\) is different.

🔗

It turns out, this same sort of theorem is true for the higher-order polynomial approximations, too:

🔗

Theorem 14.19. Taylor’s Theorem.

Let \(f\) be a function and \(a < b\) be real numbers. Assuming SOME HYPOTHESES, there must be a number \(c \in (a,b)\) where

\begin{equation*} f(b)=f(a)+f'(a)(b-a)+\frac{1}{2}f''(a)(b-a)^2+\cdots+\frac{1}{n!}f^{(n)}(a)(b-a)^n+\frac{1}{(n+1)!}f^{(n+1)}(c)(b-a)^{n+1} \end{equation*}

🔗

Checkpoint 199.

The Mean Value Theorem is just Taylor’s Theorem, if we take \(n=\).

🔗

Answer.

\(0\)

🔗

The last term -- the term involving the mysterious \(c\) -- should be thought of as an \(error\text{;}\) Taylor’s Theorem says that, if we’re willing to accept an amount of error related to the \(n+1\)st derivative, we can pretend that \(f(b)\) is a polynomial function of \(b\text{.}\)

🔗

Checkpoint 200.

You start out, at time \(t=1\text{,}\) at a position we’ll call \(f(1)=3\text{.}\) when \(t=1\text{,}\) your velocity is \(4\text{.}\) You know that, for all times \(t\leq 10\text{,}\) your acceleration is no more than \(8\text{.}\)

🔗

What is the fastest you could be going by \(t=10\text{?}\)

🔗

What is the furthest you could have traveled by \(t=10\text{?}\)

🔗

What is the greatest position you could reach by \(t=10\text{?}\)

🔗

Answer 1.

\(76\)

🔗

Answer 2.

\(360\)