12.3.2What about non-alternating series?

Lagrange Error Bound

12-99.

Examine the graphs below or use the 12-99 Student eTool (Desmos). All three curves have a tangent line of $y =\frac { 1 } { 2 }x+\frac { 1 } { 2 }$ at $(1, 1)$ . Click in the lower right corner of the graph to view it in full-screen mode.

Your teacher will provide you with a model.

At $x = 1$ , how are the curves similar?
At $x = 1$ , how are the curves different?
Even though the three curves have the same tangent line at $x = 1$ , which curve will have the best tangent line approximation at $x = 0.6$ . Explain your answer.
Recall that a tangent line can be considered a first-degree Taylor polynomial, $p_1(x) = f(a) + f^\prime(a)(x – a)$ for $f(x)$ centered at $x=a$ . Based on the three curves above, write a conjecture about how to use $p(x)$ , the Taylor series for $f(x)$ , to determine if $p_1(0.6)$ is a good approximation of $f(0.6)$ . Be prepared to share your conjecture with the class.

12-100.

In order to quantify whether a Taylor polynomial is a good or bad approximation at $x = b$ , scientists and engineers like to know the error of the approximation, $| f ( b ) - p _ { n } ( b ) |$ . Unfortunately, the error cannot be computed without knowing the actual value of the function. So instead of computing the error, we bound the error; that is, we find a range of values in which the error must lie.

In Lesson 12.3.1, you discovered how to bound the error of a Taylor polynomial whose corresponding Taylor series has terms that alternate and decrease. But, if the Taylor polynomial does not have alternating terms, bounding the error is more challenging.

Let’s examine a simple case, a first-degree Taylor polynomial, $p_1(x)$ , centered at $x = 0$ , which is used to approximate $f(x)$ at $x = a$ , where a is located near the center.

Write the general equation for $p_1(x)$ .
As shown in problem 12-99, the accuracy of a tangent line approximation depends on how curvy the graph of $y = f(x)$ is near the point of tangency, and curviness depends on the value of $f^{\prime\prime} (x)$ .

So, before we start, we have to make a decision about the minimum, $L$ , and the maximum, $M$ , value that $f^{\prime\prime} (x)$ can possibly have on the interval $[c, a]$ where $c$ represents the center (in this case, $c = 0$ ) and a represents the $x$ -value where $f(a)$ is being approximated. When these values are not known exactly, exaggerated estimates should be used for $L$ and $M$ . However, in this generic case, we will leave $L$ and $M$ in our expression as constants: $L≤f^{\prime\prime} (x)≤ M$ on the interval $[c, a]$ .

Let’s focus on the right side of the inequality for now. That is, $f^{\prime\prime} (x)≤ M$ . So, $\int _ { 0 } ^ { x } f ^ { \prime \prime } ( t ) d t \leq \int _ { 0 } ^ { x } M d t$ , for $x$ near the center of the Maclaurin, $x = 0$ .

Use the Fundamental Theorem of Calculus to evaluate the integrals and solve for $f ^\prime(x)$ .
Integrate again and solve for $f(x)$ .
Repeat the process from parts (b) and (c) to get a lower bound for $f(x)$ . Begin with $\int _ { 0 } ^ { x } L d x \leq \int _ { 0 } ^ { x } f ^ { \prime \prime } ( t ) d t$ .
Write the inequality showing $f(x)$ between its lower and upper bounds. Notice that your expression for $p_1(x)$ from part (a) appears on both sides of the inequality. Simplify each side of the inequality by substituting $p_1(x)$ .
The error of $p_1(x)$ is the difference $| f ( x ) - p _ { 1 } ( x ) |$ . Subtract $p_1(x)$ from each part of the inequality of part (e). You have just bounded the error of a first-degree Taylor polynomial! Note: Numeric values for $L$ and $M$ depend on $f^{\prime\prime }$ , the second derivative of $f$ , on the interval $[c, a]$ .
The error of a second-degree Taylor polynomial, $p_2(x)$ , depends on the size of $f^{\prime\prime} (x)$ . Go through the process above (you will need to integrate three times) to solve for the error of $p_2(x)$ . Assume there exist $L$ and $M$ such that $L≤f^{\prime\prime} (x)≤ M$ .
Generalize your results, and bound the error of $p_n(x)$ between two expressions involving $L$ and $M$ , the upper and lower bounds of $f^{ (n+1)}(x)$ for $x$ near $0$ . This method of bounding the error of a Taylor polynomial is known as the Lagrange Error Bound.

12-101.

Let $f(x) = e^x$ . Let’s calculate the error bound of $p_3(0.6)$ , the third-degree Taylor polynomial about $x = 0$ , evaluated at $x = 0.6$ .

Write an equation for $p_3(x)$ and use it to approximate $f(0.6)$ .
In order to choose appropriate values of $L$ and $M$ , we need to consider the range of possible values of $f ^{(4)}(x)$ , the fourth derivative of $e^{x}$ , on the interval $[0, 0.6]$ . Since it is known that $f ^{(4)}(x) = e^x$ and that $f(x) = e^x$ is an increasing function, it can be concluded that the maximum value of $f ^{(4)}(x)$ will happen at the right endpoint, $x = 0.6$ (not $x = 0$ or somewhere else within the interval). This maximum value of $f ^{(4)}(x)$ can be your choice for M.

However, if we were able to evaluate $e^{0.6}$ without a calculator, then we would not be using a Taylor approximation to begin with! So extend the interval to a value greater than $x = 0.6$ that is easy to evaluate. Then determine the largest value that $f ^{(4)}(x)$ can possibly be. This will be your $M$ . Use that $M$ to determine the upper bound of the error.
Use your calculator to determine the true error and compare it with your answer from part (b).

12-102.

Consider $f(x) = \cos(x)$ and its Maclaurin series, $p(x)$ .

Write an equation for $p_4(x)$ , the fourth-degree Taylor polynomial about $x = 0$ ,and use it to approximate $\cos(1)$ .
Consider the derivatives of $\cos(x)$ . What is the next non-zero term in the Taylor series for $\cos(x)$ ? Which derivative appears in that term? What is the largest value that this derivative can possibly be? This is your $M$ .
Use the error formula to calculate an upper bound for the error.
Use your calculator to find the actual value of $\cos(1)$ . What is the actual error?
How does this error bound compare to the actual error?

12-103.

Consider the function $f(x) = \sin(3x)$ . Calculate the Lagrange error bound of a third-degree Taylor polynomial that is used to approximate $f(0.1)$ .

Math Notes

The Lagrange Error Bound (for Non-Alternating Taylor Polynomials)

Suppose $p_n(x)$ is an $n$ ^th-degree Taylor polynomial centered at $x = c$ that is used to approximate $f(x)$ at some value $x = b$ within its interval of convergence. $M$ is a value that is chosen because it is equal to or greater than the maximum value between $a$ and $b$ for the next non-zero derivative of the Taylor series that is associated with $p_n(x)$ .

Then the maximum error of $p_n(b)$ is:

$| f ( b ) - p _ { n } ( b ) |$

$≤$

$\frac { M } { ( n + 1 ) ! } | c - b | ^ { ( n + 1 ) }$

error of $p_n(b)$

overestimate of the next non-zero term of $p(x)$

This is known as the Lagrange Error Bound.

Review and Preview problems below

12-104.

Write the first two non-zero terms of the Maclaurin series for each function. Homework Help ✎

$f(x) = \tan^{–1}(x)$

$g(x) = \tan^{–1}(x^2)$

12-105.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int \frac { 1 } { ( x + 1 ) ^ { 2 } + 1 } d x$

$\int \frac { \operatorname { cos } ( x ) } { \operatorname { sin } ^ { 2 } ( x ) + 1 } d x$

$\int \frac { \operatorname { sin } ( 2 x ) } { \operatorname { sin } ^ { 2 } ( x ) + 1 } d x$

$\int \operatorname { ln } ( x ^ { 2 } ) d x$

12-106.

Consider the infinite series below. For each series, decide if it converges conditionally, converges absolutely, or diverges and justify your conclusion. State the tests you used. Homework Help ✎

$\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 1 + \operatorname { cos } ( n ) } { n ^ { 2 } }$

$\displaystyle\sum _ { j = 1 } ^ { \infty } \frac { j ! } { ( 2 j + 1 ) ! }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } n e ^ { - n ^ { 2 } }$

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { 2 - k } { k \cdot 2 ^ { k } }$

12-107.

A rowboat is tethered from its bow to a dock by two ropes, each $20$ feet long. The ropes are tied to the dock at points $20$ feet apart.

The boat’s owner unties one of the ropes from the dock, and brings the boat toward the dock by walking along it away from the other tether point at a rate of $4$ ft/sec. How fast is the boat moving toward the dock when the boat’s owner has walked $10$ feet? Homework Help ✎

Triangle, bottom side is labeled dock, top vertex is tip of a boat, left & right sides, each labeled 20 foot rope.

12-108.

Multiple Choice: Which of the following integrals represents the arc length of the curve $y = \ln(\cos(x))$ from $x = 0$ to $x =\frac { \pi } { 3 }$ ? Homework Help ✎

$\int _ { 0 } ^ { \pi / 3 } \sqrt { 1 + \operatorname { sec } ^ { 2 } ( x ) } d x$

$\int _ { 0 } ^ { \pi / 3 } \sqrt { x ^ { 2 } + ( \operatorname { ln } ( \operatorname { cos } ( x ) ) ) ^ { 2 } } d x$

$\int _ { 0 } ^ { \pi / 3 } \operatorname { sec } ( x ) d x$

$\int _ { 0 } ^ { \pi / 3 } \sqrt { 1 + \operatorname { tan } ( x ) } d x$

$\int _ { 0 } ^ { \operatorname { ln } ( 0.5 ) } \sqrt { 1 + \operatorname { tan } ^ { 2 } ( x ) } d x$

12-109.

Multiple Choice: Which of the following values is the slope of the line tangent to the polar curve $r ( \theta ) = \frac { 1 } { 1 + \operatorname { sin } ( \theta ) }$ at the point $( \frac { \pi } { 6 } , \frac { 2 } { 3 } )$ ? Homework Help ✎

$- \frac { \sqrt { 3 } } { 3 }$

$- \frac { 2 \sqrt { 3 } } { 9 }$

$-\frac { \pi } { 3 }$

$\frac { 2 \sqrt { 3 } } { 3 }$

$\frac { \sqrt { 3 } } { 3 }$

12-110.

Multiple Choice: A particle moves in the plane according to the set of parametric equations $x(t) = \cos(π t)$ and $y ( t ) = \frac { t ^ { 2 } } { t + 1 }$ . What is the magnitude of the velocity vector at $t =\frac { 1 } { 2 }$ ? Homework Help ✎

$\frac { 1 } { 6 }$

$π +\frac { 5 } { 9 }$

$\frac { \sqrt { 106 } } { 9 }$

$\frac { \sqrt { 81 \pi ^ { 2 } + 25 } } { 9 }$

Calculus Third Edition

The Lagrange Error Bound (for Non-Alternating Taylor Polynomials)