🎉 The Study-to-Win Winning Ticket number has been announced! Go to your Tickets dashboard to see if you won! 🎉View Winning Ticket

University of Southern California

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80

Problem 76

Pattern Recognition Use a graphing utility to compare the graph of

$$f(x)=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x\right)$$

with the given graph. Try to improve the approximation by

adding a term to $f(x) .$ Use a graphing utility to verify that

your new approximation is better than the original. Can you

find other terms to add to make the approximation even better?

What is the pattern? (Hint: Use sine terms.)

Answer

(ANSWER NOT AVAILABLE)

You must be logged in to bookmark a video.

...and 1,000,000 more!

OR

Join 5 million HS and College students

## Discussion

## Video Transcript

all right. Question. Number 76 asks us to use a graphing utility to compare the graph of the following function. F of X equals four over pi times signed pi X plus 1/3 sign three pi X with the graph below, which is just the graph that is Ray here. Um, what we're going to want to do is find a way to improve our approximation formula, This one right here to make it match this graph and the question gives us a hint. And it says, Can you find any other terms that will improve this approximation? So we know that in order to make this function looks like this, we're going tto add some extra terms and there's a pattern. So we're not just going to add them randomly. We're gonna look at the function that they give us and see if there's any pattern we can deduce. Now, when I look right here between the 1st 2 terms were given, we see that the 1st 1 to sign Pi X and the next one is 1/3 sign three pi x. So there's a couple patterns we could think about, um, the first possible pattern could have been that for every new term you divide by three on the outside of her, tricking a metric function and multiply by three on the inside. And that's a reasonable pattern. So the next term in this in that instance, would be won over nine. Sign nine Pi X and so what you would do is you could go online to Dismas Graphing calculator or use your own graphing calculator, and you'd add that new term to this function inside these parentheses. And you do that. Justus. I didn't you'd see that. Well, that doesn't make the approximation any better. It actually makes it worse. So although that is a pattern, it's not the pattern we're looking for. So another pattern you might see is that all right, well, instead of multiplying by three, what if they're just dividing by a number that is two larger than the previous number and so and multiply our adding, adding by that same number on the inside of our children to mention function? So if that were the formula, or that we're the pattern, we see that the third term in this sequence would be 1/5 sign. Five high eggs and, um, I went along with that approximation, and I found or that pattern, and I found that it does a pretty good job of approximating. So over here this is our original function that were given to approximate. And when I add that 1/5 signed five pai ex term. You see that right here at the top form or function where there's all these peaks that a new peak is at and it gets closer to having a straight line. Now, when there's three terms, it looks like this. And then I went along and added two more terms and it gets better and better. And then, for just kicks in giggles, I added three extra terms, and you could see that the more terms we add following this pattern, the closer our approximation gets to becoming a straight line at positive one and negative one. And so, if you were to continue to adm. Or on more terms infinitely, this approximation would get closer and closer to approaching a straight line, and that's how you do it

## Recommended Questions

PATTERN RECOGNITION

(a) Use a graphing utility to graph each function.

$y_1 =\ \dfrac{4}{\pi} (sin\ \pi x\ +\ \dfrac{1}{3} sin\ 3 \pi x)$

$y_2 =\ \dfrac{4}{\pi} (sin\ \pi x\ +\ \dfrac{1}{3} sin\ 3 \pi x\ +\ \dfrac{1}{5} sin\ 5 \pi x)$

(b) Identify the pattern started in part (a) and find a function $y_3$ that continues the pattern one more term. Use a graphing utility to graph $y_3$.

(c) The graphs in parts (a) and (b) approximate the periodic function in the figure. Find a function $y_4$ that is a better approximation.

(a) Use a graphing utility to graph each function. $$y_{1}=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x\right)$$ $$y_{2}=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x+\frac{1}{5} \sin 5 \pi x\right)$$ (b) Identify the pattern in part (a) and find a function $y_{3}$ that continues the pattern one more term. Use the graphing utility to graph $y_{3}$ (c) The graphs in parts (a) and (b) approximate the periodic function in the figure. Find a function $y_{4}$ that is a better approximation.

Use a CAS or graphing calculator.

Numerically estimate $f^{\prime}(\pi)$ for $f(x)=x^{\sin x}$ and verify your answer using a CAS.

Find the linear approximation to $f(x)$ at $x=x_{0}$ Graph the function and its linear approximation.

$$f(x)=\sin x, x_{0}=\pi$$

APPROXIMATION Using calculus, it can be shown that the secant function can be approximated by the polynomial

$sec\ x\ \approx 1\ +\ \dfrac{x^2}{2!} +\ \dfrac{5x^4}{4!}$

where $x$ is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?

The approximation $\sin x \approx x$ It is often useful to know that, when $x$ is measured in radians, sin $x \approx x$ for numerically small values of $x .$ In Section $3.9,$ we will see why the approximation holds. The approximation error is less than 1 in 5000 if $|x|<0.1$ .

a. With your grapher in radian mode, graph $y=\sin x$ and $y=x$ together in a viewing window about the origin. What do you see happening as $x$ nears the origin?

b. With your grapher in degree mode, graph $y=\sin x$ and $y=x$ together about the origin again. How is the picture different from the one obtained with radian mode?

The approximation $\sin x \approx x$ It is often useful to know that,

when $x$ is measured in radians, sin $x \approx x$ for numerically small

values of $x .$ In Section $3.11,$ we will see why the approximation

holds. The approximation error is less than 1 in 5000 if $|x|<0.1$

a. With your grapher in radian mode, graph $y=\sin x$ and

$y=x$ together in a viewing window about the origin. What

do you see happening as $x$ nears the origin?

b. With your grapher in degree mode, graph $y=\sin x$ and

$y=x$ together about the origin again. How is the picture

different from the one obtained with radian mode?

Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials

$sin\ x\ \approx x\ -\ \dfrac{x^3}{3!} +\ \dfrac{x^5}{5!}$ and $cos\ x\ \approx 1\ -\ \dfrac{x^2}{2!} +\ \dfrac{x^4}{4!}$

where $x$ is in radians.

(a) Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare?

(b) Use a graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare?

(c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How did the accuracy of the approximations change when an additional term was added?

(a) Approximate $f$ by a Taylor polynomial with degree $n$ at the number $a$ .

(b) Use Taylor's Formula to estimate the accuracy of the

approximation $f(x) \approx T_{n}(x)$ when $x$ lies in the given interval.

(c) Check your result in part (b) by graphing $\left|R_{n}(x)\right|$

$$f(x)=x \sin x, \quad a=0, \quad n=4, \quad-1 \leqslant x \leqslant 1$$

(a) Approximate $f$ by a Taylor polynomial with degree $n$ at the number $a$ .

(b) Use Taylor's Formula to estimate the accuracy of the

approximation $f(x) \approx T_{n}(x)$ when $x$ lies in the given interval.

(c) Check your result in part (b) by graphing $\left|R_{n}(x)\right|$

$$f(x)=\sin x, \quad a=\pi / 6, \quad n=4, \quad 0 \leqslant x \leqslant \pi / 3$$