Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

If a ball is thrown into the air with a velocity …

03:10

Question

Answered step-by-step

Problem 4 Medium Difficulty

The point $ P(0.5, 0) $ lies on the curve $ y = \cos \pi x $.

(a) If $ Q $ is the point $ (x, \cos \pi x) $, use your calculator to find the slope of the secant line $ PQ $ (correct to six decimal places) for the following values of $ x $:

(i) $ 0 $ (ii) $ 0.4 $ (iii) $ 0.49 $
(iv) $ 0.499 $ (v) $ 1 $ (vi) $ 0.6 $
(vii) $ 0. 51 $ (viii) $ 0.501 $

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $ P(0.5, 0) $.

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $ P(0.5, 0) $.

(d) Sketch the curve, two of the secant lines, and the tangent line.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Mary Wakumoto
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Mary Wakumoto

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

07:22

Daniel Jaimes

01:13

Carson Merrill

06:31

AD

Anupa Desai

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 1

The Tangent and Velocity Problems

Related Topics

Limits

Derivatives

Discussion

You must be signed in to discuss.
CR

Cam R.

September 22, 2020

I see how that could be confusing. In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

HC

Howie C.

September 22, 2020

Anyone else confused by the slope, can someone explain?

AG

Alex G.

September 22, 2020

I know this one! The length of the adjacent side divided by the length of the hypotenuse. The abbreviation is cos. cos(?) = adjacent / hypotenuse. Well Done! Another.

LP

Lindsey P.

September 22, 2020

I'm confused by the term cosign?

ST

Samantha T.

September 22, 2020

ST

Samantha T.

September 22, 2020

Hey Nadia, A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for use as the separator. The choice of symbol also affec

NH

Nadia H.

September 22, 2020

what is decimal places?

Top Calculus 1 / AB Educators
Heather Zimmers

Oregon State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Join Course
Recommended Videos

0:00

The point $ P(0.5, 0) $ li…

0:00

The point $P(0.5,0)$ lies …

01:04

The point $P(0.5,0)$ lies …

04:51

The point $P(0.5,0)$ lies …

0:00

The point $ P(1, 0) $ lies…

01:05

The point $P\left(1, \frac…

15:13

The point $ P(2, -1) $ lie…

01:10

The point $\mathrm{P}\left…

0:00

The point $P(2,-1)$ lies o…

0:00

The point $P(1,0)$ lies on…

01:25

The point $P(1,0)$ lies on…

01:08

The point $\mathrm{P}(3,1)…

01:21

The point $P(1,0)$ lies on…

Watch More Solved Questions in Chapter 2

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9

Video Transcript

Alright so here we have Y. Equals cosine of pi acts uh the cosine graph is the one in blue. I just looked very close to the origin because we're gonna look at point P. Which is X. Value 0.0.5 Y. Value zero. And we'd like to find the tangent line at that point and that's the curve and red. And um but to find a tangent line um we're used to finding while in general to find slope of a line we do rise over run and so what we can do is do rise over run over using that point P. But another point Q. Which is um generic X, coordinate X. And Y coordinate. Co sign up high X. And by drawing a straight line between the two. This one I can probably fix a little bit. If we want this one to be a cube then we have secret lines. So for example if we have a point 01. Then in green we get the secret line and we can find the slope of that line. If we're generically a little bit over we have the brown line. So what happens is that we are able to um use a secret line slow and you can see the green lines, very different slope than the red line. The tangent line. But you can see the brown line is a tangent line. Or the secret line slope would be a little bit closer to the actual value. So we can see if as we bring point Q closer and closer to pee then we'll get a better approximation to the actual slope of the tangent line. So what we're gonna do is basically do that we're going to get closer and closer. We're going to move pink you closer and closer to p. We're gonna move closer and closer from the left side. You can see this is from the left side. Um And okay. There we go. Let's redo that one that would do it came out very well. Oh so from the left side you can see we're getting closer and closer to um point P. For the X. Value. And we're also gonna approach from the right side and we're going to approach um Also .5 but we're gonna approach from the right side. So what we're going to basically do is we're going to find secret line slopes and the secret line slopes just rise over. Run. The difference in Y values between the P and Q. Points and the difference in X values rise over run. So since our PP point has wide value zero um that we're gonna get zero minus cosine of pi X Over .5 -1. So that is our rise over run for any value of acts and all that we need to do then is plug into our calculator. Um All these different values of Acts and well then be able to see what's happening with our function and be able to approximate our tangent line. So let me go ahead and um plug into the calculator offline. And uh well then we'll get our values. Okay, so I went ahead and use my calculator and when you do it, make sure you are in radiant mode. And I went ahead. And for each of the values of X that you see on in the chart, I went ahead and plugged in to get the slope of the secret line to the rise of a run formula. And you can see the values here, you can see that we start off at -2. We quickly get to three and we can see we're really honing in on um Something that looks like -3.14159. Um well we also that's approaching from the left side. When we approach from the right side, getting closer to .5 for X, we get symmetric values and we also get something very close to pi so this looks very much like pie. So I would say that we're looking at the slope of the tangent line, A slope of tangent line uh at X equals .5. It's looking like that slope is going to be equal to pi. So let me actually rewrite this. So I'm gonna say the slope of the tangent line equals pi At x equals 2.5. Okay, so all we have left now is to go ahead and create our tangent line in general, we can write a line as y minus y. One equal slope 20 x minus x one. That's the point slope formula. So for us our point is a wide value of zero, our slope is pi And our x value is .5. So we can rearrange this as Y equals pi x minus half a pie. So I'll write it as pi over two. So that is the equation of our red tanja light. Well, hopefully that helped to have a wonderful day.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
143
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Limits

Derivatives

Top Calculus 1 / AB Educators
Heather Zimmers

Oregon State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Samuel Hannah

University of Nottingham

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Join Course
Recommended Videos

0:00

The point $ P(0.5, 0) $ lies on the curve $ y = \cos \pi x $. (a) If $ Q $ i…

0:00

The point $P(0.5,0)$ lies on the curve $y-\cos \pi x$. (a) If $Q$ is the point …

01:04

The point $P(0.5,0)$ lies on the curve $y=\cos \pi x.$ (a) If $Q$ is the point…

04:51

The point $P(0.5,0)$ lies on the curve $y=\cos \pi x .$ (a) If $Q$ is the poin…

0:00

The point $ P(1, 0) $ lies on the curve $ y = \sin (10\pi /x) $. (a) If $ Q …

01:05

The point $P\left(1, \frac{1}{2}\right)$ lies on the curve $y=x /(1+x).$ (a) I…

15:13

The point $ P(2, -1) $ lies on the curve $ y = 1/(1-x) $. (a) If $ Q $ is t…

01:10

The point $\mathrm{P}\left(1, \frac{1}{2}\right)$ lies on the curve $y=x /(1+x)…

0:00

The point $P(2,-1)$ lies on the curve $y=1 /(1-x)$. (a) If $Q$ is the point $(x…

0:00

The point $P(1,0)$ lies on the curve $y=\sin (10 \pi / x)$. (a) If $Q$ is the p…

01:25

The point $P(1,0)$ lies on the curve $y=\sin (10 \pi / x).$ (a) If $Q$ is the …

01:08

The point $\mathrm{P}(3,1)$ lies on the curve $y=\sqrt{x-2}$. (a) If $\mathrm{…

01:21

The point $P(1,0)$ lies on the curve $y=\sin (10 \pi / x)$. $$\begin{array}{l}…
Additional Mathematics Questions

01:30

An art class is making a mural for their school which has a triangle drawn i…

01:59

In Marissa's calculus course, attendance counts for 5% of the grade, qu…

01:00

An article bought for $125 was sold %175. The percentage profit

03:04

Which property of addition is shown in the equation below?
a+bi+0+0i = a …

01:07

Peter's father told Peter to buy four-tenths of a pound of chili powder…

01:14

The force F (in pounds) needed on a wrench handle to loosen a certain bolt v…

00:42

A girl walked 6km from her house to a market and discovered that she had cov…

04:18

If three sandwiches and two bags of chips cost
$22.00, and two sandwiches…

01:59

Use the equation of the water level of the river represented by the equation…

00:58

Identify the independent variable in the following scenario. A nutritionist …

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started