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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.

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07:22

Daniel Jaimes

01:13

Carson Merrill

06:31

Anupa Desai

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 1

The Tangent and Velocity Problems

Limits

Derivatives

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.

Howie C.

Anyone else confused by the slope, can someone explain?

Alex G.

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.

Lindsey P.

I'm confused by the term cosign?

Samantha T.

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

Nadia H.

what is decimal places?

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

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.

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.

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The point $ P(0.5, 0) $ li…

The point $P(0.5,0)$ lies …

01:04

04:51

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…

The point $P(2,-1)$ lies o…

The point $P(1,0)$ lies on…

01:25

01:08

The point $\mathrm{P}(3,1)…

01:21

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.

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