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Problem 9 Medium Difficulty

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

(a) If $ Q $ is the point $ (x, \sin (10\pi /x)) $, find the slope of the secant line $ PQ $ (correct to four decimal places) for $ x $ = 2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9 .
Do the slopes appear to be approaching a limit?
(b) Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at $ P $.
(c) By choosing appropriate secant lines, estimate the slope of the tangent line at $ P $.

Answer

A. (a) For the curve $y=\sin (10 \pi / x)$ and the point $P(1,0)$
$$\begin{array}{|c|l|c|}
\hline x & Q & m_{P Q} \\
\hline 2 & (2,0) & 0 \\
1.5 & (1.5,0.8660) & 1.7321 \\
1.4 & (1.4,-0.4339) & -1.0847 \\
1.3 & (1.3,-0.8230) & -2.7433 \\
1.2 & (1.2,0.8660) & 4.3301 \\
1.1 & (1.1,-0.2817) & -2.8173 \\
\hline
\end{array}$$
$$\begin{array}{|c|l|c|}
\hline x & Q & m_{P Q} \\
\hline 0.5 & (0.5,0) & 0 \\
0.6 & (0.6,0.8660) & -2.1651 \\
0.7 & (0.7,0.7818) & -2.6061 \\
0.8 & (0.8,1) & -5 \\
0.9 & (0.9,-0.3420) & 3.4202 \\
\hline
\end{array}$$
B. We see that problems with estimation are caused by the frequent oscillations of the graph. The tangent is so steep at $P$ that we need to take $x$ -values much closer to 1 in order to get accurate estimates of its slope.
C.. If we choose $x=1.001,$ then the point $Q$ is (1.001,-0.0314) and $m p Q \approx-31.3794 .$ If $x=0.993,$ then $Q$ is (0.999,0.0314) and $m p Q=-31.4422 .$ The average of these slopes is -31.4108 .So we estimate that the slope of the tangent line at $P$ is about -31.4

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Video Transcript

this is problem number nine Ah Stewart Calculus eighth edition Section two point one. The problem says the point of P one comma zero flies on the curve y equals sign of ten pie over X Party If Q is the point. Ex comer Sign over ten pie over x. Find the slope of the Secret Line p Q. Correct for tick decimal places for X equals two. One point five, one point four, one point three one point two one point one zero point five seven point six Their point seven they're pointing and zero point name. Let's interests this first part of party first and the slip of the Secret Line will be the same as taking the slope formula, which is M equals Delta y over Delta X. And since we have this function for Q, we first used about. He's given to us to find that always Why for that, Q. And then we use the Reference point p in order to find it. The daughter Raina don't X. So we're going to use X equals two. For example, that's the first X If X equals two. This equation here gives us why. So we're putting it into our signer of ten pine, divided by two that gives a zero. In this case, our delta y zero minus zero aren't up. X is two minutes one Therefore our slope zero divided by one. We continue the exact same calculation as we just discussed for all of the points mentioned. And these air the subsequent results for party find the slope of the Sikh in line for each of these X values. So the answers are the slope is zero. Next to the soap is one point seven three two one. When x is one point five. The slope is negative. One point. Oh, hate for seven. Witness X is one point for him. Soap is Nate. Two point seven for three three for next one point dream. The slope is four point three three. Cho won when x is one point two slippers. Native two point eight one seven three when x is one point one slipped zero when x zero point five soap is negative. Two point one six five one a nexus point six Slope is native to point six zero six one. My next two point seven a sloping thing in a fire the next point eight. And then finally, the slope is three point four to go to the nexus zero point nine. The second part of party asks to the soaps appear to be approaching and limit in previous problems. From this, it's Section two point one. We noticed that as X approaches approached a certain number in this case one we would see a trained trend in the slope approaching a certain number. In this case, we don't see an obvious trend. And we also recall that as we approach one from from higher numbers, we also approach one from the lower numbers, and we should agree on a number usually. But in this case, we see again that there slope it is not a hurry. There's no train and going towards one number exp especially the numbers directly surrounding one, as in one point one and zero point nine, we see that their slopes, or not an agreement at all, and so we should respond is that the slopes do not appear to be approaching a limit then, because the he points closest to one are almost completely different partying. Use a graph of the curve to explain why the slopes of the secret lines and party are not close to the actual slope of the tangent line. So to recall, this is a method. As we approach a certain number here we're approaching X equals one. We usually have slope that it is an agreement more or less. And then we can estimate this slope of the tangent line at that point that we specified. But in this case, we have no agreement. So we need to investigate the craft of the curved to see why that is so. If we look here at this graph of a sign of ten by ten point about my ex Wei have planted to seek in lines the secret line at point nine which has a slope over three point four two in red and then the secret line for X equals one point one that has a slope of negative two point eight one seven three shown here as we can see those slopes to not agree and we can see that the reason for that is that point nine point one point one are not close enough. Two point p, which is here in order for them to have a slope that resembles this tension line on green, which is the actual, I suppose, which is the actual slip of the tension line. That one at Point Pete. And so to answer the question, the reason that the slope of the Secret Line they're not close and did not agrees because they're not close enough to water point nine in one point one or not close enough to estimate the slope of the tangent line and Point Pete for part. See, we will choose appropriate seeking rings to estimate the slope of the Tenderloin and P. So point nine is not close enough. One point one is not close enough. Let's share to pick numbers at her very close to one, both less than one and a little more than one. And what we should see is that we should see that the slope of the sticky lines should be very, very close to this large negative slope for this tension that we see on this craft. So we scroll up a little bit of her spirit chief, and we're going to choose a number that's very, very close, but less than one number that's very, very close, a little more than one. Great. It's on the numbers. We see that the soaps are in agreement and that the slope of this green tension line is I was probably an average of the two. That would be our best estimate. So and we have released two. We should get a soap of a pound negative. Thirty one point four one five nine. So that will be our response. The slope that we estimate is negative. Thirty one point or one. All right? No. And we down there. By choosing them, he can live very close to one.