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.

Answer

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Video Transcript

this is problem number four Stewart A Tradition Section two point one Problem forces the point P. Terril point five comma zero lies on the curve. Why equals co sign off Pi X and we're gonna be with problem part Ay, if Q is the point X comma cosign a pi X Use your calculator to find the slope of the Secret Line P Q. Correct to Sixth decimal places for the pollen rallies, backs and the values of X are listed there. It is definitely best to make sure that your calculations are correct to do each calculation by hand. Ah, with your calculator eso follow along and confirmed the calculations from this spreadsheet. But I have prepared for calculating the slope. Uh, we recall the slope is given by the formula. M equals to rise over the runner The change in y two out of by the change in X and we have here that p zero point fine zero. And so we calculate the change in X and the change in y using the Given X is here and the calculated wise from cosign a pax. And so here we see the list of all the X values they asked us for and then the Y values air calculated and using that formula, the Delta y is the difference between the way in the P value for the point P and then the white calculated The difference in the eggs is the difference between the P value of X and the value Mac's given. And, of course, the slope is thie ratio of the two. So the answers for party R negative two four point X is our own negative three point ho named oh one seven zero for exes zero point for make it a three point one four one oh seven six, four zero point for name and they did three point one four one five eight seven. Or is there a point for a name for the remaining values? Thie Slope at one is negative, too. A slope at zero point six negative. Three point o nine. A one seven zero. So first they're one bad one, isn't it? Point one four one zero, seven six And let's look for Is there a point? Number one is thinking of three point one four one five eight seven. Now that's closer. The answer for a part a Part B says, using the results of party. Guess what the value of the slope of the tangent line would be at point zero point five zero now. This'LL last column estimates to slope of the secret line, Um, and noticed that for the first four it approaches zero point time fun the lower half. For the next four points, it approaches their point fire from the upper half. And so we're going to use this point at zero point for name, right, a little bit less than zero point five and this point zero point five for one little more than five. And we're going to use to slope here to estimate what the silver the tenderloin is. If you might be aware that the number here it's very, very close to the number pi the number making a pie as the pies three point one four one five nine that I've done so our best guess for part being will be that the slope is negative pyre, and we're going to make that note right here and is negative. Hi, and that's around to report to B for part, See, using the soap from Part B front an equation of the tension ling to the curve and point peen we use again because we have a point and we have a slope. We'LL use the point slope form for the equation of the line and we just input what we know. The slope is negative party, as we found in the previous step and we're using Point P is the reference, So that gives us our X one. Why one Beth. Our solution should be why minus one one school's negative pine. That's a slow X minus X one, which is one half. And that would be the equation of potential ring for us, that is party finally for party sketch the curve two of the secret lines and attention Lea. So we're going back to your spreadsheet and we see that from the right plant is a plot of the coastline pie of X function, which is shown in blue, and we see point peeing is noted, and zero point zero there's two seeking lines that were drawn. Remember that these are the points that we were asked to find the slope for the signaling. The second Kraft here on the left is the one that goes through Q is equal to one zero one, and then cue overhears. A second ticket line is equal to one negative one. So the sick in here and red is the first second has a slope of negative, too. We can see that from here, and we see that for exes. One slip is also needed to, and that's something we can see clearly here for the second, second and finally the Green Line and represents D tangent line to pee noticed that its slope is steeper. Then this red or black line, and we see that steeper means more negative. And the exact slope that we found from part being is that this slope negative Hi, and that completes the responses and results for all of the parts.