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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07:22

Daniel J.

01:13

Carson M.

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Video Transcript

And this question we're going to approximate the slope of the tangent line at point B. By approaching point P. From another point Q. And then calculating the slip of second line peak you And getting the X. here smaller and smaller and closer to .5. That we will be able to approximate the tangent line. The slope of the tangent line at P. So for that our first sub question We have X equals zero. And that makes her cute zero comma co sign of by comma zero. uh by time zero which is zero times cosine of zero which is one. And that gives us a slope of one minus 0/0 minus 10.5. Using the slope formula here. And that would be negative too. For our second sub question we have X equals 0.4 and plugging that into Q. Here we have q equaling 0.4 comma co sign of pi times 0.4. Which when we plug into the calculator We get zero three 09. Using the slope formula here again, Her slope would be zero 309 0 Over .4 -15 Which is negative 3.09. For a third question, X is equal to 0.49. We're getting closer. Rx is getting closer and closer to .5. Uh marquis was going to be .49 comma co sign of thai times 49 .49 which is using the calculator 3.0314. And that gives us a slow both negative three point one for using the steps. The Question one and 2 For our 4th question Our X is equal to zero four 99. Whoops. Yeah, so we get a cube of 0.499 comma co sign of play time. 0.499 which is 3.14 times 10. Raised to -3. And that gives us a slope of negative 3.14. Again for question five Our X is equal to one that gives us a queue of one comma co sign of pie which is negative one. And that's a slope of negative 1 0 Over 1 -15 which is negative two for question six. Our X is equal to .6. That gives us a queue of 0.6 comma co sign of pi times 0.6 which is negative zero point There is 09. Mhm. And so our slope is going to be negative zero 30,990 Divided by .6 -15 Which is negative 3.09 for our seventh question X is equal to 0.51. That gives us a queue of 0.51 comma co sign of High time, 0.51 which is 0.0 314, Point 0314. And that gives us a slow bus negative 3.14. Notice how the slope values that we were getting after Question five for question 5 6 and seven are mirroring the Slips slope values for question 1, 2, three and four For question eight similarly Um X is equal to 0.501. And so our Q. Is going to be is your point 501 comma coastline of pi times 0.5 0 1. Which is going to be -3.14 Times 10 raised -3. That gives us a slope of negative 3.14. So notice how as we're getting closer and closer two and X equals 2.5 or slope is approaching negative 3.14 or Pie. So negative pi would be the slope of are tangent line act .5. Just be for the equation of our change of mind. The formula bertha blind prefer is equal to mm times x minus x one Plugging the coordinates of p. and the value of flow of tangent line into our equation we have Y -0 0 is equal to negative pi Times X -100. Yeah. Which is why is equal to negative pi X plus bye. Yes. Uh huh.