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Problem 3 Medium Difficulty

The point $ P(2, -1) $ lies on the curve $ y = 1/(1-x) $.

(a) If $ Q $ is the point $ (x, 1/(1-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) $ 1.5 $ (ii) $ 1.9 $ (iii) $ 1.99 $ (iv) $ 1.999 $
(v) $ 2.5 $ (vi) $ 2.1 $ (vii) $ 2.01 $ (viii) $ 2.001 $

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $ P(2, -1) $.

(c) Using the slope from part (b), find an equation of the tangent line to the curve at $ P(2, -1) $.

Answer

A.(a) $y=\frac{1}{1-x^{2}}, P(2,-1)$
$$\begin{array}{|r|l|l|l|}
\hline & x & Q(x, 1 /(1-x)) & m_{P Q} \\
\hline(i) & 1.5 & (1.5,-2) & 2 \\
(i i) & 1.9 & (1.9,-1.111111) & 1.111111 \\
(\text { ii }) & 1.99 & (1.99,-1.010101) & 1.010101 \\
(i v) & 1.999 & (1.999,-1.001001) & 1.001001 \\
(v) & 2.5 & (2.5,-0.666667) & 0.666667 \\
(v i) & 2.1 & (2.1,-0.909091) & 0.509091 \\
(\text { vin }) & 2.01 & (2.01,-0.900099) & 0.990099 \\
( \text { viii) } & 2.001 & (2.001,-0.999001) & 0.999001 \\
\hline
\end{array}$$
B.1
C.$y=x-3$

Discussion

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EV

Eric V.

September 22, 2020

What is slope intercept form?

EG

Erica G.

September 22, 2020

The first of the forms for a linear equation is slope-intercept form. Equations in slope-intercept form look like this: y = mx + b. where m is the slope of the line and b is the y-intercept of the line, or the y-coordinate of the point at which the line c

DF

Doug F.

September 22, 2020

What is a tangent?

JM

Julia M.

September 22, 2020

it's an honor to answer this Doug! The tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the c

BS

Brad S.

September 22, 2020

what is a slope?

SH

Sarah H.

September 22, 2020

Hello Brad! In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinc

EA

Etsubdink A.

October 21, 2020

Video Transcript

Mrs Problem number three of Stuart Ate Edition section two point one. Problem says the point, Pete. Two. Negative one lies on the curve Y equals one, divided by the quantity, one minus x. Ah, and we're gonna go through this problem part by part. First part party if Q. Is the point. Ex comer one divided by the quantity. One minus X. Use your calculator to find the slope of the Secret Line. P. Q. Correct to sixty seven places for the calling Values of X and the values of X are provided below, so it's gonna take quite a bit of a calculation. But here we have a spreadsheet ready, Um, that does appropriate calculations, and we refer again to the slope formula, Mme. Where we take the difference in the Y values directed by the difference in the excise. So let's take the first point. For an example again, the reference points point p right. We're taking I second line between all of the Seacon lines between Point p M point. Q. There are eight, two different point cues, but there's only one point peen. So what we do is we take the difference between the Y values of point peen shown here. That's why I value you get one and then Q. It's white value is provided by this function, one divided by the quantity one minus X and taking that difference. We see that the difference in wine or the rise for this function is the one that X equals one point five. And then we take the difference in the x Point p the Valium access to Point Q. At one point, time to let us one point five different. There's zero point five, and then the slope is the difference. I'm sorry. The slope is thie ratio between the rise and the run. Delta y, directed by Delta X, gives us two as the sloper for this first part of party, and we just repeat with each subsequent new value of X for Q. Each of these new values using this formula gives us new values of why for Q and we repeat the same calculations bearing up the difference in the wine, the difference in X and then this Come here is the result of the slope of the secret lines for every combination of point p and break you and so The answer for Number one of Party is that the soap is too. At one point in the service, one point one repeating and one point nine name the soap is one point zero one, repeating at one point nine nine nine. The soap is one point zero there, one repeating and so on. At two point five of the slope is zero point six or beating and two point one the soap is zero point nine zero nine zero. Repeating the sober two point one is point nine nine zero zero repeating and a two point zero zero one slip its point nine nine zero zero one. To those of the solutions for part part B, it has to use his results to get the value of the super potential into the curve at point people. Important part here is to know where Point P s when Pia is at X equals two and the specific points that we did for part of air, or to show what the trend is of the slope as you go towards too. So is he has X increases from one point five towards, too. We see that the slope decreases until it is about one. Likewise. Here we started a high value two point five. We decrease from two point five two point one two, two point Oh one. Getting closer to the value of two. And we see that the soaps get closer to one. So between this line and this line, which had the eyes close, is two two. We see that their slopes are very close to one. Our best guess is to say that the slope at P, the slope at X equals two is one. So we're going to write that down. Our best guess here is that the slope for party is the one and party finally using the slope from party. Slope is one. Find an equation with potential into the curve at point P two. Negative one. So what we're gonna use here, we're going to use Thie slope intercept form. I'm sorry when you use the point slope form because we have a point and we have a soap and that looks like this. Why, man, it's one one equals m times explaining six one where why one thinks one represent the X and y values of the point that we have. So we play gin. Why? Minus why one is negative. One m is a slope, which we guessed as one X one is two and then friendly. We simplify our equation and our solution for part. See, it's why. Plus one he cools X minus two, and this is thie equation of the tension line to the curve at that point P.