The point ( P(5,-3) ) lies on the curve ( y=frac34-x ). (a)...

The point ( P(5,-3) ) lies on the curve ( y=frac{3}{4-x} ). (a) If ( Q ) is the point ( left(x, frac{3}{4-x} ight) ), find the slope of the secant line ( P Q ) (correct to six decimal places) for the following values of ( x ). (i) 4.9 ( m_{P Q}= ) (ii) 4.99 ( m_{P Q}= ) (iii) 4.999 ( m_{P Q}= ) (iv) 4.9999 ( m_{P Q}= ) (v) 5.1 ( m_{P Q}= ) (vi) 5.01 ( m_{P Q}= ) (vii) 5.001 ( m_{P Q}= ) (viii) 5.0001 ( m_{P Q}= ) (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at ( P(5,-3) ). ( m= )

Transcript

00:01 Okay, i'm going to translate that problem for you.

00:03 So here's what's going on.

00:05 You're given a function, this 3 over 4 minus x, and then p is a point that's on the graph of this.

00:15 Because if you plug in x equals 5, you get y equals minus 3.

00:21 Okay, now, q is going to be another point on the graph, and that point q is near to p.

00:29 So they're picking an x value for you that's near to five, and then you're going to find out what y is by plugging it in, and then they want to know what's the slope of the line between this point p, 5 -93, and this new point q that you just found right here.

00:54 Okay, now you don't have to just find one q, you have to find eight different cues, because they want you to get closer closer and closer to p.

01:05 So you really got eight problems to do here, and each problem has two parts, and then you have to answer two more questions after that.

01:15 Okay, so here's what you do.

01:17 You take the x value of q, which is 4 .9, and you plug it in to f, f of 4 .9.

01:25 That's 3 over 4 minus 4 .9.

01:30 So that's 3 over minus 0 .9 .9.

01:34 And so then you get out your calculator and then you divide three divide by point by point oh no oops three divide by point nine okay and you get minus three followed by a bunch of three so let's go ahead and keep like five of them okay so that's the y value of your point and now they want you to find the slope between this point, 5 negative 3, and this point right here.

02:21 Okay, so slope is y minus y over x minus x.

02:35 So on the top you get minus 0 .33333 and on the bottom you get minus point one.

02:45 So that will give you 3 .3333 .3.

02:49 3333.

02:51 Okay, so that'll be your slope.

02:55 3 .3333.

02:59 All right.

03:02 Now you do it again.

03:04 Only this time, q is going to be 4 .9, or the x value of q is going to be 4 .99.

03:12 So first you find f of 4 .99.

03:16 So that gives you 3 over 4 minus 4 .99.

03:25 So 3 over minus .99, 3 divided by .99.

03:36 So this time you get minus 3 .0303.

03:44 And some more 03, so keep however many you want.

03:52 Okay, now find the slope again...

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