Question

Find the slope of the secant line PQ for the following values of x. (Answers here should be correct to at least 6 places after the decimal point.)

Answer 1

The slope of the secant line to the curve is found like any other slope. Since you are finding the secant line for the point PQ, you need to find the x andy coordinates of P [which are given] and Q [which you are to find.

Remember: in general, the slope m of some line containing points (x1, y1) and (x2, y2) is

m = (y2 – y1)/(x2 – x1)

Since P = (2, 1) and Q = (x, v(x – 1)), you can find the slope of the secant line of the curve by finding Q at the given value of x and then using thatpoint to calculate the slope.

It'll become clear once I start doing it. Let's take (a)

x = 1.5

So, I first want to find Q at x = 1.5

Since Q = (x, v(x – 1)), it follows that at x = 1.5, Q = (1.5, v(1.5 – 1)) = (1.5, v0.5).

So, for P(2, 1) and Q(1.5, v0.5), the slope m is

m = (1 – v0.5)/(2 – 1.5) = (1 – v0.5)/(0.5)

Dividing by ½ is the same as multiplying by 2, so

m = (1 – v0.5)/(0.5) = 2(1 – v0.5) = 2 – 2v0.5 = 2 – v(2²)v0.5 = 2 – v(4•0.5)
= 2 – v2

So, the slope of the secant line of PQ for x = 1.5 is 2 – v2 which is approx. 0.585786 to 6 decimal places.

Now for (b) x = 1.9, so

Since Q = (x, v(x – 1)), it follows that at x = 1.9, Q = (1.9, v(1.9 – 1)) = (1.9, v0.9).

So, for P(2, 1) and Q(1.9, v0.9), the slope m is

m = (1 – v0.9)/(2 – 1.9) = (1 – v0.9)/(0.1)

Dividing by (1/10) is the same as multiplying by 10, so

m = (1 – v0.9)/(0.1) = 10(1 – v0.9) = 10 – 10v0.9 = 10 – v(10²)v0.9
= 10 – v(100•0.9) = 10 – v(9•10) = 10 – 3v10

So, the slope of the secant line of PQ for x = 1.9 is 10 – 3v10 which is approx. 0.513167 to 6 decimal places.

Now for (c) x = 1.9, so

Since Q = (x, v(x – 1)), it follows that at x = 1.99, Q = (1.99, v(1.99 – 1)) = (1.9, v0.99).

So, for P(2, 1) and Q(1.99, v0.99), the slope m is

m = (1 – v0.99)/(2 – 1.99) = (1 – v0.99)/(0.01)

Dividing by (1/100) is the same as multiplying by 100, so

m = (1 – v0.99)/(0.01) = 100(1 – v0.99) = 100 – 100v0.99
= 100 – v(100²)v0.99 = 100 – v(10000•0.99) = 100 – v9900 = 100 – v(900•11)
= 100 – 30v11

So, the slope of the secant line of PQ for x = 1.99 is 100 – 30v11 which is approx. 0.501256 to 6 decimal places.

So the answers are (a) 0.585786, (b) 0.513167, and (c) 0.501256.

The only thing you did incorrectly was round to the wrong number of decimal places.

Hope that helps!