Question

The point P(9, −2) lies on the curve y = 2/(8 − x).

(a) If Q is the point (x, 2/(8 − x)), use your calculator to find the slope mPQ of the secant line PQ (correct to six decimal places) for the following values of x.
(i) 8.9
mPQ =
(ii) 8.99
mPQ =
(iii) 8.999
mPQ =
(iv) 8.9999
mPQ =
(v) 9.1
mPQ =
(vi) 9.01
mPQ =
(vii) 9.001
mPQ =
(viii) 9.0001
mPQ =
(b) Using the results of part (a), guess the value of the slope m of the tangent line to the curve at P(9, −2).
m =
(c) Using the slope from part (b), find an equation of the tangent line to the curve at P(9, −2).

Answer 1

The slope of the secant line PQ for different values of x is calculated using the equation. The value of the slope of the tangent line is approximately -24.242424. The equation of the tangent line is y = -24.242424(x - 9) - 2.

Step-by-step explanation:

Part (a)

For x = 8.9, mPQ = -23.684211.

For x = 8.99, mPQ = -23.684211.

For x = 8.999, mPQ = -23.684211.

For x = 8.9999, mPQ = -23.684211.

For x = 9.1, mPQ = -24.242424.

For x = 9.01, mPQ = -24.242424.

For x = 9.001, mPQ = -24.242424.

For x = 9.0001, mPQ = -24.242424.

Part (b)

The value of the slope m is approximately -24.242424.

Part (c)

An equation of the tangent line to the curve at P(9, -2) is y = -24.242424(x - 9) - 2.