The point P(7, −2) lies on the curve y = 2/(6 − x). (a) If Q is the point (x, 2/(6 − 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) 6.9
mPQ = 1
(ii) 6.99
mPQ = 2
(iii) 6.999
mPQ = 3
(iv) 6.9999
mPQ = 4
(v) 7.1
mPQ = 5
(vi) 7.01
mPQ = 6
(vii) 7.001
mPQ = 7
(viii) 7.000
mPQ = 8
(b) Using the results of part (a), guess the value of the slope m of the tangent line to the curve at
P(7, −2).
m = 9
(c) Using the slope from part (b), find an equation of the tangent line to the curve at
P(7, −2).

a) (i) , (ii) , (iii) , (iv) , (v) , (vi) , (vii) , (viii) ; b) ; c) The equation of the tangent line to curve at P (7, -2) is .

Step-by-step explanation:

a) The slope of the secant line PQ is represented by the following definition of slope:

(i) :

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)  

b) The slope at P (7,-2) can be estimated by using the following average:

The slope of the tangent line to the curve at P(7, -2) is 2.

c) The equation of the tangent line is a first-order polynomial with the following characteristics:

Where:

- Independent variable.

- Depedent variable.

- Slope.

- x-Intercept.

The slope was found in point (b) (m = 2). Besides, the point of tangency (7,-2) is known and value of x-Intercept can be obtained after clearing the respective variable:

The equation of the tangent line to curve at P (7, -2) is .

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step-by-step explanation: