The point

P(7, −4)
lies on the curve
y = 4/(6 − x).
(a) If Q is the point
(x, 4/(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 =

(ii) 6.99
mPQ =

(iii) 6.999
mPQ =

(iv) 6.9999
mPQ =

(v) 7.1
mPQ =

(vi) 7.01
mPQ =

(vii) 7.001
mPQ =

(viii) 7.0001
mPQ =

(b) Using the results of part (a), guess the value of the slope m of the tangent line to the curve at
P(7, −4).

m =

(c) Using the slope from part (b), find an equation of the tangent line to the curve at
P(7, −4).

(a)

(i) mPQ = -0.058824
(ii) mPQ = -0.058823
(iii) mPQ = -0.058823
(iv) mPQ = -0.058823
(v) mPQ = -0.059524
(vi) mPQ = -0.058824
(vii) mPQ = -0.058824
(viii) mPQ = -0.058824

(b) The value of the slope m of the tangent line at P(7, -4) is approximately -0.058824 (based on the results from part (a)).

(c) Using the slope from part (b), the equation of the tangent line to the curve at P(7, -4) is y = -0.058824(x - 7) - 4.

To find the slope of the secant line PQ, we need to calculate the slope between points P(7, -4) and Q(x, 4/(6 - x)) for different values of x.

(a) To find the slope, we will use the formula:

mPQ = (yQ - yP) / (xQ - xP)

(i) For x = 6.9:
Substitute the given values into the formula:
mPQ = (4/(6 - 6.9) - (-4)) / (6.9 - 7)
= (4/(-0.9) + 4) / (-0.1)
= (-4.444444 + 4) / (-0.1)
= (-0.444444) / (-0.1)
≈ 4.444444

(ii) For x = 6.99:
mPQ = (4/(6 - 6.99) - (-4)) / (6.99 - 7)
= (4/(-0.99) + 4) / (-0.01)
= (-4.040404 + 4) / (-0.01)
= (-0.040404) / (-0.01)
≈ 4.040404

(iii) For x = 6.999:
mPQ = (4/(6 - 6.999) - (-4)) / (6.999 - 7)
= (4/(-0.999) + 4) / (-0.001)
= (-4.004004 + 4) / (-0.001)
= (-0.004004) / (-0.001)
≈ 4.004004

(iv) For x = 6.9999:
mPQ = (4/(6 - 6.9999) - (-4)) / (6.9999 - 7)
= (4/(-0.9999) + 4) / (-0.0001)
= (-4.00040004 + 4) / (-0.0001)
= (-0.00040004) / (-0.0001)
≈ 4.00040004

(v) For x = 7.1:
mPQ = (4/(6 - 7.1) - (-4)) / (7.1 - 7)
= (4/(-1.1) + 4) / (0.1)
= (-3.636363 + 4) / (0.1)
= (0.363636) / (0.1)
≈ 3.636363

(vi) For x = 7.01:
mPQ = (4/(6 - 7.01) - (-4)) / (7.01 - 7)
= (4/(-1.01) + 4) / (0.01)
= (-3.960396 + 4) / (0.01)
= (0.039604) / (0.01)
≈ 3.960396

(vii) For x = 7.001:
mPQ = (4/(6 - 7.001) - (-4)) / (7.001 - 7)
= (4/(-1.001) + 4) / (0.001)
= (-3.996004 + 4) / (0.001)
= (0.003996) / (0.001)
≈ 3.996004

(viii) For x = 7.0001:
mPQ = (4/(6 - 7.0001) - (-4)) / (7.0001 - 7)
= (4/(-1.0001) + 4) / (0.0001)
= (-3.99960004 + 4) / (0.0001)
= (0.00039996) / (0.0001)
≈ 3.99960004

(b) By observing the results from part (a), we can see that as x approaches 7, the values of mPQ approach 4. Therefore, we can guess that the value of the slope m of the tangent line to the curve at P(7, -4) is 4.

(c) To find the equation of the tangent line at P(7, -4), we can use the point-slope form of a line. The equation is:

y - y1 = m(x - x1)

Substituting the values of P(7, -4) and m = 4:

y - (-4) = 4(x - 7)
y + 4 = 4x - 28
y = 4x - 32

Therefore, the equation of the tangent line to the curve at P(7, -4) is y = 4x - 32.

well,

(a)(i)
4/(6-6.9) = 4/-.9 = -4.44
mPQ = (-4-(-4.444))/(7-6.9) = .444/.1 = 4.444

similarly for the other points

(b) I expect you will guess m=4
(c) now you have a point and a slope, so the line is

y+4 = 4(x-7)