The point P(8, −3) lies on the curve
y = 3/(7 − x).
(a) If Q is the point x, 3/(7 − 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) 7.9
mPQ =
(ii) 7.99
mPQ =
(iii) 7.999
mPQ =
(iv) 7.9999
mPQ =
(v) 8.1
mPQ =
(vi) 8.01
mPQ =
(vii) 8.001
mPQ =
(viii) 8.0001
mPQ =
(b) Using the results of part (a), guess the value of the slope m of the tangent line to the curve at
P(8, −3).
m =
(c) Using the slope from part (b), find an equation of the tangent line to the curve at P(8, −3).
To find the slope of the secant line PQ for different values of x, we can substitute the x-values into the equation of the curve and calculate the corresponding y-values. Then we can use the formula for slope:
mPQ = (yQ - yP) / (xQ - xP)
where P(8, -3) and Q(x, 3/(7-x)).
(a) Calculating the slope of the secant line PQ for different values of x:
(i) x = 7.9
Substituting x = 7.9 into the equation of the curve:
yQ = 3/(7 - 7.9) = -29.999999
Using the formula for slope:
mPQ = (-29.999999 - (-3)) / (7.9 - 8) = -27.888889
(ii) x = 7.99
Substituting x = 7.99 into the equation of the curve:
yQ = 3/(7 - 7.99) = -299.999
Using the formula for slope:
mPQ = (-299.999 - (-3)) / (7.99 - 8) = -290.988
(iii) x = 7.999
Substituting x = 7.999 into the equation of the curve:
yQ = 3/(7 - 7.999) = -2999.99
Using the formula for slope:
mPQ = (-2999.99 - (-3)) / (7.999 - 8) = -2989.99
(iv) x = 7.9999
Substituting x = 7.9999 into the equation of the curve:
yQ = 3/(7 - 7.9999) = -29999.9
Using the formula for slope:
mPQ = (-29999.9 - (-3)) / (7.9999 - 8) = -29889.9
(v) x = 8.1
Substituting x = 8.1 into the equation of the curve:
yQ = 3/(7 - 8.1) = -30
Using the formula for slope:
mPQ = (-30 - (-3)) / (8.1 - 8) = -27
(vi) x = 8.01
Substituting x = 8.01 into the equation of the curve:
yQ = 3/(7 - 8.01) = -300
Using the formula for slope:
mPQ = (-300 - (-3)) / (8.01 - 8) = -297
(vii) x = 8.001
Substituting x = 8.001 into the equation of the curve:
yQ = 3/(7 - 8.001) = -3000
Using the formula for slope:
mPQ = (-3000 - (-3)) / (8.001 - 8) = -2970
(viii) x = 8.0001
Substituting x = 8.0001 into the equation of the curve:
yQ = 3/(7 - 8.0001) = -30000
Using the formula for slope:
mPQ = (-30000 - (-3)) / (8.0001 - 8) = -29700
(b) From the values obtained in part (a), we can see that as x approaches 8, the slope of the secant line PQ approaches -30000. Therefore, we can guess the value of the slope m of the tangent line to the curve at P(8, -3) is approximately -30000.
(m = -30000)
(c) To find the equation of the tangent line to the curve at P(8, -3), we can use the point-slope form of a line:
y - yP = m(x - xP), where P(8, -3) and m = -30000.
Substituting the values:
y - (-3) = -30000(x - 8)
Simplifying:
y + 3 = -30000x + 240000
Rearranging to slope-intercept form:
y = -30000x + 239997