THE POINT (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 OF THE SECANT LINE PQ CORRECT TO SIX DECIMAL PLACES FOR THE VALUE OF X.
(i)7.9 Mpq?
(ii)7.99 mPQ?
(iii)7.999Mpq?
(iV)7.9999 MPQ?
P = (8,-3)
The slope of PQ is
∆y/∆x = (3/(7-x)-(-3))/(x-8)
= (3+3(7-x))/((7-x)(x-8))
= (24-3x)/((7-x)(x-8))
You have provided no data about Q or the desired slope, but just plug in the numbers you have.
To find the slope of the secant line PQ, we need to use the formula for slope, which is given by:
slope = (y2 - y1) / (x2 - x1)
Given that P is the point (8, -3) and Q is the point (x, 3 / (7 - x)), we can substitute the values into the formula to find the slopes.
(i) For x = 7.9,
slope = (3 / (7 - 7.9) - (-3)) / (7.9 - 8)
= (3 / (-0.9)) / (-0.1)
= -30 / 0.09
= -333.333333
Therefore, the slope of the secant line PQ for x = 7.9 is approximately -333.333333.
(ii) For x = 7.99,
slope = (3 / (7 - 7.99) - (-3)) / (7.99 - 8)
= (3 / (-0.99)) / (-0.01)
= -300 / 0.0099
= -30303.030303
Therefore, the slope of the secant line PQ for x = 7.99 is approximately -30303.030303.
(iii) For x = 7.999,
slope = (3 / (7 - 7.999) - (-3)) / (7.999 - 8)
= (3 / (-0.999)) / (-0.001)
= -3000 / 0.000999
= -3003003.003003
Therefore, the slope of the secant line PQ for x = 7.999 is approximately -3003003.003003.
(iV) For x = 7.9999,
slope = (3 / (7 - 7.9999) - (-3)) / (7.9999 - 8)
= (3 / (-0.9999)) / (-0.0001)
= -30000 / 0.00009999
= -3000300.300030
Therefore, the slope of the secant line PQ for x = 7.9999 is approximately -3000300.300030.
To find the slope of the secant line PQ, we need to use the point-slope formula:
m(PQ) = (y(Q) - y(P)) / (x(Q) - x(P))
Given that point P is (8, -3), we can substitute the coordinates of P into the formula:
m(PQ) = (y(Q) - (-3)) / (x(Q) - 8)
Now we need to find the coordinates of point Q for different values of x.
(i) For x = 7.9:
Substitute x = 7.9 into the equation of the curve:
y = 3 / (7 - 7.9)
Calculate the value of y using a calculator:
y ≈ 29.7
Substitute the values into the slope formula:
m(PQ) = (29.7 - (-3)) / (7.9 - 8)
m(PQ) ≈ 32.7 / (-0.1)
m(PQ) ≈ -327
(ii) For x = 7.99:
Substitute x = 7.99 into the equation of the curve:
y = 3 / (7 - 7.99)
Calculate the value of y using a calculator:
y ≈ 299.7
Substitute the values into the slope formula:
m(PQ) = (299.7 - (-3)) / (7.99 - 8)
m(PQ) ≈ 302.7 / (-0.01)
m(PQ) ≈ -30270
(iii) For x = 7.999:
Substitute x = 7.999 into the equation of the curve:
y = 3 / (7 - 7.999)
Calculate the value of y using a calculator:
y ≈ 2999.7
Substitute the values into the slope formula:
m(PQ) = (2999.7 - (-3)) / (7.999 - 8)
m(PQ) ≈ 3002.7 / (-0.001)
m(PQ) ≈ -3002700
(iV) For x = 7.9999:
Substitute x = 7.9999 into the equation of the curve:
y = 3 / (7 - 7.9999)
Calculate the value of y using a calculator:
y ≈ 29999.7
Substitute the values into the slope formula:
m(PQ) = (29999.7 - (-3)) / (7.9999 - 8)
m(PQ) ≈ 30002.7 / (-0.0001)
m(PQ) ≈ -300027000
The slopes of the secant line PQ, correct to six decimal places, are approximately:
(i) -327
(ii) -30270
(iii) -3002700
(iV) -300027000