Answer:
a) (i) [tex]m = 2.22[/tex], (ii) [tex]m = 2[/tex], (iii) [tex]m = 2[/tex], (iv) [tex]m = 2[/tex], (v) [tex]m = 1.82[/tex], (vi) [tex]m = 2[/tex], (vii) [tex]m = 2[/tex], (viii) [tex]m = 2[/tex]; b) [tex]m \approx 2[/tex]; c) The equation of the tangent line to curve at P (7, -2) is [tex]y = 2\cdot x + 12[/tex].
Step-by-step explanation:
a) The slope of the secant line PQ is represented by the following definition of slope:
[tex]m = \frac{\Delta y}{\Delta x} = \frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}[/tex]
(i) [tex]x_{Q} = 6.9[/tex]:
[tex]y_{Q} =\frac{2}{6-6.9}[/tex]
[tex]y_{Q} = -2.222[/tex]
[tex]m = \frac{-2.222 + 2}{6.9-7}[/tex]
[tex]m = 2.22[/tex]
(ii) [tex]x_{Q} = 6.99[/tex]
[tex]y_{Q} =\frac{2}{6-6.99}[/tex]
[tex]y_{Q} = -2.020[/tex]
[tex]m = \frac{-2.020 + 2}{6.99-7}[/tex]
[tex]m = 2[/tex]
(iii) [tex]x_{Q} = 6.999[/tex]
[tex]y_{Q} =\frac{2}{6-6.999}[/tex]
[tex]y_{Q} = -2.002[/tex]
[tex]m = \frac{-2.002 + 2}{6.999-7}[/tex]
[tex]m = 2[/tex]
(iv) [tex]x_{Q} = 6.9999[/tex]
[tex]y_{Q} =\frac{2}{6-6.9999}[/tex]
[tex]y_{Q} = -2.0002[/tex]
[tex]m = \frac{-2.0002 + 2}{6.9999-7}[/tex]
[tex]m = 2[/tex]
(v) [tex]x_{Q} = 7.1[/tex]
[tex]y_{Q} =\frac{2}{6-7.1}[/tex]
[tex]y_{Q} = -1.818[/tex]
[tex]m = \frac{-1.818 + 2}{7.1-7}[/tex]
[tex]m = 1.82[/tex]
(vi) [tex]x_{Q} = 7.01[/tex]
[tex]y_{Q} =\frac{2}{6-7.01}[/tex]
[tex]y_{Q} = -1.980[/tex]
[tex]m = \frac{-1.980 + 2}{7.01-7}[/tex]
[tex]m = 2[/tex]
(vii) [tex]x_{Q} = 7.001[/tex]
[tex]y_{Q} =\frac{2}{6-7.001}[/tex]
[tex]y_{Q} = -1.998[/tex]
[tex]m = \frac{-1.998 + 2}{7.001-7}[/tex]
[tex]m = 2[/tex]
(viii) [tex]x_{Q} = 7.0001[/tex]
[tex]y_{Q} =\frac{2}{6-7.0001}[/tex]
[tex]y_{Q} = -1.9998[/tex]
[tex]m = \frac{-1.9998 + 2}{7.0001-7}[/tex]
[tex]m = 2[/tex]
b) The slope at P (7,-2) can be estimated by using the following average:
[tex]m \approx \frac{f(6.9999)+f(7.0001)}{2}[/tex]
[tex]m \approx \frac{2+2}{2}[/tex]
[tex]m \approx 2[/tex]
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:
[tex]y = m\cdot x + b[/tex]
Where:
[tex]x[/tex] - Independent variable.
[tex]y[/tex] - Depedent variable.
[tex]m[/tex] - Slope.
[tex]b[/tex] - 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:
[tex]-2 = 2 \cdot 7 + b[/tex]
[tex]b = -2 + 14[/tex]
[tex]b = 12[/tex]
The equation of the tangent line to curve at P (7, -2) is [tex]y = 2\cdot x + 12[/tex].
