Answer:
The derivative of the function at [tex]x = -16[/tex] equals [tex]\frac{32}{2209}[/tex].
The equation of the tangent line that passes through the point [tex]\left(-16, \frac{32}{47} \right)[/tex] is [tex]y = \frac{32}{2209}\cdot x +\frac{2016}{2209}[/tex].
Step-by-step explanation:
From Differential Calculus we remember the following definition of derivative, [tex]f'(x)[/tex]:
[tex]f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex] (Eq. 1)
Where [tex]f(x+h)[/tex] is the function evaluated at [tex]x = x+h[/tex], dimensionless.
If we know that [tex]f(x) = \frac{32}{31-x}[/tex], then [tex]f(x+h)[/tex] is:
[tex]f(x+h) = \frac{32}{31-x-h}[/tex] (Eq. 2)
Now we proceed to expand (Eq. 1):
[tex]f'(x) = \lim_{h \to 0} \frac{\frac{32}{31-x-h}-\frac{32}{31-x} }{h}[/tex] (Eq. 1b)
[tex]f'(x) = 32\cdot \lim_{h \to 0} \frac{31-x-31+x+h}{h\cdot (31-x-h)\cdot (31-x)}[/tex]
[tex]f'(x) = 32\cdot \lim_{h \to 0} \frac{1}{(31-x-h)\cdot (31-x)}[/tex]
[tex]f'(x) = \frac{32}{(31-x)^{2}}[/tex] (Eq. 3)
And the derivative is evaluated at [tex]x = -16[/tex]:
[tex]f'(-16) = \frac{32}{(31+16)^{2}}[/tex]
[tex]f'(-16) = \frac{32}{2209}[/tex]
The derivative of the function at [tex]x = -16[/tex] equals [tex]\frac{32}{2209}[/tex].
This value represents the slope of the tangent line that passes through [tex]x = -16[/tex], and value of [tex]y[/tex] is now found:
[tex]f(-16) = \frac{32}{47}[/tex]
The tangent line is represented by the following model:
[tex]y = m\cdot x + b[/tex] (Eq. 4)
Where:
[tex]m[/tex] - Slope, dimensionless.
[tex]b[/tex] - y-Intercept, dimensionless.
[tex]x[/tex] - Independent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
If we know that [tex]m = \frac{32}{2209}[/tex], [tex]x = -16[/tex] and [tex]y = \frac{32}{47}[/tex], the y-Intercept is:
[tex]b = y-m\cdot x[/tex] (Eq. 4b)
[tex]b = \frac{32}{47}-\frac{32}{2209}\cdot (-16)[/tex]
[tex]b = \frac{2016}{2209}[/tex]
The equation of the tangent line that passes through the point [tex]\left(-16, \frac{32}{47} \right)[/tex] is [tex]y = \frac{32}{2209}\cdot x +\frac{2016}{2209}[/tex].
