Answer:
a) The slope of the tangent line to the graph of the function [tex]g(x)=2x[/tex] at [tex]x=9[/tex] is 2.
b) The tangent line is [tex]y=2x[/tex].
Step-by-step explanation:
a) Tho find the slope of the tangent line to the graph of a given function [tex]f(x)[/tex] at a point [tex]x_0[/tex] we only need to calculate the derivative of [tex]f(x)[/tex] at [tex]x_0[/tex], i.e., [tex]f'(x_0)[/tex].
For the given function [tex]g(x)=2x[/tex], its derivative is [tex]g'(x)=2[/tex]. So, in particular, for [tex]x=9[/tex]: [tex]g'(9)=2[/tex]. Thus, the slope of the tangent is 2.
b) The equation of the tangent line of the graph of a function [tex]f(x)[/tex] at a point [tex]x_0[/tex] is given by the formula
[tex] y-f(x_0) = f'(x_0)(x-x_0)[/tex].
In this exercise we have the function [tex]g(x)=2x[/tex] and [tex]x_0=9[/tex]. Then,
[tex]g(x_0) = 2x_0 = 2\cdot 9=18[/tex]
[tex]g'(x_0) = 2[/tex] (from the previous answer)
So, the equation of the tangent line is
[tex]y-18 = 2(x-9)[/tex]
which is equivalent to
[tex]y-18 = 2x-18[/tex]
that yields
[tex]y=2x[/tex].
