Respuesta :
Answer:
The tangent line to the given curve at the given point is [tex]y=9x-26[/tex].
Step-by-step explanation:
To find the slope of the tangent line we to compute the derivative of [tex]y=2x^2-7x+6[/tex] and then evaluate it for [tex]x=4[/tex].
[tex](y=2x^2-7x+6)'[/tex] Differentiate the equation.
[tex](y)'=(2x^2-7x+6)'[/tex] Differentiate both sides.
[tex]y'=(2x^2)'-(7x)'+(6)'[/tex] Sum/Difference rule applied: [tex](f(x)\pmg(x))'=f'(x)\pm g'(x)[/tex]
[tex]y'=2(x^2)'-7(x)'+(6)'[/tex] Constant multiple rule applied: [tex](cf)'=c(f)'[/tex]
[tex]y'2(2x)-7(1)+(6)'[/tex] Applied power rule: [tex](x^n)'=nx^{n-1}[/tex]
[tex]y'=4x-7+0[/tex] Simplifying and apply constant rule: [tex](c)'=0[/tex]
[tex]y'=4x-7[/tex] Simplify.
Evaluate y' for x=4:
[tex]y'=4(4)-7[/tex]
[tex]y'=16-7[/tex]
[tex]y'=9[/tex] is the slope of the tangent line.
Point slope form of a line is:
[tex]y-y_1=m(x-x_1)[/tex]
where [tex]m[/tex] is the slope and [tex](x_1,y_1)[/tex] is a point on the line.
Insert 9 for [tex]m[/tex] and (4,10) for [tex](x_1,y_1)[/tex]:
[tex]y-10=9(x-4)[/tex]
The intended form is [tex]y=mx+b[/tex] which means we are going need to distribute and solve for [tex]y[/tex].
Distribute:
[tex]y-10=9x-36[/tex]
Add 10 on both sides:
[tex]y=9x-26[/tex]
The tangent line to the given curve at the given point is [tex]y=9x-26[/tex].
------------Formal Definition of Derivative----------------
The following limit will give us the derivative of the function [tex]f(x)=2x^2-7x+6[/tex] at [tex]x=4[/tex] (the slope of the tangent line at [tex]x=4[/tex]):
[tex]\lim_{x \rightarrow 4}\frac{f(x)-f(4)}{x-4}[/tex]
[tex]\lim_{x \rightarrow 4}\frac{2x^2-7x+6-10}{x-4}[/tex] We are given f(4)=10.
[tex]\lim_{x \rightarrow 4}\frac{2x^2-7x-4}{x-4}[/tex]
Let's see if we can factor the top so we can cancel a pair of common factors from top and bottom to get rid of the x-4 on bottom:
[tex]2x^2-7x-4=(x-4)(2x+1)[/tex]
Let's check this with FOIL:
First: [tex]x(2x)=2x^2[/tex]
Outer: [tex]x(1)=x[/tex]
Inner: [tex](-4)(2x)=-8x[/tex]
Last: [tex]-4(1)=-4[/tex]
---------------------------------Add!
[tex]2x^2-7x-4[/tex]
So the numerator and the denominator do contain a common factor.
This means we have this so far in the simplifying of the above limit:
[tex]\lim_{x \rightarrow 4}\frac{2x^2-7x-4}{x-4}[/tex]
[tex]\lim_{x \rightarrow 4}\frac{(x-4)(2x+1)}{x-4}[/tex]
[tex]\lim_{x \rightarrow 4}(2x+1)[/tex]
Now we get to replace x with 4 since we have no division by 0 to worry about:
2(4)+1=8+1=9.
