Question

g Use this to find the equation of the tangent line to the parabola y = 2 x 2 − 7 x + 6 at the point ( 4 , 10 ) . The equation of this tangent line can be written in the form y = m x + b where m is: 9 Correct and where b is: Incorrect LicensePoints possible: 1 Unlimited attempts. Score on last attempt: (0.33, 0.33, 0), Score in gradebook: (0.33, 0.33, 0), Out of: (0.33, 0.33, 0.34) Message instructor about this question

Answer 1

Answer:

The tangent line to the given curve at the given point is $y=9x-26$.

Step-by-step explanation:

To find the slope of the tangent line we to compute the derivative of $y=2x^2-7x+6$ and then evaluate it for $x=4$.

$(y=2x^2-7x+6)'$          Differentiate the equation.

$(y)'=(2x^2-7x+6)'$       Differentiate both sides.

$y'=(2x^2)'-(7x)'+(6)'$    Sum/Difference rule applied: $(f(x)\pmg(x))'=f'(x)\pm g'(x)$

$y'=2(x^2)'-7(x)'+(6)'$  Constant multiple rule applied: $(cf)'=c(f)'$

$y'2(2x)-7(1)+(6)'$        Applied power rule: $(x^n)'=nx^{n-1}$

$y'=4x-7+0$               Simplifying and apply constant rule: $(c)'=0$

$y'=4x-7$                    Simplify.

Evaluate y' for x=4:

$y'=4(4)-7$

$y'=16-7$

$y'=9$ is the slope of the tangent line.

Point slope form of a line is:

$y-y_1=m(x-x_1)$

where $m$ is the slope and $(x_1,y_1)$ is a point on the line.

Insert 9 for $m$ and (4,10) for $(x_1,y_1)$:

$y-10=9(x-4)$

The intended form is $y=mx+b$ which means we are going need to distribute and solve for $y$.

Distribute:

$y-10=9x-36$

Add 10 on both sides:

$y=9x-26$

The tangent line to the given curve at the given point is $y=9x-26$.

------------Formal Definition of Derivative----------------

The following limit will give us the derivative of the function $f(x)=2x^2-7x+6$ at $x=4$ (the slope of the tangent line at $x=4$):

$\lim_{x \rightarrow 4}\frac{f(x)-f(4)}{x-4}$

$\lim_{x \rightarrow 4}\frac{2x^2-7x+6-10}{x-4}$  We are given f(4)=10.

$\lim_{x \rightarrow 4}\frac{2x^2-7x-4}{x-4}$

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:

$2x^2-7x-4=(x-4)(2x+1)$

Let's check this with FOIL:

First: $x(2x)=2x^2$

Outer: $x(1)=x$

Inner: $(-4)(2x)=-8x$

Last: $-4(1)=-4$

---------------------------------Add!

$2x^2-7x-4$

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:

$\lim_{x \rightarrow 4}\frac{2x^2-7x-4}{x-4}$

$\lim_{x \rightarrow 4}\frac{(x-4)(2x+1)}{x-4}$

$\lim_{x \rightarrow 4}(2x+1)$

Now we get to replace x with 4 since we have no division by 0 to worry about:

2(4)+1=8+1=9.