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

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

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)(f(x)-f(4))/(x-4)`

`\lim_(x \rightarrow 4)(2x^2-7x+6-10)/(x-4)` We are given f(4)=10.

`\lim_(x \rightarrow 4)(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)(2x^2-7x-4)/(x-4)`

`\lim_(x \rightarrow 4)((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.