Guse 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

The tangent line to the given curve at the given point is .

Step-by-step explanation:

To find the slope of the tangent line we to compute the derivative of and then evaluate it for .

         Differentiate the equation.

      Differentiate both sides.

   Sum/Difference rule applied:

 Constant multiple rule applied:

       Applied power rule:

              Simplifying and apply constant rule:

                   Simplify.

Evaluate y' for x=4:

is the slope of the tangent line.

Point slope form of a line is:

where is the slope and is a point on the line.

Insert 9 for and (4,10) for :

The intended form is which means we are going need to distribute and solve for .

Distribute:

Add 10 on both sides:

The tangent line to the given curve at the given point is .

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

The following limit will give us the derivative of the function at (the slope of the tangent line at ):

 We are given f(4)=10.

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:

Let's check this with FOIL:

First:

Outer:

Inner:

Last:

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

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:

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

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