Question

Question is in the picture, trying to find the slope.

Answer 1

108

Explanation:

So the slope of the tangent line to the graph of y=4x² is:

`\lim_(x \to 3) (4x^3-108)/(x-3)`

As directed, try values near 3 to estimate the slope. So, using a calculator, I'm going to try 2.9, 2.99, and 2.999:

2.9:

`(4(2.9)^3-108)/((2.9)-3)\approx104.44`

2.99:

`(4(2.99)^3-108)/((2.99)-3)\approx107.6404`

2.999:

`(4(2.999)^3-108)/((2.999)-3)\approx107.96`

And let's also try 2.9999 and 2.99999. So:

`(4(2.9999)^3-108)/((2.9999)-3)\approx107.9964`

And:

`(4(2.9999)^3-108)/((2.9999)-3)\approx107.99964`

Let's also check by coming from the right with 3.001:

`(4(3.001)^3-108)/((3.001)-3)\approx108.036`

Therefore, our limit or slope is:

`\lim_(x \to 3) (4x^3-108)/(x-3)\approx108`

And we're done!

The answer is 108