Answer:
108
Step-by-step explanation:
So the slope of the tangent line to the graph of y=4x² is:
[tex]\lim_{x \to 3} \frac{4x^3-108}{x-3}[/tex]
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:
[tex]\frac{4(2.9)^3-108}{(2.9)-3}\approx104.44[/tex]
2.99:
[tex]\frac{4(2.99)^3-108}{(2.99)-3}\approx107.6404[/tex]
2.999:
[tex]\frac{4(2.999)^3-108}{(2.999)-3}\approx107.96[/tex]
And let's also try 2.9999 and 2.99999. So:
[tex]\frac{4(2.9999)^3-108}{(2.9999)-3}\approx107.9964[/tex]
And:
[tex]\frac{4(2.9999)^3-108}{(2.9999)-3}\approx107.99964[/tex]
Let's also check by coming from the right with 3.001:
[tex]\frac{4(3.001)^3-108}{(3.001)-3}\approx108.036[/tex]
Therefore, our limit or slope is:
[tex]\lim_{x \to 3} \frac{4x^3-108}{x-3}\approx108[/tex]
And we're done!
The answer is 108
