Answer:
The slope of the line tangent to the function at x = 1 is 2.01 ≅2.
Step-by-step explanation:
Using the formula of derivative, it can be easily shown that, [tex]\frac{d f(x)}{dx} = 2[/tex] where [tex]f(x) = x^{2}[/tex].
Here we need to show that as per the instructions in the given table.
Δy = f(x + Δx) - f(x) = f(1 + 0.01) - f(1) = [tex](1 + 0.01)^{2} - 1^{2} = 0.0201[/tex].
In the above equation, we have put x = 1 because we need to find the slope of the line tangent at x = 1.
Hence, dividing Δy by Δx, we get, [tex]\frac{0.0201}{0.01} = 2.01[/tex].
Let's examine this taking a smaller value.
If we take Δx = 0.001, then Δy = [tex]1.001^{2} - 1^{2} = 0.002001[/tex].
Thus, [tex]\frac{0.002001}{0.001} = 2.001[/tex].
The more smaller value of Δx is taken, the slope of the tangent will be approach towards the value of 2.
