Activity 1.8.3.
This activity concerns a function \(f(x)\) about which the following information is known:
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\(\displaystyle f(2) = -1\)
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\(y = f'(x)\) has its graph given in the following figure.
Your overall task is to determine as much information as possible about \(f\) (especially near the value \(a = 2\)) by responding to the questions below.
(a)
Find a formula for the tangent line approximation, \(L(x)\text{,}\) to \(f\) at the point \((2,-1)\text{.}\)
(b)
Use the tangent line approximation to estimate the value of \(f(2.07)\text{.}\) Show your work carefully and clearly.
(c)
Sketch a graph of \(y = f''(x)\) on the righthand grid in the provided figure; label it appropriately. Write a sentence to explain why your graph looks the way it does.
(d)
Is the slope of the tangent line to \(y = f(x)\) increasing, decreasing, or neither when \(x = 2\text{?}\) Explain.
(e)
Sketch a possible graph of \(y = f(x)\) near \(x = 2\) on the lefthand grid in the provided figure. Include a sketch of \(y=L(x)\) (found in part (a)). Explain how you know the graph of \(y = f(x)\) looks like you have drawn it.
(f)
Does your estimate in (b) over- or under-estimate the true value of \(f(2.07)\text{?}\) Why?
