Questions: Use the given graph to estimate the value of each derivative. (i) (a) f'(0) (b) f'(1) (c) f'(2) (d) f'(3) (e) f'(4) (f) f'(5)

Transcript text: Use the given graph to estimate the value of each derivative. (i) (a) $f^{\prime}(0)$ $\square$ (b) $f^{\prime}(1)$ $\square$ (c) $f^{\prime}(2)$ $\square$ (d) $f^{\prime}(3)$ $\square$ (e) $f^{\prime}(4)$ $\square$ (f) $f^{\prime}(5)$ $\square$

Solution

Solution Steps

Step 1: Understanding the Problem

We need to estimate the value of the derivative $ f'(x) $ at specific points using the given graph of $ y = f(x) $.

Step 2: Estimating $ f'(0) $

To estimate $ f'(0) $, we look at the slope of the tangent line to the curve at $ x = 0 $. The graph shows that at $ x = 0 $, the slope is negative and steep. Estimating visually, $ f'(0) \approx -2 $.

Step 3: Estimating $ f'(1) $

To estimate $ f'(1) $, we look at the slope of the tangent line to the curve at $ x = 1 $. The graph shows that at $ x = 1 $, the slope is positive and less steep. Estimating visually, $ f'(1) \approx 1 $.

Step 4: Estimating $ f'(2) $

To estimate $ f'(2) $, we look at the slope of the tangent line to the curve at $ x = 2 $. The graph shows that at $ x = 2 $, the slope is zero (the curve has a horizontal tangent). Therefore, $ f'(2) = 0 $.

Final Answer

$ f'(0) \approx -2 $
$ f'(1) \approx 1 $
$ f'(2) = 0 $

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