2. Intro to Derivatives

Based on the graph of $f\left(x\right)$, describe the graph of the derivative $f^{\prime}\left(x\right)$ on the interval $\left(0,\infty\right)$.

Based on the graph of $f\left(x\right)$, describe the graph of the derivative $f^{\prime}\left(x\right)$at the point $x=-1$.

Based on the graph of $f\left(x\right)$, describe where the derivative curve $f^{\prime}\left(x\right)$ would be below the x-axis.

Based on the graph $f\left(x\right)$, describe all points where the derivative $f^{\prime}\left(x\right)$would have a jump.

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

a. At what time is the rate of change of the charge Q' the greatest?

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

b. Is Q′ positive or negative for t≥0?

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

c. Sketch a graph of f'.

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

a. Estimate L' (1.5) and state the physical meaning of this quantity.

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

b. Estimate the value of L'(a) for a ≥ 4 . What does this tell you about the talon lengths on these birds? (Source: ZooKeys, 132, 2011)

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.

Match the graphs of the functions in a–d with the graphs of their derivatives in A–D. <MATCH A-D IMAGE>

Analyzing Graphs

Each of the figures in Exercises 91 and 92 shows two graphs, the graph of a function 𝔂 = ƒ(x) together with the graph of its derivative ƒ'(x). Which graph is which? How do you know?

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Recovering a function from its derivative

b. Repeat part (a), assuming that the graph starts at (−2, 0) instead of (−2, 3).