2. Intro to Derivatives

Suppose the graph of the derivative $f^{\prime }$ of a continuous function $f$ is shown and $f^{\prime }$ is continuous everywhere. Which of the following statements is necessarily true about the function $f$?

Suppose the graph of the derivative $f^{\prime }$ of a function $f$ is shown. At which of the following points does the function $f$ have a local maximum?

Given the graph of the function $f\left(x\right)$ below, which of the following graphs best represents its derivative $f^{\prime }\left(x\right)$?

Given that the graph of the derivative $f^{\prime }$ of a function $f$ is shown, which of the following statements is true about the function $f$?

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>