15.3.1 Instantaneous Rate of Change

This subsection is not completed yet, but in the meantime here are some practice questions.

Remember:
When graphing the derivative of a graph, take note of the slope. Where $f(x)$ is increasing, the derivative is positive, and where $f(x)$ is decreasing, the derivative is negative. Horizontal points on the graph have a slope (derivative) of zero. Points of inflection have a local maximum or minimum slope. Using this information, you can graph the derivative: If you’re going the opposite way and graphing $f(x)$ knowing what the graph of $f’(x)$ looks like, remember that where $f’(x)$ is negative, $f(x)$ is decreasing; where $f’(x)=0$, $f(x)$ is stationary; and where $f’(x)$ is positive, $f(x)$ is increasing. It doesn’t matter what value of $f(x)$ you choose to start with – your graph may be shifted up or down compared to one drawn by someone else.

Remember:
The derivative of displacement with respect to time is velocity. Speed is the magnitude of velocity. Acceleration is the derivative of velocity with respect to time (which makes it the second derivative of displacement with respect to time). \begin{align*} v &= \frac{\dif x}{\dif t} = \dot x\\ \mathrm{speed} &= \abs{v}\\ a &= \frac{\dif v}{\dif t} = \dot v \\ &= \frac{\dif^2 x}{\dif t^2} = \ddot x \end{align*}

Question 15.3.1

If $\dot x = -3\,\mathrm{m/s}$ and $\ddot x = 1\,\mathrm{m/s^2}$, is the speed increasing or decreasing? Show answer

Remember:
Since speed is the magnitude of velocity, it increases when velocity gets further away from zero, and decreases when velocity gets closer to zero.