Worksheet for Section 2.5

Secton 2.5 is about the second derivative, and what information it gives about the graph of a function. First, note that the sign of the derivative $f'(x)$ of a function $f(x)$ indicates whether the function $f(x)$ is increasing or decreasing:

• A function $f(x)$ is increasing on an interval if for any two numbers $x_1$ and $x_2$ in the interval, $x_1<x_2$ implies $f(x_1)<f(x_2)$.
• A function $f(x)$ is decreasing on an interval if for any two numbers $x_1$ and $x_2$ in the interval, $x_1<x_2$ implies $f(x_1)>f(x_2)$.

More intuitively, a function is increasing if it is going up from left to right in its graph, decreasing if it is going down from left to right. This description, though, suggests a connection with derivatives -- if $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then the derivative $f'(x)$ gives the following information:

• If $f'(x)>0$ for all $x$ in $(a,b)$, then $f(x)$ is increasing on $[a,b]$.
• If $f'(x)<0$ for all $x$ in $(a,b)$, then $f(x)$ is decreasing on $[a,b]$.
• If $f'(x)=0$ for all $x$ in $(a,b)$, then $f(x)$ is constant (neither increasing nor decreasing) on $[a,b]$.

Determine the intervals where each of the following functions are increasing or decreasing:

$$f(x)=x^2\qquad g_1(x)=\frac1x \qquad h(x)=x^3-3x$$

(All of these are functions for which you computed derivative functions in Section 2.3.)

• A function $f(x)$ is concave upward on an open interval $I$ if $f'(x)$ is increasing on $I$.
• A function $f(x)$ is concave downward on an open interval $I$ if $f'(x)$ is decreasing on $I$.

Since the information about whether $f'(x)$ is increasing or decreasing can be determined from the sign of the derivative of $f'(x)$, i.e. from the sign of the second derivative $f''(x)$, the concavity (concave upward or concave downward) of $f(x)$ can be determined from the sign of $f''(x)$:

• If $f''(x)>0$ for all $x$ in the open interval $I$, then $f(x)$ is concave upward on $I$.
• If $f''(x)<0$ for all $x$ in the open interval $I$, then $f(x)$ is concave downward on $I$.

Here are some examples -- for each function, find the intervals where the function is concave upward or concave downward.

$$f(x)=x^2\qquad g_1(x)=\frac1x \qquad h(x)=x^3-3x$$

(All of these are functions for which you computed derivative functions in Section 2.3.) An inflection point is a point on the graph of a function where the graph's concavity changes -- so where the sign of $f''(x)$ changes. Where do the functions above have inflection points?

Regarding the derivative as a rate of change gives a physical interpretation of the second derivative as well. Recall that the instantaneous rate of change in the position of an object is its velocity -- this is the derivative of the position function. The second derivative, then is the instantaneous rate of change in the velocity function -- in other words, the acceleration. For example, recall the falling object problem from Section 2.1, where the position (in this case, height) of the object at time $t$ was given by the function $s(t)=10-16t^2$. What is the acceleration of the object, as a function of time $t$? (At this point, you will need to compute this using the limit definition of the derivative twice.)