§1.6 The second derivative

Introduction

Increasing or decreasing

The Second Derivative

Concavity

Summary

The position of a car driving along a straight road at time $t$ in minutes is given by the function $y = s(t)$ whose graph is provided. The car's position function has units measured in thousands of feet. For instance, the point $(2,4)$ on the graph indicates that after 2 minutes, the car has traveled 4000 feet.

In everyday language, describe the behavior of the car over the provided time interval. In particular, you should carefully discuss what is happening on each of the time intervals $[0,1]$, $[1,2]$, $[2,3]$, $[3,4]$, and $[4,5]$, plus provide commentary overall on what the car is doing on the interval $[0,12]$.

On what intervals is the position function $y = s(t)$ increasing? decreasing? Why?

On which intervals is the velocity function $y = v(t) = s'(t)$ increasing? decreasing? neither? Why?

Acceleration is defined to be the instantaneous rate of change of velocity, as the acceleration of an object measures the rate at which the velocity of the object is changing. Say that the car's acceleration function is named $a(t)$. How is $a(t)$ computed from $v(t)$? How is $a(t)$ computed from $s(t)$? Explain.

What can you say about $s''$ whenever $s'$ is increasing? Why?

Using only the words increasing, decreasing, constant, concave up, concave down, and linear, complete the following sentences. For the position function $s$ with velocity $v$ and acceleration $a$, - on an interval where $v$ is positive, $s$ is . - on an interval where $v$ is negative, $s$ is . - on an interval where $v$ is zero, $s$ is . - on an interval where $a$ is positive, $v$ is . - on an interval where $a$ is negative, $v$ is . - on an interval where $a$ is zero, $v$ is . - on an interval where $a$ is positive, $s$ is . - on an interval where $a$ is negative, $s$ is . - on an interval where $a$ is zero, $s$ is .

What are the units on $F'(t)$? What is the precise meaning of the value $F'(20) = 3.645$?

Use a central difference to estimate the value of $F''(30)$.

What is the meaning of the value of $F''(30)$ that you have computed in (b) in terms of the potato's temperature? Write several careful sentences that describe the overall behavior of the potato's temperature at this point in time. In particular, you should cite the values of $F(30)$, $F'(30)$, and $F''(30)$, each with appropriate units. Be sure to explicitly discuss what you expect to happen in the minute that transpires from $t = 30$ to $t = 31$.

On the interval from $t = 10$ to $t = 40$, is the potato's temperature increasing at an increasing rate, increasing at a constant rate, or increasing at a decreasing rate? Why?

Suppose that $y = f(x)$ is a twice-differentiable function such that $f''$ is continuous for which the following information is known: $f(2) = -3$, $f'(2) = 1.5$, $f''(2) = -0.25$. - Is $f$ increasing or decreasing near $x = 2$? Is $f$ concave up or concave down near $x = 2$? - Do you expect $f(2.1)$ to be greater than $-3$, equal to $-3$, or less than $-3$? Why? - Do you expect $f'(2.1)$ to be greater than $1.5$, equal to $1.5$, or less than $1.5$? Why? - Sketch a graph of $y = f(x)$ near $(2,f(2))$ and include a graph of the tangent line.

For a certain function $y = g(x)$, its derivative is given by the function pictured in Figure 1.6.

• What is the approximate slope of the tangent line to $y = g(x)$ at the point $(2,g(2))$? - How many real number solutions can there be to the equation $g(x) = 0$? Justify your conclusion fully and carefully by explaining what you know about how the graph of $g$ must behave based on the given graph of $g'$. - On the interval $-3 \lt x \lt 3$, how many times does the concavity of $g$ change? Why? - Use the provided graph to estimate the value of $g''(2)$.

A bungee jumper's height $h$ (in feet ) at time $t$ (in seconds) is given in part by the table:

$t$ | $0.0$ | $0.5$ | $1.0$ | $1.5$ | $2.0$ | $2.5$ | $3.0$ | $3.5$ | $4.0$ | $4.5$ | $5.0$

$h(t)$ | $200$ | $184.2$ | $159.9$ | $131.9$ | $104.7$ | $81.8$ | $65.5$ | $56.8$ | $55.5$ | $60.4$ | $69.8$

$t$ | $5.5$ | $6.0$ | $6.5$ | $7.0$ | $7.5$ | $8.0$ | $8.5$ | $9.0$ | $9.5$ | $10.0$

$h(t)$ | $81.6$ | $93.7$ | $104.4$ | $112.6$ | $117.7$ | $119.4$ | $118.2$ | $114.8$ | $110.0$ | $104.7$

• Use the given data to estimate $h'(4.5)$, $h'(5)$, and $h'(5.5)$. At which of these times is the bungee jumper rising most rapidly? - Use the given data and your work in (a) to estimate $h''(5)$. - What physical property of the bungee jumper does the value of $h''(5)$ measure? What are its units? - Based on the data, on what approximate time intervals is the function $y = h(t)$ concave down? What is happening to the velocity of the bungee jumper on these time intervals?

For each prompt that follows, sketch a possible graph of a function on the interval $-3 \lt x \lt 3$ that satisfies the stated properties. -$y = f(x)$ such that $f$ is increasing on $-3 \lt x \lt 3$, concave up on $-3 \lt x \lt 0$, and concave down on $0 \lt x \lt 3$. -$y = g(x)$ such that $g$ is increasing on $-3 \lt x \lt 3$, concave down on $-3 \lt x \lt 0$, and concave up on $0 \lt x \lt 3$. -$y = h(x)$ such that $h$ is decreasing on $-3 \lt x \lt 3$, concave up on $-3 \lt x \lt -1$, neither concave up nor concave down on $-1 \lt x \lt 1$, and concave down on $1 \lt x \lt 3$. -$y = p(x)$ such that $p$ is decreasing and concave down on $-3 \lt x \lt 0$ and is increasing and concave down on $0 \lt x \lt 3$.