Introduction

Welcome to AP Calculus AB/BC Topic 4.3 - Rates of Change in Applied Contexts Other Than Motion! Topic 4.3 will teach us that derivatives allow us to solve real-world problems involving rates of change.

Increasing or Decreasing?

To determine whether something is increasing or decreasing, check its derivative. If its derivative is greater than $0$, then it is increasing. If its derivative is less than $0$, then it is decreasing. If its derivative is equal to $0$, then it is not changing (or it is constant). With this in mind, answer the following questions.

Questions

1. How can we show, mathematically, that $f(x)$ is increasing?

2. How can we show, mathematically, that $g(x)$ is decreasing at $x=4$?

3. How can we show, mathematically, that $h^{\prime }(x)$ is increasing at $x=2$?

4. How can we show, mathematically, that $j^{\prime \prime }(x)$ is constant?

Solutions

1.
$f^{\prime }(x)>0$

2.
$g^{\prime }(4)<0$

3.
$h^{\prime \prime }(2)>0$

4.
$j^{\prime \prime \prime }(x)=0$

Using Our Calculator

Recall that we can use our calculator (either the physical one or Desmos) to compute the derivative at a single point.

Questions

1. (Calculator Active) If $f(x)$ is given by $f(x)=5\cos \left(\frac{\pi }{x^2}\right)+{\tan }^{-1}(2x)$, determine if $f(x)$ is increasing or decreasing at $x=3.5$.

2. (Calculator Active) If $g(x)$ is given by $g(x)=\frac{5\sin (3x)}{\ln x}$, determine if $g^{\prime }(x)$ is increasing or decreasing at $x=3.5$.

Solutions

1.
$f^{\prime }(3.5)\approx 0.225>0$, therefore $f(x)$ is increasing at $x=3.5$.

2.
For this problem, we are trying to see if $g^{\prime }(x)$ is increasing or decreasing, not $g(x)$! Therefore, we must look at whether $g^{\prime \prime }(x)$ is greater than or less than $0$ at $x=3.5$.

$g^{\prime \prime }(3.5)\approx 36.602>0$, therefore $g^{\prime }(x)$ is increasing at $x=3.5$.

Functions that are already Rates of Change

Questions

1. If $p(t)$ represents the pelican population after $t$ years, what does $p^{\prime }(t)$ represent? What are the units for $p^{\prime }(t)$?

2. If $p(t)$ represents the rate of change of the pelican population per year, what does $p^{\prime }(t)$ represent? What are the units for $p^{\prime }(t)$?

Solutions

1.
$p^{\prime }(t)$ represents the rate of change of the pelican population per year. The units are pelicans per year.

2.
$p^{\prime }(t)$ represents the rate of change of the rate of change of the pelican population per year. The units are pelicans per year per year or pelicans per year${}^2$.

Practice Problems

Questions

1. (Calculator Active) A $4,000$-liter tank of water is filled to its capacity. At time $t=0$, water begins to drain out of the tank at a rate modeled by $r(t)$, measured in liters per minute, where $r(t)=\frac{2t^3}{4^t}$. Find $r^{\prime }(6)$. Using correct units, explain the meaning of this value in the context of this problem.

2. (Calculator Active) Cars are entering the highway at a rate modeled by the function $E$ given by $E(t)=15+\frac{\ln (t-2)}{4t^3}$. Cars are leaving the highway at a rate modeled by the function $L$ given by $L(t)=12t^6+4^t$. Both $E(t)$ and $L(t)$ are measured in cars per hour, and $t$ is measured in hours since midnight ($t=0$).

(A) What mathematical expression gives the rate at which cars are entering the highway at 4 AM ($t=4$)?

(B) What mathematical expression gives the rate of change of the rate at which cars are leaving the highway at 4 AM ($t=4$)?

(C) Is the rate of change of the number of cars on the highway increasing or decreasing at 4 AM ($t=4$)? Justify your answer.

Note: These functions are completely made up, and do not necessarily represent realistic numbers.

Solutions

1.
$r^{\prime }(6)=-0.093$ liters per minute per minute (or liters per minute${}^2$). This shows the rate at which the water is draining from the tank is decreasing by $0.093$ liters per minute${}^2$ at time $t=6$.

2.

(A)
$E(4)$

(B)
$L^{\prime }(4)$

(C)
Let $R(t)$ represent the rate of change of the number of cars on the highway $t$ hours since midnight ($t=0$).

$R(t)=E(t)-L(t)$

To determine if the rate of change of the number of cars on the highway is increasing or decreasing at $t=4$, we need to check the sign of $R^{\prime }(4)$.

$R^{\prime }(4)=E^{\prime }(4)-L^{\prime }(4)\approx -74,082<0$, therefore the rate of change of the number of cars on the highway is decreasing at 4 AM.