Derivatives as Rates of Change: Fresh Take

Lumen Learning

Derivatives as Rates of Change: Fresh Take

Calculate how quantities change on average over time
Use rates of change to figure out how an object’s position, speed, and acceleration are changing over time
Estimate future population sizes using current data and how fast the population is growing
Use derivatives to determine the cost and revenue of producing one more unit in a business

Amount of Change Formula

The Main Idea

Amount of Change:
- Change in $y$ -values over an interval $[a, a+h]$
- Given by $f(a+h) - f(a)$
Average Rate of Change:
- Ratio of amount of change to change in $x$ -values
- Formula: $\frac{f(a+h) - f(a)}{h}$
Amount of Change Formula:
- Approximates $f(a+h)$ using $f(a)$ and $f'(a)$
- Formula: $f(a+h) \approx f(a) + f'(a)h$
Accuracy depends on the size of $h$ and the behavior of $f'(x)$

Given $f(10)=-5$ and $f^{\prime}(10)=6$ , estimate $f(10.1)$ .

$-4.4$

Given $f(3) = 2$ and $f'(3) = 5$ , estimate $f(3.2)$ .

Identify known values:

$a = 3$ , $f(a) = 2$ , $f'(a) = 5$

Calculate $h$ :

$h = 3.2 - 3 = 0.2$

Apply the Amount of Change Formula:

$\begin{array}{rcl} f(3.2) &\approx& f(3) + f'(3)(0.2) \\ &\approx& 2 + 5(0.2) \\ &\approx& 2 + 1 \\ &\approx& 3 \end{array}$

Rate of Change Applications

The Main Idea

Motion Along a Line:
- Position function: $s(t)$
- Velocity: $v(t) = s'(t)$
- Speed: $|v(t)|$
- Acceleration: $a(t) = v'(t) = s''(t)$
Population Change:
- Population function: $P(t)$
- Population growth rate: $P'(t)$
Changes in Cost and Revenue:
- Marginal Cost: $MC(x) = C'(x)$
- Marginal Revenue: $MR(x) = R'(x)$
- Marginal Profit: $MP(x) = P'(x) = MR(x) - MC(x)$

Application Techniques

Motion Analysis:
- Use position function to find velocity and acceleration
- Analyze sign of velocity for direction of motion
- Find zeros of velocity for points where object is at rest
Population Estimation:
- Use current population and growth rate to estimate future population
- Apply Amount of Change Formula: $P(t+h) \approx P(t) + P'(t)h$
Economic Analysis:
- Use marginal functions to estimate changes in cost, revenue, or profit
- Approximate change: $f(x+1) - f(x) \approx f'(x)$

A particle moves along a coordinate axis in the positive direction to the right. Its position at time $t$ is given by $s(t)=t^3-4t+2$ . Find $v(1)$ and $a(1)$ and use these values to answer the following questions.

Is the particle moving from left to right or from right to left at time $t=1$ ?
Is the particle speeding up or slowing down at time $t=1$ ?

Begin by finding $v(t)$ and $a(t)$ .

$v(t)=s^{\prime}(t)=3t^2-4$ and $a(t)=v^{\prime}(t)=s''(t)=6t$ .

Evaluating these functions at $t=1$ , we obtain $v(1)=-1$ and $a(1)=6$ .

Because $v(1)<0$ , the particle is moving from right to left.
Because $v(1)<0$ and $a(1)>0$ , velocity and acceleration are acting in opposite directions. In other words, the particle is being accelerated in the direction opposite the direction in which it is traveling, causing $|v(t)|$ to decrease. The particle is slowing down.

A particle moves along a coordinate axis. Its position at time $t$ is given by $s(t)=t^2-5t+1$ . Is the particle moving from right to left or from left to right at time $t=3$ ?

Find $v(3)$ and look at the sign.

Given the position function $s(t) = t^3 - 9t^2 + 24t + 4$ for $t \geq 0$ :

Find the velocity function.
When is the particle at rest?

Velocity function: $v(t) = s'(t) = 3t^2 - 18t + 24$
The particle is at rest when $v(t) = 0$ :

$3t^2 - 18t + 24 = 0$

$3(t-2)(t-4) = 0$

$t = 2$ or $t = 4$

A city’s population triples every $5$ years. The current population is $10,000$ . Estimate the population after $2$ years.

Estimate growth rate:

$P'(0) \approx \frac{P(5) - P(0)}{5-0} = \frac{30,000 - 10,000}{5} = 4,000$ per year

Estimate population after $2$ years:

$P(2) \approx P(0) + 2P'(0) = 10,000 + 2(4,000) = 18,000$

Estimated population after $2$ years: $18,000$

The current population of a mosquito colony is known to be $3,000$ ; that is, $P(0)=3,000$ . If $P^{\prime}(0)=100$ , estimate the size of the population in $3$ days, where $t$ is measured in days.

Use $P(3)\approx P(0)+3P^{\prime}(0)$ .

$3,300$

Given the revenue function $R(x) = -0.03x^2 + 9x$ for $0 \leq x \leq 300$ :

Find the Marginal Revenue function.
Estimate the revenue from selling the $101$ st item.
Calculate the actual revenue change from the $100$ th to the $101$ st item.

Marginal Revenue function:
$MR(x) = R'(x) = -0.06x + 9$
Estimated revenue from $101$ st item:
$MR(100) = -0.06(100) + 9 = 3$

Estimated additional revenue: $$3$
Actual revenue change:
$R(101) - R(100) = 602.97 - 600 = 2.97$

Actual additional revenue: $$2.97$

Suppose that the profit obtained from the sale of $x$ fish-fry dinners is given by $P(x)=-0.03x^2+8x-50$ . Use the marginal profit function to estimate the profit from the sale of the $101$ ^st fish-fry dinner.

Use $P^{\prime}(100)$ to approximate $P(101)-P(100)$ .

$$2$