- 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 [latex]y[/latex]-values over an interval [latex][a, a+h][/latex]
- Given by [latex]f(a+h) - f(a)[/latex]
- Average Rate of Change:
- Ratio of amount of change to change in [latex]x[/latex]-values
- Formula: [latex]\frac{f(a+h) - f(a)}{h}[/latex]
- Amount of Change Formula:
- Approximates [latex]f(a+h)[/latex] using [latex]f(a)[/latex] and [latex]f'(a)[/latex]
- Formula: [latex]f(a+h) \approx f(a) + f'(a)h[/latex]
- Accuracy depends on the size of [latex]h[/latex] and the behavior of [latex]f'(x)[/latex]
Given [latex]f(10)=-5[/latex] and [latex]f^{\prime}(10)=6[/latex], estimate [latex]f(10.1)[/latex].
Given [latex]f(3) = 2[/latex] and [latex]f'(3) = 5[/latex], estimate [latex]f(3.2)[/latex].
Rate of Change Applications
The Main Idea
- Motion Along a Line:
- Position function: [latex]s(t)[/latex]
- Velocity: [latex]v(t) = s'(t)[/latex]
- Speed: [latex]|v(t)|[/latex]
- Acceleration: [latex]a(t) = v'(t) = s''(t)[/latex]
- Population Change:
- Population function: [latex]P(t)[/latex]
- Population growth rate: [latex]P'(t)[/latex]
- Changes in Cost and Revenue:
- Marginal Cost: [latex]MC(x) = C'(x)[/latex]
- Marginal Revenue: [latex]MR(x) = R'(x)[/latex]
- Marginal Profit: [latex]MP(x) = P'(x) = MR(x) - MC(x)[/latex]
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: [latex]P(t+h) \approx P(t) + P'(t)h[/latex]
- Economic Analysis:
- Use marginal functions to estimate changes in cost, revenue, or profit
- Approximate change: [latex]f(x+1) - f(x) \approx f'(x)[/latex]
A particle moves along a coordinate axis in the positive direction to the right. Its position at time [latex]t[/latex] is given by [latex]s(t)=t^3-4t+2[/latex]. Find [latex]v(1)[/latex] and [latex]a(1)[/latex] and use these values to answer the following questions.
- Is the particle moving from left to right or from right to left at time [latex]t=1[/latex]?
- Is the particle speeding up or slowing down at time [latex]t=1[/latex]?
A particle moves along a coordinate axis. Its position at time [latex]t[/latex] is given by [latex]s(t)=t^2-5t+1[/latex]. Is the particle moving from right to left or from left to right at time [latex]t=3[/latex]?
Given the position function [latex]s(t) = t^3 - 9t^2 + 24t + 4[/latex] for [latex]t \geq 0[/latex]:
- Find the velocity function.
- When is the particle at rest?
A city’s population triples every [latex]5[/latex] years. The current population is[latex]10,000[/latex]. Estimate the population after [latex]2[/latex] years.
The current population of a mosquito colony is known to be [latex]3,000[/latex]; that is, [latex]P(0)=3,000[/latex]. If [latex]P^{\prime}(0)=100[/latex], estimate the size of the population in [latex]3[/latex] days, where [latex]t[/latex] is measured in days.
Given the revenue function [latex]R(x) = -0.03x^2 + 9x[/latex] for [latex]0 \leq x \leq 300[/latex]:
- Find the Marginal Revenue function.
- Estimate the revenue from selling the [latex]101[/latex]st item.
- Calculate the actual revenue change from the [latex]100[/latex]th to the [latex]101[/latex]st item.
Suppose that the profit obtained from the sale of [latex]x[/latex] fish-fry dinners is given by [latex]P(x)=-0.03x^2+8x-50[/latex]. Use the marginal profit function to estimate the profit from the sale of the [latex]101[/latex]st fish-fry dinner.