Rates of Change, Related Rates, and Differentials in Applied Calculus

Rates of Change in Motion and Marginal Analysis

Rate of Change

The concept of rate of change is fundamental in calculus, describing how a quantity varies with respect to another. It is widely used in physics, economics, and other fields to analyze dynamic systems.

• Increment/Rate of Change: If a function f(x) represents a quantity, the change in f as x changes from a to b is given by: $ext{Increment} = f(b) - f(a)$

• Average Rate of Change: The average rate of change of f(x) over the interval [a, b] is: $ext{Average Rate} = rac{f(b) - f(a)}{b - a}$

Example: If a factory's production changes from 100 units to 120 units over 2 hours, the average rate of change is $rac{120 - 100}{2} = 10$ units/hour.

Redefining the Derivative

The derivative provides the instantaneous rate of change of a function. It is the slope of the tangent line to the curve at a point.

• Derivative Definition: $f'(x) = rac{d}{dx}f(x) = ext{lim}_{h o 0} rac{f(x+h) - f(x)}{h}$

• Interpretation:

  • If $rac{dy}{dx} > 0$, y increases as x increases.

  • If $rac{dy}{dx} < 0$, y decreases as x increases.

Example (Functional Growth): The rate at which the price of a commodity increases can be modeled by the derivative of the price function with respect to time.

Rectilinear Motion

Motion Along a Straight Line and the Position Function

Motion along a straight line is called rectilinear motion. The position of a particle at time t is given by s(t).

• Average Velocity: $ext{Average Velocity} = rac{s(t_2) - s(t_1)}{t_2 - t_1}$

• Instantaneous Velocity: $v(t) = s'(t) = rac{ds}{dt}$

Remarks:

• The sign of velocity indicates direction of movement.

• If $v(t) > 0$, the particle moves in the positive direction; if $v(t) < 0$, it moves in the negative direction.

Acceleration and Instantaneous Acceleration

Acceleration measures the rate of change of velocity with respect to time.

• Instantaneous Acceleration: $a(t) = v'(t) = s''(t) = rac{dv}{dt} = rac{d^2s}{dt^2}$

Remarks:

• The sign of acceleration indicates change in velocity.

• If velocity and acceleration have the same sign, the particle is speeding up; if different, slowing down.

Example: If a ball is thrown off a cliff, its position and velocity functions can be used to determine when it hits the ground and its speed at impact.

Marginal Analysis

Cost Functions

Marginal analysis is used in economics to study how cost, revenue, and profit change as production varies.

• Cost Function: $C(x)$ is the total cost of producing x units.

• Marginal Cost: $MC(x) = C'(x)$ is the instantaneous rate of change of cost with respect to units produced.

• Average Cost: $AC(x) = rac{C(x)}{x}$

Revenue and Profit Functions

• Total Revenue: $R(x) = p imes x$

• Marginal Revenue: $MR(x) = R'(x)$

• Total Profit: $P(x) = R(x) - C(x)$

• Marginal Profit: $MP(x) = P'(x) = MR(x) - MC(x)$

Example: If a manufacturer’s average cost function is $AC(x) = 2x + 5$, then $MC(x) = 2$.

Related Rates

Strategy for Solving Related Rates Problems

Related rates problems involve finding the rate of change of one variable in terms of another, often using implicit differentiation.

• Draw a diagram and assign variables.

• Write an equation relating the variables.

• Differentiate both sides with respect to time.

• Substitute known values and solve for the unknown rate.

Example: If sand is dropped to form a pile, relate the rate of change of volume to the rate of change of height.

Differentials and Approximations

Differentials

Differentials are used to approximate changes in functions and estimate errors.

• Differential Definition: If $y = f(x)$, then $dy = f'(x)dx$

• Relative Error: $ext{Relative Error} = rac{dy}{y}$

• Percentage Error: $ext{Percentage Error} = ext{Relative Error} imes 100 ext{ extpercent}$

Example: Estimate the error in the volume of a sphere if the radius is measured with a small error.

Tables and Data

Motion Analysis Table

The following table summarizes the direction, speed, and acceleration of a particle at various times:

t	s(t)	v(t)	a(t)	Direction	Speed	Acceleration
0	0	10	-10	Right	Speeding up	Slowing down
1	10	0	-10	Right	Slowing down	Slowing down
2	20	-10	-10	Left	Speeding up	Speeding up
3	30	-20	-10	Left	Speeding up	Speeding up

Summary

• Rates of change and derivatives are essential for analyzing motion, economics, and related rates problems.

• Marginal analysis applies calculus to cost, revenue, and profit functions.

• Differentials provide tools for error estimation and approximation.

Additional info: These notes cover topics from Chapters 2, 3, and 5 of a standard Calculus curriculum, including derivatives, applications of differentiation, and related rates.