Module 4 Study Notes: Analyzing Rates of Change

Instantaneous Rate of Change

The derivative f'(x) represents the rate at which the dependent variable (y) changes with respect to the independent variable (x) at a specific instant.

Units of the Derivative

The units of the derivative are always a ratio: Units of f'(x) = Units of f(x) / Units of x.

Interpretation Template for Derivative

To interpret a derivative: "At [input value with units], the [name of quantity using context] is [increasing/decreasing] at a rate of [absolute value of output] [units of y per unit of x]."

Position (s(t) or x(t))

The location of the particle relative to the origin at time t.

Velocity (v(t))

How fast and in what direction the position is changing; v(t) = s'(t).

Acceleration (a(t))

How fast and in what direction the velocity is changing; a(t) = v'(t) = s''(t).

Velocity vs. Speed

Velocity is a vector indicating magnitude and direction; Speed is a scalar indicating magnitude only.

Speeding Up

When v(t) and a(t) have the SAME sign.

Slowing Down

When v(t) and a(t) have OPPOSITE signs.

Marginal Cost (C'(x))

The approximate cost of producing the (x+1)th item; C'(x) ≈ C(x+1) - C(x).

Flow Rate

The derivative dV/dt measures how volume changes with respect to time.

Linear Density

If M(x) is the mass of a wire of length x, then dM/dx is the linear density (mass per unit length).

Identify Variables

In word problems, read the text and assign variables to quantities involved.

Identify Given Data

Determine known values and known rates in a problem.

Check Units

Ensure input units match the derivative's denominator and output units match the numerator.

Common Mistake: Decreasing Negative Error

Avoid saying "The temperature is decreasing at a rate of -5 degrees." Instead, say "Decreasing at a rate of 5 degrees."

Common Mistake: Confusing Velocity and Speed

Do not assume if acceleration is negative that the particle is slowing down; check velocity.

Common Mistake: Ignoring Units

Always provide numerical answers with units in an FRQ.

Common Mistake: Over the Interval vs. At Time t

Do not interpret a derivative f'(5) as applying over an interval; it applies only at an exact moment.

PRT (Position, Rate, Time)

A method to analyze motion by considering position over time and the rates of change.

Marginal Revenue

In economics, the derivative of the revenue function indicates the additional revenue per product sold.

Chain Rule

A formula for computing the derivative of the composition of two or more functions.

Product Rule

A formula for finding the derivative of the product of two functions.

Quotient Rule

A formula used to find the derivative of the quotient of two functions.

Critical Points

Points on a graph where the derivative is zero or undefined; important for identifying maxima and minima.

Concavity

The direction of the curvature of a graph, determined by the second derivative.

Inflection Point

A point on a curve where the concavity changes.