Comprehensive Guide to Unit 2: Differentiation

Average Rate of Change (AROC)

The slope of the secant line connecting two points on a curve, calculated with the formula: AROC = (f(b) - f(a)) / (b - a).

Secant Line

A line that intersects a curve at two distinct points.

Instantaneous Rate of Change (IROC)

The exact rate of change of a quantity at a specific moment, represented by the slope of the tangent line.

Tangent Line

A line that touches a curve at exactly one point, representing the instantaneous rate of change.

Derivative

A function that gives the instantaneous rate of change of a function at any input x.

Limit Definition of Derivative

The formula: f'(x) = lim(h -> 0)(f(x+h) - f(x))/h used to calculate derivatives.

Constant Rule

The derivative of a constant function k is zero, f'(x) = 0.

Power Rule

The rule that states: d/dx(x^n) = n*x^(n-1) for any real number exponent n.

Product Rule

A rule for differentiating products of two functions: f'(x) = u(x)v'(x) + v(x)u'(x).

Quotient Rule

A rule for differentiating the quotient of two functions: f'(x) = (v(x)u'(x) - u(x)v'(x)) / [v(x)]^2.

Differentiability implies Continuity

If f'(c) exists, then the function f(x) must be continuous at x=c.

Cusp

A pointed end or corner on a curve where the slope is undefined.

Corner

A sharp turn on a graph where the left-hand limit of the derivative does not equal the right-hand limit.

Vertical Tangent

A point on a curve where the slope of the tangent line is undefined or infinite.

Natural Log Derivative

The derivative of ln(x) is 1/x.

Exponential Derivative

The derivative of e^x is e^x, meaning it is its own derivative.

sin x Derivative

The derivative of sin x is cos x.

cos x Derivative

The derivative of cos x is -sin x.

tan x Derivative

The derivative of tan x is sec^2 x.

Velocity Function v(t)

The first derivative of the position function, representing the rate of change of position.

Acceleration Function a(t)

The derivative of the velocity function, representing the rate of change of velocity.

Symmetric Difference Quotient

An average rate of change approximation using points surrounding a value c.

Using Limits for IROC

The process of finding IROC by taking the limit as h approaches zero in the average rate of change formula.

Order of Derivative Notation

When using the limit definition: keep writing 'lim' until substituting h=0.

Discontinuous Function

A function that has holes, jumps, or vertical asymptotes; not differentiable at those points.

Misusing Power Rule

The common mistake of applying the power rule to constants, which results in 0.

Derivative Notation

Includes f'(x), y', dy/dx, and d/dx[f(x)], all representing the derivative of f.

Differentiability Failures

Reasons why a function may be continuous but not differentiable, like cusps or corners.

tangent line

A straight line that touches a curve at just one point.

tan & sec relationship

The rules for tan x and sec x derivatives, both involved in trigonometric differentiation.

C Functions Rule

All trigonometric functions starting with 'C' (cos, csc, cot) have negative derivatives.

Function Height vs. Slope

Confusing the value of a function at x=c with its slope (derivative) at that point.

AP Exam Estimation

Using given data tables or graphs to estimate derivatives, typically using AROC.

Key Derivative of 1/x

The derivative of 1/x is -1/x^2.

Long Division in Derivatives

Used to simplify fractions in the context of the quotient rule.

Limit Process for Derivative

The method of approaching the instantaneous rate of change using limits.

Position Function

Describes the location of an object over time, denoted as x(t) or s(t).

Differentiation Process

The overall method of applying various rules to find derivatives of equations.

Common Mistake in Differentiation

Forgetting parentheses in product/quotient rules, which can lead to algebra errors.

Example of AROC Calculation

To find AROC, use AROC = (f(4) - f(2)) / (4 - 2) to estimate f'(3).

Function Naming in Context

Recognizing that f(x) and f'(x) serve different roles in calculus.

Mathematical Concept of Limits

The foundational idea in calculus that allows for the definition of derivatives through approaching values.

Derivative of x^3

The derivative of x^3 is 3x^2, thus illustrating the Power Rule in action.