Functions and Rates of Change

Average and Instantaneous Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. The instantaneous rate of change at a point is the derivative at that point, representing the slope of the tangent line.

• Average Rate of Change: for in

• Instantaneous Rate of Change:

• Example: For , average rate from to is ; instantaneous at is .

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as the input approaches a certain value. One-sided limits consider approach from only one direction.

• Limit:

• One-sided Limits: (from left), (from right)

• Special Limit:

• Limits at Infinity: Used to find horizontal asymptotes. For example,

Continuity and Discontinuities

A function is continuous at if . Discontinuities can be classified as removable, jump, or infinite.

• Removable Discontinuity: Limit exists, but function is not defined or not equal to the limit at that point.

• Jump Discontinuity: Left and right limits exist but are not equal.

• Infinite Discontinuity: Function approaches infinity near the point.

Derivatives and Differentiation

Definition and Calculation of Derivatives

The derivative of a function at a point measures the instantaneous rate of change. It can be defined as a limit.

• Derivative at a Point:

• Derivative as a Function:

• Units: If is in meters and in seconds, is in meters per second.

Higher Order Derivatives and Notation

• Second Derivative:

• Notation: , , ,

Rules of Differentiation

• Power Rule:

• Exponential Rule:

• Trigonometric Rules: ,

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Implicit Differentiation

• Used when is defined implicitly by an equation involving and .

• Example: For ,

• Second Derivative: Differentiate with respect to again, using implicit differentiation.

Derivatives of Inverse, Logarithmic, and Inverse Trig Functions

• Inverse Function:

• Logarithmic Function:

• Logarithmic Differentiation: Take of both sides to simplify differentiation, especially for products and powers.

• Inverse Trig:

Tangent Lines and Linear Approximation

• Tangent Line:

• Linearization: Approximates near using the tangent line.

• Differentials:

Related Rates

• Problems involving rates at which related variables change over time.

• Example: If and are related by , and is known, find using .

Applications of Derivatives

Critical Points, Extrema, and Tests

• Critical Points: Where or is undefined.

• Absolute/Local Max/Min: Highest/lowest values on a domain or in a neighborhood.

• First Derivative Test: Determines if a critical point is a max or min by sign changes in .

• Second Derivative Test: If , local min at ; if , local max at .

Increasing/Decreasing and Concavity

• Increasing:

• Decreasing:

• Concave Up:

• Concave Down:

• Inflection Point: Where concavity changes, and sign changes.

Graph Sketching

• Use critical points, intervals of increase/decrease, concavity, and inflection points to sketch graphs.

Optimization

• Set up a function to model the quantity to be optimized.

• Find critical points and determine maxima or minima.

• Justify solution and include correct units.

L'Hôpital's Rule and Indeterminate Forms

• For or forms: (if limit exists).

• Other indeterminate forms (e.g., , ) can often be manipulated into or .

Newton's Method

• Iterative method to approximate roots of .

• Formula:

Mean Value Theorem (MVT) for Derivatives

• If is continuous on and differentiable on , then such that

Integrals and Their Applications

Antiderivatives and Indefinite Integrals

• Antiderivative: such that

• General Form:

• Examples:

  • (for )

Riemann Sums and Definite Integrals

• Riemann Sum: Approximates area under a curve using rectangles.

• Summation Notation:

• Definite Integral: is the net area under from to .

• Properties: Linearity, additivity, reversing limits, etc.

Average Value and Mean Value Theorem for Integrals

• Average Value:

• Mean Value Theorem for Integrals: such that

Fundamental Theorem of Calculus (FTC)

• Part 1: If , then

• Part 2: , where is any antiderivative of

Substitution and Integration Techniques

• Substitution: For , let ,

Area and Applications

• Net Area: (areas below -axis are negative)

• Area Between Curves: , where

• Applications: Use integrals to relate acceleration, velocity, and position: ,

Differential Equations

Separable Differential Equations

• Can be written as

• Separate variables:

• Integrate both sides to solve for .

• Apply initial conditions if given to find particular solutions.

Additional info: This guide covers all major Calculus I topics as outlined in the provided syllabus, including foundational concepts, computational techniques, and applications. For each topic, students should practice both conceptual understanding and problem-solving skills.