Calculus - Differentiation

• Calculus is the study of rates of change and accumulation.

• It has two main branches: Differential Calculus (rates of change) and Integral Calculus (accumulation).

• Limits: The value a function approaches as its input approaches a certain value.

• A limit exists if the function can be “boxed in” near and .

• Formally, for every , there exists a such that if , then .

• One-Sided Limits: The behavior of a function as approaches from only one side (left or right).

• (from the right)

• (from the left)

• A limit exists if and only if .

• Continuity: A function is continuous at if:

1. is defined.

2. exists.

3. .

• This can be summarized as .

• Properties and Rules of Limits: Used to evaluate limits of combinations of functions.

• Sum/Difference Rule:

• Scalar Multiple Rule:

• Product Rule:

• Quotient Rule: , provided .

• Power Rule:

• For polynomials, .

• Indeterminate Forms and “Scrubbing Out” Discontinuities:

• Forms like or require further analysis.

• Techniques include factoring and canceling common terms, or multiplying by a conjugate to remove square roots.

Differential Calculus: Rate of Change and Derivatives

• Rate of Change: Measures how one quantity changes with respect to another.

• Average Rate of Change between and : .

• Instantaneous Rate of Change (Derivative): The rate of change at a single point.

• The derivative represents:

• The slope of the tangent line to the graph of at the point .

• The sensitivity of to small changes in .

• If a function is differentiable at a point, it is also continuous at that point. The converse is not always true (e.g., at ).

Properties and Rules of Derivatives

• Constant Rule: for any constant .

• Power Rule: for any real number .

• Constant Multiple Rule: .

• Sum/Difference Rule: .

• Product Rule: For two differentiable functions and :

• Example: Find the derivative of . Let and . Then and . Using the product rule:

• Quotient Rule: For two differentiable functions and , where :

• Chain Rule: For composite functions :

• Example: Find the derivative of . Let and . Then and . Using the chain rule:

Derivatives of Common Functions:

• for

• (for )

• for

• Implicit Differentiation: Used for equations where is not explicitly defined as a function of . Treat as and differentiate both sides of the equation with respect to , using the chain rule for terms involving .

• Example: Find for . Differentiate both sides with respect to :

Applications of Differentiation 1: Newton-Raphson and L’Hôpital’s Rule

• Roots of a Function: Values of for which .

• Newton-Raphson Method: An iterative numerical method to find roots of .

1. Start with an initial guess .

2. Calculate the next approximation using the formula:

1. Repeat step 2 until or is smaller than a specified tolerance .

• Example: Find a root of . Let .

• .

• . .

• . . . . The root is approximately .

• L’Hôpital’s Rule (LHR): Used to evaluate limits of indeterminate forms or .

• If is of the form or , then:

• This rule can be applied repeatedly if the resulting limit is still indeterminate.

• Example: Evaluate . As and . This is an indeterminate form. Apply LHR: The derivative of the numerator is . The derivative of the denominator is . So, the limit becomes:

Applications of Differentiation 2: Extrema and Concavity

Extrema of a Function

• Extrema (singular: extremum) are points where a function reaches its highest or lowest values. These can be local or global.

• Global Maximum: The highest value such that for all in the domain.

• Global Minimum: The lowest value such that for all in the domain.

• Local Maximum: A value such that for all in some open interval around .

• Local Minimum: A value such that for all in some open interval around .

• Critical Points: Points where the derivative or is undefined. Local extrema often occur at critical points.

• To find local and global extrema for a continuous function on a closed interval :

1. Find all critical points within the interval .

2. Evaluate the function at these critical points.

3. Evaluate the function at the endpoints of the interval ( and ).

4. The largest value is the global maximum, and the smallest value is the global minimum. Other values at critical points are local extrema.

Shape of Graph: Concavity and Inflection Points

• Concavity: Describes the curvature of the graph of a function.

• Concave Up (U shape): is increasing. This means .

• Concave Down (n shape): is decreasing. This means .

• Inflection Point: A point where the concavity of the graph changes (from concave up to concave down, or vice versa). This typically occurs where or is undefined.

• Fermat’s Theorem (Second Derivative Test): Used to classify critical points:

• If and , then has a local minimum at .

• If and , then has a local maximum at .

• If and , the test is inconclusive (might be an inflection point or a local extremum, requiring further analysis like the first derivative test).

Infinite Limits and Asymptotes

Infinite Limits and Vertical Asymptotes

• Infinite Limits: Occur when the function’s output grows without bound (positively or negatively) as the input approaches a certain value.

• or .

• Vertical Asymptote: A vertical line that the graph of approaches as . This happens when:

• or .

• Vertical asymptotes often occur where the denominator of a rational function is zero, provided the numerator is non-zero at that point.

• Example: For , the denominator is zero at and . Since the numerator (1) is non-zero, and are vertical asymptotes.

Limits at Infinity and Horizontal Asymptotes

• Limits at Infinity: Describe the behavior of a function as becomes very large (positive or negative).

• or .

• Horizontal Asymptote: A horizontal line that the graph of approaches as or . This happens when:

• or .

• Evaluating Limits at Infinity for Polynomials: The limit is determined by the term with the highest power of .

• Evaluating Limits at Infinity for Rational Functions: Compare the degrees of the numerator and denominator.

• If degree(numerator) < degree(denominator), the limit is 0 .

• If degree(numerator) = degree(denominator), the limit is the ratio of the leading coefficients.

• If degree(numerator) degree(denominator), the limit is (determined by the signs of the leading terms).

• Example: Evaluate . The degrees of the numerator and denominator are both 2 . The ratio of leading coefficients is . So, . Thus, is a horizontal asymptote.

• Switching Between and : To evaluate a limit as , substitute , where .

• Example: Calculate . Let . As .

Summary of Key Concepts

• Derivatives: Measure instantaneous rates of change and slopes of tangent lines.

• Found using the limit definition or differentiation rules (power, product, quotient, chain, derivatives of common functions).

• Used to find critical points, local extrema, and analyze function behavior (increasing/decreasing).

• Second Derivatives: Measure the rate of change of the first derivative.

• Used to determine concavity and classify critical points as local minima or maxima (Second Derivative Test).

• Points where concavity changes are inflection points.

• Limits at Infinity and Asymptotes: Describe the long-term behavior of functions.

• Vertical asymptotes occur where the function approaches .

• Horizontal asymptotes occur where the function approaches a finite value as .

• Newton-Raphson Method: A numerical technique for approximating roots of functions.

• L’Hôpital’s Rule: A method for evaluating limits of indeterminate forms or .