Limits and Derivatives

Learning Objectives

• Understand the concept of derivatives as the instantaneous rate of change, using average velocity as an example.
• Calculate limits using formal definitions, including left-hand and right-hand limits, and perform algebraic operations involving limits.
• Derive the definition of the derivative using the limit of the difference quotient and understand its geometric interpretation.
• Apply rules for derivatives including sum, difference, product, and quotient rules.
• Find derivatives of polynomial functions and basic trigonometric functions using standard rules.
• Use the Sandwich Theorem to evaluate limits, especially for trigonometric functions.
• Evaluate limits involving trigonometric functions using geometric proofs and identities.
• Apply direct substitution for polynomial limits and factorization/cancellation for rational limits, especially in 0/0 forms.
• Utilize important algebraic limit formulas such as $\lim_{x \to a} \frac{x^n - a^n}{x-a} = na^{n-1}$ and related rational/surd limits.
• Use derivatives to determine the slope of tangents and the rate of change in simple motion/function problems.

Chapter Notes

Intuitive Idea of Derivatives

• Instantaneous Rate of Change: The derivative represents the instantaneous rate of change of a function.
• Example: Consider the distance $S = 4.9t^2$ for a body dropped from a cliff. The velocity at $t = 2$ seconds is approximated by calculating average velocities over smaller intervals.
• Conclusion: As intervals decrease, the average velocity approaches the instantaneous velocity.

Limits and Their Algebra

• Formal Definition: The limit of $f(x)$ as $x$ approaches $a$ is denoted as $\lim_{x \to a} f(x) = L$.
• Left-Hand and Right-Hand Limits: Defined by approaching $a$ from the left ($x \to a^-$) and right ($x \to a^+$).
• Algebraic Operations: Limits respect addition, subtraction, multiplication, and division.

Derivative from First Principles

• Definition: The derivative of $f(x)$ at $x = a$ is $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.
• Geometric Interpretation: The derivative is the slope of the tangent to the curve at a point.

Algebra of Derivatives

• Sum Rule: $(f + g)' = f' + g'$.
• Difference Rule: $(f - g)' = f' - g'$.
• Product Rule: $(fg)' = f'g + fg'$.
• Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

Derivative of Polynomial and Trigonometric Functions

• Polynomials: The derivative of $x^n$ is $nx^{n-1}$.
• Trigonometric Functions: Derivatives include $(\sin x)' = \cos x$ and $(\cos x)' = -\sin x$.

Sandwich Theorem

• Theorem: If $f(x) \leq g(x) \leq h(x)$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

Limits of Trigonometric Functions

• Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ using the Sandwich Theorem.

Limits of Polynomial and Rational Functions

• Direct Substitution: For polynomials, $\lim_{x \to a} f(x) = f(a)$.
• Factorization: For rational functions, factor and cancel common terms to evaluate limits, especially for $\frac{0}{0}$ forms.

Important Algebraic Limit Formula

• Formula: $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$.

Tangent and Rate-of-Change Applications

• Tangent: The derivative at a point gives the slope of the tangent to the curve at that point.
• Rate of Change: Derivatives can be used to calculate rates of change in various contexts, such as velocity in physics.