Calculus I Exam #1 Review: Limits, Continuity, and the Derivative

Exam #1 Review: Limits, Continuity, and the Derivative

Introduction

This study guide covers foundational topics in Calculus I, focusing on limits, continuity, and the derivative. These concepts are essential for understanding the behavior of functions and their rates of change, which are central to calculus. The guide includes definitions, properties, and problem-solving strategies, as well as example problems and explanations.

How to Study for the Exam

• Start early: Begin reviewing lecture notes and practice problems well before the exam date.

• Active practice: Work through problems without looking at solutions first. Check your answers after attempting each problem.

• Understand concepts: Focus on understanding the reasoning behind each solution, not just memorizing steps.

• Review mistakes: Analyze errors to avoid repeating them.

• Use resources: Attend office hours, join study groups, and use available help centers.

Learning Outcomes

Students should be able to demonstrate proficiency in the following areas:

Learning Outcome	Confidence Level
Calculate slopes of secant and tangent lines
Calculate average and instantaneous velocity
Define a limit
Determine the limit of a function at a point graphically and numerically
Find left- and right-hand limits at a point graphically and numerically
Determine limits of basic functions
Apply the limit laws to evaluate limits
Evaluate limits using algebraic manipulation
Evaluate one-sided limits
Describe the relationship between one-sided and two-sided limits
Find limits using the Squeeze Theorem
Find limits (including infinity, graphically and analytically)
Find and graph vertical asymptotes
Find and graph horizontal asymptotes
Develop the real behavior of a function and the connection to horizontal asymptotes
Recognize limits at infinity
Find points of (dis)continuity in intervals of continuity
Use the Intermediate Value Theorem
Compute the slope of a tangent line
Describe the relationship between the slope of a tangent and instantaneous velocity
Evaluate derivatives at a point using the limit definition of the derivative
Relate differentiability to continuity

Key Concepts and Definitions

Limits

• Definition: The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$.

• Notation: $\lim_{x \to a} f(x) = L$

• One-sided limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

• Limit Laws: Sum, difference, product, quotient, and power rules for limits.

• Special Limits: Limits involving infinity, indeterminate forms, and the Squeeze Theorem.

Example:

Evaluate $\lim_{x \to 2} (x^2 - 4)/(x - 2)$.

Solution: Factor numerator: $x^2 - 4 = (x-2)(x+2)$. So, $\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$.

Continuity

• Definition: A function $f(x)$ is continuous at $x=a$ if:

  • $f(a)$ is defined

  • $\lim_{x \to a} f(x)$ exists

  • $\lim_{x \to a} f(x) = f(a)$

• Types of Discontinuity: Removable, jump, and infinite discontinuities.

Example:

Is $f(x) = \frac{x^2 - 1}{x - 1}$ continuous at $x=1$?

Solution: $f(1)$ is undefined, so $f(x)$ is not continuous at $x=1$.

Intermediate Value Theorem (IVT)

• If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = N$.

Example:

Show that $f(x) = x^3 - x - 2$ has a root in $[1, 2]$.

Solution: $f(1) = -2$, $f(2) = 4$. Since $0$is between$-2$and $4$, IVT guarantees a root in $[1, 2]$.

The Derivative

• Definition: The derivative of $f$ at $x=a$ is $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

• Interpretation: The derivative represents the instantaneous rate of change or the slope of the tangent line at $x=a$.

Example:

Find the derivative of $f(x) = x^2$ at $x=3$ using the limit definition.

Solution:

$\begin{align*} f'(3) &= \lim_{h \to 0} \frac{(3+h)^2 - 9}{h} \\ &= \lim_{h \to 0} \frac{9 + 6h + h^2 - 9}{h} \\ &= \lim_{h \to 0} \frac{6h + h^2}{h} \\ &= \lim_{h \to 0} (6 + h) = 6 \end{align*}$

Common Problem Types

• Evaluating limits algebraically and graphically

• Determining continuity at a point

• Finding vertical and horizontal asymptotes

• Applying the Intermediate Value Theorem

• Using the limit definition to compute derivatives

• Sketching graphs based on limit and continuity information

Sample True/False and Short Answer Questions

• If $\lim_{x \to a} f(x)$ exists, then $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ must exist and be equal. (True)

• If $\lim_{x \to a} f(x) = \infty$, what does this mean? (The function increases without bound as $x$ approaches $a$.)

• What are the three conditions for continuity at a point? (See above under Continuity.)

Key Formulas and Theorems

• Limit Definition of the Derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Intermediate Value Theorem: If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = N$.

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

Additional info:

• This guide is based on a comprehensive exam review covering Chapters 1 and 2 of a typical Calculus I course, focusing on functions, limits, continuity, and the introduction to derivatives.