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

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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 as approaches is the value that approaches as gets arbitrarily close to .

  • Notation:

  • One-sided limits: (from the left), (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 .

Solution: Factor numerator: . So, .

Continuity

  • Definition: A function is continuous at if:

    • is defined

    • exists

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

Example:

Is continuous at ?

Solution: is undefined, so is not continuous at .

Intermediate Value Theorem (IVT)

  • If is continuous on and is between and , then there exists in such that .

Example:

Show that has a root in .

Solution: , . Since $0-2, IVT guarantees a root in .

The Derivative

  • Definition: The derivative of at is .

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

Example:

Find the derivative of at using the limit definition.

Solution:

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 exists, then and must exist and be equal. (True)

  • If , what does this mean? (The function increases without bound as approaches .)

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

Key Formulas and Theorems

  • Limit Definition of the Derivative:

  • Intermediate Value Theorem: If is continuous on and is between and , then there exists in such that .

  • Squeeze Theorem: If for all near , and , then .

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.

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