Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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Multiple Choice

Given the graph of a function $f$ , find a number $ε > 0$ such that if $| x - 1 | < ε$ , then $| f (x) - 1 | < 0.2$ .

ε = 0.05

ε = 0.3

ε = 0.1

ε = 0.5

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Verified step by step guidance

Step 1: Understand the problem. The goal is to find a value of $ \epsilon > 0 $ such that if $ |x - 1| < \epsilon $, then $ |f(x) - 1| < 0.2 $. This is related to the concept of continuity and the definition of limits.

Step 2: Analyze the graph of $ f(x) $. Look at the behavior of $ f(x) $ near $ x = 1 $. Specifically, observe how $ f(x) $ changes as $ x $ approaches 1 and determine the range of $ x $ values that keep $ |f(x) - 1| < 0.2 $.

Step 3: Determine the interval around $ x = 1 $ where $ |f(x) - 1| < 0.2 $. This involves finding the values of $ x $ such that $ f(x) $ stays within 0.2 units of 1. The interval will help you identify the corresponding $ \epsilon $.

Step 4: Relate the interval to $ \epsilon $. The condition $ |x - 1| < \epsilon $ defines the distance from $ x = 1 $. Choose the largest $ \epsilon $ that satisfies $ |f(x) - 1| < 0.2 $ for all $ x $ within the interval.

Step 5: Verify your choice of $ \epsilon $. Ensure that for the chosen $ \epsilon $, the inequality $ |f(x) - 1| < 0.2 $ holds true for all $ x $ satisfying $ |x - 1| < \epsilon $. This confirms the correctness of your solution.