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

Continuity

Calculus

1. Limits and Continuity

Continuity

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2:33 minutes

Problem 2.5.27

Textbook Question

At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³

Verified step by step guidance

Step 1: Understand the concept of continuity. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.

Step 2: Identify the type of function given. The function y = (2x - 1)^(1/3) is a cube root function, which is continuous everywhere in its domain.

Step 3: Determine the domain of the function. The cube root function is defined for all real numbers, so the domain of y = (2x - 1)^(1/3) is all real numbers.

Step 4: Since the function is defined for all real numbers and cube root functions are continuous over their entire domain, y = (2x - 1)^(1/3) is continuous for all x in the real number set.

Step 5: Conclude that the function y = (2x - 1)^(1/3) is continuous everywhere on the real number line, meaning it does not have any points of discontinuity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.

Cube Root Function

The cube root function, denoted as y = (x)^(1/3), is defined for all real numbers. Unlike square roots, which are only defined for non-negative numbers, cube roots can take any real number as input, including negative values. This characteristic ensures that the cube root function is continuous everywhere on the real number line.

Domain of the Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function y = (2x – 1)^(1/3), the expression inside the cube root can take any real number, meaning the domain is all real numbers. Understanding the domain is crucial for determining where the function is continuous.