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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2:21 minutes

Problem 2.7b

Textbook Question

Limits and Continuity
On what intervals are the following functions continuous?

b. g(x) = x³/⁴

Verified step by step guidance

Identify the type of function: The function g(x) = x^(3/4) is a power function, which is generally continuous wherever it is defined.

Determine the domain of the function: Since the exponent 3/4 is a positive rational number, g(x) is defined for all x ≥ 0. This is because the fourth root of a negative number is not real.

Check for any points of discontinuity: For power functions like g(x) = x^(3/4), there are no discontinuities within their domain. Therefore, the function is continuous on its entire domain.

Conclude the intervals of continuity: Since g(x) is defined and continuous for all x ≥ 0, the interval of continuity is [0, ∞).

Summarize the findings: The function g(x) = x^(3/4) is continuous on the interval [0, ∞), as it is a power function with a domain of all non-negative real numbers.

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Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions, especially at points where they may not be explicitly defined. They help determine continuity and differentiability, which are essential for evaluating functions over intervals.

Continuity

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. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is vital for understanding how functions behave and ensuring that there are no breaks, jumps, or holes in their graphs.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Analyzing these functions requires checking continuity at the boundaries of the pieces to ensure that the function behaves consistently. Understanding how to evaluate these functions across their defined intervals is essential for determining where they are continuous.