Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 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
1. Limits and Continuity
Continuity
2:00 minutes
Problem 2.6.25
Textbook Question
Textbook QuestionDetermine the interval(s) on which the following functions are continuous.
p(x)=4x^5−3x^2+1
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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. For a function to be continuous over an interval, it must be continuous at every point within that interval. This means there are no breaks, jumps, or asymptotes in the function's graph.
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Polynomial Functions
Polynomial functions, like p(x) = 4x^5 - 3x^2 + 1, are continuous everywhere on the real number line. This is because they are composed of terms that are powers of x with real coefficients, which do not introduce any discontinuities. Understanding the nature of polynomial functions is crucial for determining their continuity.
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Intervals of Continuity
Intervals of continuity refer to the ranges of x-values over which a function remains continuous. For polynomial functions, the interval of continuity is typically all real numbers, denoted as (-∞, ∞). Identifying these intervals involves analyzing the function's behavior and ensuring it meets the criteria for continuity across the specified range.
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