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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2:00 minutes

Problem 2.6.25

Textbook Question

Determine the interval(s) on which the following functions are continuous.
p(x)=4x^5−3x^2+1

Verified step by step guidance

p>Step 1: Recognize that the function $ p(x) = 4x^5 - 3x^2 + 1 $ is a polynomial function.

p>Step 2: Recall that polynomial functions are continuous everywhere on the real number line.

p>Step 3: Conclude that the function $ p(x) $ is continuous for all real numbers.

p>Step 4: Express the interval of continuity for $ p(x) $ as $(-\infty, \infty)$.

p>Step 5: Verify that there are no restrictions or discontinuities in the function $ p(x) $.

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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.

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.

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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Textbook Question

Use the graph of $f$ in the figure to determine the values of $x$ in the interval $open paren negative 3 comma 5 close paren$ at which f fails to be continuous. Justify your answers using the continuity checklist.

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Textbook Question

Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see figure). Consider 0≤t≤60. <FIGURE>

Textbook Question

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.

f(x)=2x^2+3x+1 / x^2+5x; a=−5

Textbook Question

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.

f(x)= √x−2; a=1

Textbook Question

Determine the interval(s) on which the following functions are continuous.

p(x)=3x^2−6x+7 / x^2+x+1

Textbook Question

Determine the interval(s) on which the following functions are continuous.

f(x)=x^5+6x+17 / x^2−9