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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1:16 minutes

Problem 2.5.6a

Textbook Question

Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
<IMAGE>
a. Does f (1) exist?

Verified step by step guidance

Identify the piecewise function definition for f(x) at x = 1. According to the problem, f(x) is defined as 1 when x = 1.

To determine if f(1) exists, check if there is a specific value assigned to f(x) at x = 1 in the piecewise function.

Since the piecewise function explicitly defines f(x) = 1 for x = 1, f(1) exists and is equal to 1.

In general, for a function value to exist at a specific point, the function must be defined at that point in the piecewise definition.

Conclude that f(1) exists because the piecewise function provides a specific value for f(x) at x = 1.

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Key Concepts

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

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has distinct formulas for various intervals of x, which means that the output depends on which interval the input falls into. Understanding how to evaluate piecewise functions is crucial for determining function values at specific points.

Function Existence

For a function to exist at a certain point, it must have a defined output for that input. This involves checking if the input falls within the domain of the function and if the corresponding expression yields a real number. In the context of the question, we need to evaluate f(1) to see if it is defined according to the piecewise conditions.

Continuity at a Point

Continuity at a point means that the function's value at that point matches the limit of the function as it approaches that point from both sides. While the question specifically asks about the existence of f(1), understanding continuity helps in analyzing the behavior of the function around that point, which can be important for further questions regarding limits or derivatives.