The antiderivative of an absolute value function, denoted as |x|, depends on the given interval of x values. In general, the antiderivative of |x| is a piecewise-defined function.
To understand the antiderivative of an absolute value function, let’s consider the two separate cases: when x is greater than or equal to zero, and when x is less than zero.
Table of Contents
- Frequently Asked Questions:
- 1. What is an antiderivative?
- 2. How is the absolute value function defined?
- 3. Why is the antiderivative of an absolute value function piecewise-defined?
- 4. Can the antiderivative of an absolute value function be expressed using absolute value notation?
- 5. How do you find the antiderivative of an absolute value function on a specific interval?
- 6. What is the derivative of the antiderivative of an absolute value function?
- 7. Can the antiderivative of |x| be a linear function?
- 8. Does the antiderivative of |x| have any discontinuities?
- 9. How does the constant of integration affect the antiderivative of an absolute value function?
- 10. Can the antiderivative of an absolute value function have a negative coefficient?
- 11. Are there other methods to find the antiderivative of an absolute value function?
- 12. Can the antiderivative of an absolute value function be visualized graphically?
Case 1: When x ≥ 0
When x is nonnegative, the absolute value function |x| is equal to x itself. Therefore, the antiderivative of |x| in this case is another piecewise-defined function, denoted as ½x² + C, where C is the constant of integration.
Case 2: When x < 0
When x is negative, the absolute value function |x| becomes -x. In this scenario, the antiderivative of |x| is given by -½x² + C, where C represents the constant of integration.
Combining both cases, we obtain the complete antiderivative of an absolute value function as follows:
The antiderivative of an absolute value function is:
½x² + C, if x ≥ 0
-½x² + C, if x < 0. This piecewise-defined function captures the behavior of the absolute value function and allows us to find the antiderivative on different intervals.
Frequently Asked Questions:
1. What is an antiderivative?
An antiderivative, also known as an indefinite integral, represents a function whose derivative would give the original function.
2. How is the absolute value function defined?
The absolute value function |x| is defined as the nonnegative value of x, regardless of its sign; |x| equals x if x is nonnegative, and -x if x is negative.
3. Why is the antiderivative of an absolute value function piecewise-defined?
The antiderivative of an absolute value function is piecewise-defined because its value changes depending on the interval of x. For x greater than or equal to zero, the antiderivative is different from when x is negative.
4. Can the antiderivative of an absolute value function be expressed using absolute value notation?
No, the antiderivative of an absolute value function cannot be expressed solely using absolute value notation. It requires piecewise notation to accurately represent the function on different intervals.
5. How do you find the antiderivative of an absolute value function on a specific interval?
To find the antiderivative on a specific interval, follow the steps of splitting the interval into cases (positive x and negative x) and apply the appropriate formula accordingly.
6. What is the derivative of the antiderivative of an absolute value function?
The derivative of the antiderivative of an absolute value function is the original absolute value function itself.
7. Can the antiderivative of |x| be a linear function?
No, the antiderivative of |x| cannot be a linear function. It is a quadratic function since it contains x² terms.
8. Does the antiderivative of |x| have any discontinuities?
No, the antiderivative of |x| does not have any discontinuities. However, the absolute value function itself does exhibit a discontinuity at x = 0.
9. How does the constant of integration affect the antiderivative of an absolute value function?
The constant of integration, denoted as C, determines the vertical shift of the antiderivative graph. It does not affect the shape or slope of the graph.
10. Can the antiderivative of an absolute value function have a negative coefficient?
Yes, the antiderivative of an absolute value function can have a negative coefficient, as demonstrated in the case when x < 0.
11. Are there other methods to find the antiderivative of an absolute value function?
No, there are no alternative methods to find the antiderivative of an absolute value function. The piecewise-defined formulas provided are the standard and only ways to calculate it.
12. Can the antiderivative of an absolute value function be visualized graphically?
Yes, the antiderivative of an absolute value function can be visualized graphically as a piecewise quadratic function. The graph will have a vertex at the point (0, 0) and different slopes on either side of zero.
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