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Chapter 4: Problem 9

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\frac{2}{3} x^{-1 / 3} \quad\) b. \(\frac{1}{3} x^{-2 / 3} \quad\) c. \(-\frac{1}{3} x^{-4 / 3}\)

Short Answer

Expert verified
(a) \( x^{2/3} + C \); (b) \( x^{1/3} + C \); (c) \( x^{-1/3} + C \).

Step by step solution

01

Understanding the Antiderivative

The antiderivative is essentially the reverse process of differentiation. For a given function, finding the antiderivative means finding a function whose derivative is the original function.
02

Using the Power Rule for Antiderivatives

To find the antiderivative of a power function of the form \( ax^n \), we use the formula: \( F(x) = \frac{a}{n+1}x^{n+1} + C \), where \(C\) is the constant of integration.
03

Finding the Antiderivative for (a)

For the function \( \frac{2}{3} x^{-1/3} \), apply the power rule: \[ F(x) = \frac{2/3}{-1/3 + 1} x^{-1/3 + 1} = \frac{2/3}{2/3} x^{2/3} = x^{2/3} + C. \]
04

Finding the Antiderivative for (b)

For the function \( \frac{1}{3} x^{-2/3} \), apply the power rule: \[ F(x) = \frac{1/3}{-2/3 + 1} x^{-2/3 + 1} = \frac{1/3}{1/3} x^{1/3} = x^{1/3} + C. \]
05

Finding the Antiderivative for (c)

For the function \( -\frac{1}{3} x^{-4/3} \), apply the power rule: \[ F(x) = \frac{-1/3}{-4/3 + 1} x^{-4/3 + 1} = \frac{-1/3}{-1/3} x^{-1/3} = x^{-1/3} + C. \]
06

Check by Differentiation for (a)

Differentiate \( F(x) = x^{2/3} + C \): \[ \frac{d}{dx} (x^{2/3}) = \frac{2}{3}x^{-1/3}. \] This matches the original function.
07

Check by Differentiation for (b)

Differentiate \( F(x) = x^{1/3} + C \): \[ \frac{d}{dx} (x^{1/3}) = \frac{1}{3}x^{-2/3}. \] This matches the original function.
08

Check by Differentiation for (c)

Differentiate \( F(x) = x^{-1/3} + C \): \[ \frac{d}{dx} (x^{-1/3}) = -\frac{1}{3}x^{-4/3}. \] This matches the original function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule for Antiderivatives
Finding an antiderivative can sometimes feel like detective work, and the power rule is your trusty magnifying glass. The power rule simplifies the process of integration for functions of the form \( ax^n \). To find the antiderivative, we use the formula:
  • \( F(x) = \frac{a}{n+1} x^{n+1} + C \), where \( C \) is the constant of integration.
This method involves adding one to the exponent \( n \) of the variable \( x \), and then dividing by the new exponent. Don't forget to add \( C \) at the end to account for any constant that might have been lost during differentiation.
For example, to find the antiderivative of \( \frac{2}{3} x^{-1/3} \), we
  • Add 1 to the exponent: -1/3 + 1 = 2/3.
  • Divide the coefficient by the new exponent: \( \frac{2/3}{2/3} \).
  • Write down the result: \( x^{2/3} + C \).
Integration Process
Integration is the process of finding an antiderivative or "undoing" differentiation. It's about identifying a function whose derivative gives you back the function you started with.
Unlike differentiation, which is pretty straightforward, integration can have multiple interpretations due to the constant of integration. In the above examples, the power rule gave us the antiderivatives, but it's important to understand that integration is essentially summing up infinitely small quantities.
This means that:
  • Each function can have many antiderivatives, differing only by a constant.
  • The integral sign \( \int \) represents the process of integration.
  • It's important to become familiar with different types of integrals and techniques to solve them.
Mastering these skills provides a solid foundation for tackling various integrals in calculus.
Differentiation as a Check
Differentiation helps us verify our work by cross-checking an antiderivative. It's the mathematical tool that ensures accuracy. When you differentiate an antiderivative and get back the original function, you know you've found the correct antiderivative.
In practice:
  • After finding an antiderivative, differentiate it to compare with the original function.
  • If the two match, your antiderivative is accurate.
  • If they don't, check your steps, especially the application of the power rule and the constants involved.
For example, differentiating \( F(x) = x^{2/3} + C \) results in \( \frac{2}{3} x^{-1/3} \), which matches the initial function. This confirms that the antiderivative was correctly calculated.
The Constant of Integration
The constant of integration, often symbolized by \( C \), plays a vital role in antiderivatives. It's the reminder that every function has an infinite number of antiderivatives due to constants that can be added.
  • When finding an antiderivative, always append \( C \) at the end.
  • \( C \) represents any constant that may not affect the function's derivative but alters its integral.
  • In applied problems, boundary conditions can help identify the actual value of \( C \), giving a specific solution.
For example, if you know an initial condition like \( F(1) = 5 \), you can solve for \( C \) by substituting into your antiderivative equation. This gives you a tailored solution for that particular problem.

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Most popular questions from this chapter

How we cough a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity \(v\) can be modeled by the equation $$ v=c\left(r_{0}-r\right) r^{2} \mathrm{cm} / \mathrm{sec}, \quad \frac{r_{0}}{2} \leq r \leq r_{0} $$ where \(r_{0}\) is the rest radius of the trachea in centimeters and \(c\) is a positive constant whose value depends in part on the length of the trachea. Show that \(v\) is greatest when \(r=(2 / 3) r_{0},\) that is, when the trachea is about 33\(\%\) contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough. b. Take \(r_{0}\) to be 0.5 and \(c\) to be 1 and graph \(v\) over the interval \(0 \leq r \leq 0.5\) . Compare what you see with the claim that \(v\) isat a maximum when \(r=(2 / 3) r_{0} .\)

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int x^{-3}(x+1) d x $$

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