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Chapter 4: Problem 5

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{5 / 4} $$

Short Answer

Expert verified
The general antiderivative is \( F(x) = \frac{4}{9}x^{9/4} + C \).

Step by step solution

01

Identify the Function to Integrate

The function given is \( f(x) = x^{5/4} \). We want to find its antiderivative.
02

Apply the Power Rule for Integration

The power rule for integration states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \) where \( n eq -1 \). Here, \( n = \frac{5}{4} \).
03

Calculate the Antiderivative

Using the power rule, increase the exponent by one: \( \frac{5}{4} + 1 = \frac{9}{4} \). Now apply the rule: the antiderivative of \( f(x) = x^{5/4} \) is \( \frac{x^{9/4}}{9/4} + C \).
04

Simplify the Expression

Simplify the expression obtained: \( \frac{x^{9/4}}{9/4} \) can be rewritten as \( \frac{4}{9}x^{9/4} + C \) by multiplying the numerator by the reciprocal of the fraction in the denominator.

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Key Concepts

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

Antiderivatives
In calculus, finding an antiderivative means determining a function whose derivative gives the original function. This process is also known as integration, and it's the reverse operation of differentiation. When we talk about finding the general antiderivative of a function, we refer to a family of functions, all differing by a constant, often represented as \( C \).

  • For an antiderivative of a function \( f(x) \), you may encounter it expressed as \( F(x) + C \).
  • The constant \( C \) represents any fixed number since when you take a derivative, constants vanish.
Understanding antiderivatives is crucial because it forms the basis of solving differential equations, calculating definite integrals, and understanding the accumulation of quantities represented by integrals.
Power Rule for Integration
The Power Rule for Integration is a straightforward guideline used to find antiderivatives of power functions, which are of the form \( x^n \). This rule allows you to integrate without needing more complex techniques.

  • According to this rule, if you have a function \( x^n \), its antiderivative is \( \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
  • This rule simply involves increasing the exponent by one, then dividing by the new exponent.
This simplicity is what makes the power rule essential for introductory calculus problems, especially when direct application is possible, just like in our given problem with the function \( f(x) = x^{5/4} \). The antiderivative is found by raising the exponent by one from 5/4 to 9/4 and then dividing by this new exponent.
Exponents in Integration
When dealing with exponents in integration, it's essential to carefully manage the arithmetic involving fractional (or even negative) exponents. This is key when applying the power rule and simplifying the results.

  • Fractional exponents arise frequently, especially in roots and polynomial expressions. Here, we deal with \( x^{5/4} \), a common kind of fractional exponent.
  • We must handle these exponents carefully: when incrementing the exponent by one, ensure precision in adding fractions (e.g., \( \frac{5}{4} + \frac{4}{4} = \frac{9}{4} \)).
Once the power is increased, simplifying the coefficients involved in division, such as checking that \( \frac{x^{9/4}}{9/4} \) becomes \( \frac{4}{9} x^{9/4} \), is crucial for clarity and accuracy. As such, correctly handling exponents and their arithmetic paves the way for successful integration.

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Most popular questions from this chapter

Suppose that every customer order taken by the XYZ Company requires exacty 5 hours of labor for handling the paperwork; this length of time is fixed and does not vary from lot to lot. The total number of hours \(y\) required to manufacture and sell a lot of size \(x\) would then be \(y=(\) number of hours to produce a lot of size \(x)+5\) Some data on XYZ's bookcases are given in the following table. $$ \begin{array}{ccc} \hline & & \text { Total Labor } \\ \text { Order } & \text { Lot Size } x & \text { Hours } y \\ \hline 1 & 11 & 38 \\ 2 & 16 & 52 \\ 3 & 8 & 29 \\ 4 & 7 & 25 \\ 5 & 10 & 38 \\ \hline \end{array} $$ (a) From the description of the problem, the least-squares line should have 5 as its \(y\) -intercept. Find a formula for the value of the slope \(b\) that minimizes the sum of squares $$ S=\sum_{i=1}^{n}\left[y_{i}-\left(5+b x_{i}\right)\right]^{2} $$ (b) Use this formula to estimate the slope \(b\). (c) Use your least-squares line to predict the total number of labor hours to produce a lot consisting of 15 bookcases.

An object thrown from the edge of a 42 -foot cliff follows the path given by \(y=-\frac{2 x^{2}}{25}+x+42\) (Figure 10\()\). An observer stands 3 feet from the bottom of the cliff. (a) Find the position of the object when it is closest to the observer. (b) Find the position of the object when it is farthest from the observer.

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d s}{d t}=16 t^{2}+4 t-1 ; s=100 \text { at } t=0 $$

A car is stationary at a toll booth. Eighteen minutes later at a point 20 miles down the road the car is clocked at 60 miles per hour. Sketch a possible graph of \(v\) versus \(t\). Sketch a possible graph of the distance traveled \(s\) against \(t .\) Use the Mean Value Theorem to show that the car must have exceeded the 60 mile per hour speed limit at some time after leaving the toll booth, but before the car was clocked at 60 miles per hour.

If the brakes of a car, when fully applied, produce a constant deceleration of 11 feet per second per second, what is the shortest distance in which the car can be braked to a halt from a speed of 60 miles per hour?
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