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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.4b
Chapter 4, Problem 4.7.4b

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁻³/2 + x²

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1
Identify the function to find the antiderivative of: \(f(x) = x^{-\frac{3}{2}} + x^{2}\).
Recall the power rule for antiderivatives: for \(f(x) = x^{n}\), an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Apply the power rule to each term separately: for \(x^{-\frac{3}{2}}\), add 1 to the exponent to get \(-\frac{3}{2} + 1 = -\frac{1}{2}\), so the antiderivative is \(\frac{x^{-\frac{1}{2}}}{-\frac{1}{2}}\).
For the second term \(x^{2}\), add 1 to the exponent to get \(2 + 1 = 3\), so the antiderivative is \(\frac{x^{3}}{3}\).
Combine the antiderivatives of both terms and add the constant of integration \(C\): \(F(x) = \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + \frac{x^{3}}{3} + C\).

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Key Concepts

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

Antiderivative (Indefinite Integral)

An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed with an arbitrary constant since differentiation of a constant is zero.
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Power Rule for Integration

The power rule states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is fundamental for integrating polynomial terms and applies directly to terms like x^2 and x^(-3/2).
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Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps identify any mistakes in the integration process.
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Related Practice
Textbook Question

Identifying Extrema


In Exercises 19–40:


b. Identify the function’s local extreme values, if any, saying where they occur.


f(θ) = 3θ² − 4θ³

Textbook Question

[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.

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b. Find the domain of V for the problem situation and graph V over this domain.

Textbook Question

Identifying Extrema


In Exercises 19–40:


b. Identify the function’s local extreme values, if any, saying where they occur.


f(r) = 3r³ + 16r

Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


b. Does f'(3) exist?

Textbook Question

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:



b. On what open intervals is f increasing or decreasing?


f′(x) = (x − 1)(x + 2)(x − 3)

Textbook Question

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


b. On what open intervals is f increasing or decreasing?


f′(x) = 1− 4/x², x ≠ 0