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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 4.7.6b
Chapter 4, Problem 4.7.6b

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
1 / 2x³

Verified step by step guidance
1
Identify the function to find the antiderivative of, which is \(\frac{1}{2}x^{3}\).
Recall the power rule for antiderivatives: for \(x^{n}\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration and \(n \neq -1\).
Apply the power rule to \(x^{3}\): increase the exponent by 1 to get 4, then divide by the new exponent, so the antiderivative of \(x^{3}\) is \(\frac{x^{4}}{4} + C\).
Since the original function is multiplied by \(\frac{1}{2}\), multiply the antiderivative by \(\frac{1}{2}\) to get \(\frac{1}{2} \cdot \frac{x^{4}}{4} + C\).
Simplify the expression to write the antiderivative as \(\frac{x^{4}}{8} + C\). Remember to verify your answer by differentiating it to check if you get back the original function.

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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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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule for integration states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for finding antiderivatives of polynomial functions like x³.
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Power Rule for Indefinite Integrals

Verification by Differentiation

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

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1/(3³√x)

1
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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

114. Parabolas

b. When is the parabola concave up? Concave down?

Textbook Question

Identifying Extrema


In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).


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b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

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