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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 4.7.7a
Chapter 4, Problem 4.7.7a

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

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1
Rewrite the given function in a form that is easier to integrate. Recall that \(\sqrt{x} = x^{1/2}\). So, the function becomes \(\frac{3}{2} x^{1/2}\).
Use the power rule for integration, which states that for any real number \(n \neq -1\), the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Apply the power rule to \(\frac{3}{2} x^{1/2}\). Increase the exponent by 1: \(\frac{1}{2} + 1 = \frac{3}{2}\). Then divide by the new exponent: \(\frac{3}{2} \cdot \frac{x^{3/2}}{3/2}\).
Simplify the expression by multiplying \(\frac{3}{2}\) by the reciprocal of \(\frac{3}{2}\), which will simplify the coefficient.
Add the constant of integration \(C\) to the antiderivative. To check your answer, differentiate your result and verify that you get back the original function \(\frac{3}{2} x^{1/2}\).

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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 often expressed using the integral symbol without limits. Finding antiderivatives helps solve problems involving accumulation and area.
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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule for integration states that the integral of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for integrating polynomial and root functions by rewriting roots as fractional exponents and applying the formula.
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Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This verification step ensures the correctness of the antiderivative and reinforces understanding of the relationship between differentiation and integration.
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Related Practice
Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sec²x − 2tan x, −π/2 < x < π/2

Textbook Question

Theory and Examples


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


Let f(x) = (x − 2)²ᐟ³.


a. Does f′(2) exist?

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.

csc x cot x

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Textbook Question

Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫tanθ sec²θ dθ = sec³θ / 3 + C

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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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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.