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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 4.7.65b
Chapter 4, Problem 4.7.65b

Checking Antiderivative Formulas


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


∫3(2x + 1)² dx = (2x + 1)³ + C

Verified step by step guidance
1
Identify the integral given: \(\int 3(2x + 1)^2 \, dx\).
Recognize that the integrand is a composite function, so consider using substitution. Let \(u = 2x + 1\).
Compute the derivative of \(u\) with respect to \(x\): \(\frac{du}{dx} = 2\), which implies \(dx = \frac{du}{2}\).
Rewrite the integral in terms of \(u\): \(\int 3u^2 \cdot \frac{du}{2} = \frac{3}{2} \int u^2 \, du\).
Integrate \(u^2\) with respect to \(u\): \(\int u^2 \, du = \frac{u^3}{3} + C\). Multiply by \(\frac{3}{2}\) to get \(\frac{3}{2} \cdot \frac{u^3}{3} + C = \frac{u^3}{2} + C\). Finally, substitute back \(u = 2x + 1\) to express the antiderivative in terms of \(x\).

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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 f(x) is another function F(x) whose derivative is f(x). It is represented by the indefinite integral ∫f(x) dx = F(x) + C, where C is an arbitrary constant. Understanding this helps verify if a given formula correctly reverses differentiation.
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Introduction to Indefinite Integrals

Chain Rule and Its Reverse (Substitution Method)

The chain rule in differentiation handles composite functions. Its reverse, used in integration, often requires substitution to simplify the integral. Recognizing when to apply substitution is key to correctly finding antiderivatives of functions like (2x + 1)².
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Intro to the Chain Rule

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. When integrating expressions like (2x + 1)², the power rule applies after appropriate substitution, ensuring the integral is computed correctly rather than just raising the inner function to a higher power.
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Power Rule for Indefinite Integrals
Related Practice
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

Theory and Examples


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

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.

4 sec 3x tan 3x

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Textbook Question

38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 1?

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(x − 1)

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.

x⁷

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