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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 4.7.1c
Chapter 4, Problem 4.7.1c

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

Verified step by step guidance
1
Identify the function to find the antiderivative of: \(x^{2} - 2x + 1\).
Recall that the antiderivative (indefinite integral) of a function \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\). We will integrate each term separately.
Use the power rule for integration: For any term \(x^{n}\), the antiderivative is \(\frac{x^{n+1}}{n+1}\), provided \(n \neq -1\).
Integrate each term: \(\int x^{2} \, dx = \frac{x^{3}}{3}\), \(\int (-2x) \, dx = -2 \cdot \frac{x^{2}}{2} = -x^{2}\), and \(\int 1 \, dx = x\).
Combine the results and add the constant of integration \(C\): \(F(x) = \frac{x^{3}}{3} - x^{2} + x + C\). This is the general antiderivative of the given 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 often expressed with an arbitrary constant, C, 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) + C. This rule is essential for integrating polynomial terms like x² and x.
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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.

√x + 1/√x

Textbook Question

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[−f(x)] dx

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Textbook Question

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.

c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)

Textbook Question

Analyzing Functions from Derivatives


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

c. At what points, if any, does f assume local maximum or minimum values?


f′(x) = (x − 1)(x + 2)(x − 3)

Textbook Question

Dependence on Initial Point

8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations

c. x_0=2

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.

sin πx − 3sin 3x

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