Textbook Question
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.10b
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/2)x⁻³ᐟ²
Verified step by step guidance1
Identify the function to find the antiderivative of: \(-\frac{1}{2} x^{-\frac{3}{2}}\).
Recall the power rule for antiderivatives: For \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Apply the power rule by increasing the exponent by 1: \(-\frac{3}{2} + 1 = -\frac{1}{2}\).
Write the antiderivative as \(-\frac{1}{2} \cdot \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + C\).
Simplify the expression and add the constant of integration \(C\) to complete the antiderivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivatives (Indefinite Integrals)
An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often using known integral formulas. The result includes a constant of integration since differentiation of a constant is zero.
Recommended video:
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, for any real number n ≠ -1. This rule is essential for integrating functions with variable exponents, including negative and fractional powers, by increasing the exponent by one and dividing by the new exponent.
Recommended video:
04:04Power 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 identify any mistakes in the integration process.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)
Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = (x − 2)²ᐟ³.
b. Show that the only local extreme value of f occurs at x = 2.
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
Theory and Examples
Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.
b. How many local extreme values can f have?
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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