Book cover for Calculus Early Transcendental Functions

Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

ISBN #9781285774770

6th Edition

8,973 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the intimate connection between differentiation and integration, highlighting that integration is essentially the inverse process of differentiation. Topics covered include finding antiderivatives using basic integration rules, representing the family of antiderivatives by including a constant of integration, and solving differential equations to find both general and particular solutions. Additionally, techniques like rewriting integrals to simplify the integration process and the use of sigma notation set the stage for understanding area calculations and the Fundamental Theorem of Calculus.

Learning Objectives

1

Write the general solution of a differential equation using antiderivatives and indefinite integral notation.

2

Apply basic integration rules such as the power rule, constant multiple rule, and sum rule to find antiderivatives.

3

Determine particular solutions of differential equations by incorporating initial conditions.

4

Rewrite integrals into forms that simplify the integration process through algebraic or trigonometric manipulations.

5

Understand the inverse relationship between differentiation and integration as highlighted by the Fundamental Theorem of Calculus.

Key Concepts

CONCEPT

DEFINITION

Antiderivative

A function F(x) whose derivative is f(x); there are infinitely many antiderivatives for a function, differing by a constant.

Indefinite Integral

The notation ∫f(x) dx representing the family of all antiderivatives of f(x), expressed with an added constant of integration, C.

Constant of Integration

An arbitrary constant, represented by C, added to the antiderivative to account for all possible vertical translations of the antiderivative function.

Differential Equation

An equation that involves a function and its derivatives; solving it often involves finding its antiderivative or general solution.

Initial Condition

A specified value of the function for a certain x, used to determine the particular solution from the general antiderivative.

Basic Integration Rules

Rules such as the power rule, constant multiple rule, and sum/difference rule, which are applied to simplify and compute antiderivatives.

Rewriting Before Integrating

The process of algebraically manipulating an integrand, such as rewriting fractions as products or using trigonometric identities, to bring it into a form that fits a basic integration rule.

Example Problems

Example 1

Verify the statement by showing that the derivative of the right side equals the integrand of the left side. $$\int\left(-\frac{6}{x^{4}}\right) d x=\frac{2}{x^{3}}+C$$

Example 2

Verify the statement by showing that the derivative of the right side equals the integrand of the left side. $$\int\left(8 x^{3}+\frac{1}{2 x^{2}}\right) d x=2 x^{4}-\frac{1}{2 x}+C$$

Example 3

Find the general solution of the differential equation and check the result by differentiation. $$\frac{d y}{d t}=9 t^{2}$$

Example 4

Find the general solution of the differential equation and check the result by differentiation. $$\frac{d y}{d t}=5$$

Example 5

Find the general solution of the differential equation and check the result by differentiation. $$\frac{d y}{d x}=x^{3 / 2}$$
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Step-by-Step Explanations

QUESTION

Find the general antiderivative of f(x) = 2x.

STEP-BY-STEP ANSWER:

Step 1: Recognize that the function 2x fits the power rule for integration.
Step 2: Apply the integration rule ∫x^n dx = x^(n+1)/(n+1) for n = 1, so ∫2x dx = 2 * (x^2/2).
Step 3: Simplify to obtain x^2.
Step 4: Include the constant of integration to represent the family of antiderivatives.
Final Answer: x^2 + C.

Antiderivative Example

QUESTION

Solve the differential equation dy/dx = 3x^2 - 1 given that y(2) = 4.

STEP-BY-STEP ANSWER:

Step 1: Integrate the differential equation to obtain the general solution: ∫(3x^2 - 1) dx = x^3 - x + C.
Step 2: Apply the initial condition by substituting x = 2 and y = 4: 4 = 2^3 - 2 + C = 8 - 2 + C.
Step 3: Simplify the equation: 4 = 6 + C, so C = -2.
Final Answer: y = x^3 - x - 2.

Particular Solution Using an Initial Condition

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Common Mistakes

  • Forgetting to add the constant of integration (C) when writing the antiderivative.
  • Incorrect application of the power rule, especially when dealing with n = -1 (which requires the natural logarithm).
  • Attempting to separately integrate the numerator and denominator in a quotient instead of properly rewriting the integrand.
  • Overcomplicating the constant of integration by not recognizing that any constant will suffice.
  • Failing to check the antiderivative by differentiating it to see if it recovers the original function.