Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=3 x^{2}-\pi x $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = x^3 - \frac{\pi}{2}x^2 + C \).

Step 1: Understand the Problem

We are asked to find the general antiderivative of the function \( f(x) = 3x^2 - \pi x \). The goal is to find a function \( F(x) \) such that the derivative \( F'(x) = f(x) \).

Step 2: Apply Antiderivative Rules

Recall that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \). Also, the antiderivative of a constant \( a \) multiplied by \( x \) is \( \frac{a}{2} x^2 \). We will apply these rules to each term in \( 3x^2 - \pi x \).

Step 3: Find Antiderivative of \(3x^2\)

For the term \( 3x^2 \), the antiderivative is obtained by using the rule \( \int x^n \,dx = \frac{x^{n+1}}{n+1} + C \). Therefore, \( \int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3 \).

Step 4: Find Antiderivative of \(-\pi x\)

For the term \(-\pi x\), the antiderivative is found by applying the rule for \( ax^1 \). Thus, \( \int -\pi x \, dx = -\pi \cdot \frac{x^2}{2} = -\frac{\pi}{2} x^2 \).

Step 5: Combine Antiderivatives

Combine the antiderivatives of each term to get \( F(x) = x^3 - \frac{\pi}{2}x^2 \). Add the constant \( C \) to represent the general antiderivative.Therefore, \( F(x) = x^3 - \frac{\pi}{2}x^2 + C \).

Step 6: Write the General Antiderivative

The general antiderivative of the function \( f(x) = 3x^2 - \pi x \) is given by:\[ F(x) = x^3 - \frac{\pi}{2}x^2 + C \]

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on studying change. It's divided into two main parts: differentiation and integration. Differentiation is concerned with finding the rate at which a quantity changes, while integration is about accumulating quantities. Together, they form the fundamental theorem of calculus which connects these two concepts.
In simpler terms, calculus helps us understand how things develop over time, such as speed in physics or growth rates in biology. It's an essential tool for scientists and engineers.

• Differentiation: Finding rates of change.
• Integration: Calculating accumulated change.

These tools allow us to solve complex problems involving curves, optimize functions, and find areas under curves, all of which are vital for real-life applications.

Integration

Integration is the process of finding the integral of a function. It's the reverse operation of differentiation. In basic terms, if differentiation tells us how a function's output varies with its input, integration helps us find the original function from its rate of change.
Integration is used to compute things like areas under curves or total accumulated values over time. There are two main types of integrals:

• Definite Integrals: Give a specific numerical value, often representing areas.
• Indefinite Integrals: Represent a family of functions and include a constant of integration, typically denoted as "C".

Understanding integration is vital because it helps in reversing the process of differentiation. It broadens our ability to solve problems involving continuous change.

Indefinite Integrals

Indefinite integrals, unlike definite integrals, do not have limits of integration. They represent a family of functions whose derivative gives the original function. Essentially, an indefinite integral is the reverse of a derivative.
The result of an indefinite integral is called the antiderivative or integral of the function, denoted by a capital letter such as \( F(x) \). An indefinite integral always includes a constant \( C \) because differentiating \( F(x) + C \) with respect to \( x \) gives \( f(x) \), no matter what value \( C \) assumes.
Key formula:

• For a function \( f(x) = x^n \), the antiderivative is \( F(x) = \frac{x^{n+1}}{n+1} + C \).

This formula helps in finding the general form of the antiderivative, which is crucial in solving calculus problems.

Polynomial Functions

Polynomial functions are expressions that involve sums of powers of variables with coefficients. The most simple ones are linear, but they can rise in complexity with higher degree terms. Polynomial functions are smooth, continuous, and easy to differentiate and integrate.
They play a significant role in calculus because they serve as building blocks for more complex functions and are often used in approximation through Taylor series.

• Standard Form: \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)
• Degree: The highest power of the variable, which determines the function's behavior.

Finding the antiderivative of polynomials often involves integrating each term separately, which is both straightforward and vital. This makes them perfect for practicing and understanding the basic operations of calculus.