Problem 1 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The derivative of the right-hand side of the equation is \(x^2 - 4\), which is equal to the integrand of the left-hand side, verifying the given integral equation.

Step 1: Recognize the Problem Type

The exercise is asking to verify an integral by using the Fundamental Theorem of Calculus, which basically says that the integral of a function f(x) from a to x is the anti-derivative F(x) of f. Therefore, to verify the given integral, the derivative of the right side of the equation, which is F(x), needs to be found and compared to f(x), which is (x-2)(x+2).

Step 2: Find the Derivative of the Right Side

The right-hand side of the equation is the function F(x) = (1/3)*x^3 - 4x + C. The derivative of this function, denoted as F'(x), can be found using the power rule for derivatives, which states that the derivative of x^n is n*x^(n-1), and the constant rule, which says the derivative of a constant is zero. Therefore, F'(x) = (1/3)*3*x^(3-1) - 4 + 0 = x^2 - 4.

Step 3: Compare F'(x) with f(x)

Now, the derivative of the right-hand side, F'(x) = x^2 - 4, is compared to the integrand of the left-hand side, f(x) = (x-2)(x+2). Notice that the expression (x-2)(x+2) is a difference of squares and can be simplified to x^2 - 2^2 = x^2 - 4. Therefore, F'(x) is indeed equal to f(x).

Key Concepts

Understanding Derivative

The concept of a derivative is of central importance in calculus. When we talk about the derivative of a function, we mean the rate at which the function's value changes as its input changes. Think of it as the slope or steepness of the curve of the function at any given point.

To find a derivative, you often use specific rules, like the power rule, which simplifies the process of differentiation.

• A derivative tells us how fast a function's output changes.
• It is used to calculate tangents, and to optimize problems by finding maximum and minimum values.

In calculus, derivatives can be represented in notation as \( f'(x) \) or \( \frac{dy}{dx} \), where \( y \) is a function of \( x \). Understanding the basic concept will help in solving more advanced problems involving rates of change or motion.

Exploring Anti-Derivative

An anti-derivative, also known as an indefinite integral, is essentially the reverse of a derivative.

While the derivative of a function tells us about changes in values, the anti-derivative helps us find the original function given its derivative. It "undoes" the process of differentiation.

To compute an anti-derivative, you integrate a function. However, unlike differentiation, there is often no one single way to find an anti-derivative. Integrals can often be adjusted by adding a constant (C), since the derivative of a constant is zero.

• Understanding anti-derivatives is crucial for solving problems involving area under curves.
• It helps in recovering a function from its rate of change.

Integration is not just reversing differentiation, it allows us to solve real-world problems involving cumulative totals or averages.

The Power Rule for Derivatives

The power rule is a simple yet vital tool for finding derivatives. The rule applies to any power of \( x \) and makes the tedious process of differentiation nearly instant.

The power rule states that if you have a term \( x^n \), then its derivative is calculated as \( n \times x^{n-1} \). This is applicable for any real number exponent.

• For example, if \( f(x) = x^3 \), then using the power rule, \( f'(x) = 3x^2 \).
• If \( f(x) = x^{-2} \), then \( f'(x) = -2x^{-3} \).

This rule is particularly useful because it simplifies the process of differentiation. It helps to quickly derive values in polynomial forms, thereby saving time while solving complex calculus problems. Using the power rule can reduce multi-step calculations into a single, easy-to-follow step.