Problem 20 - Calculus of a Single Variable

Question

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=x^{3}+x^{2} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^3 + x^2\) is \(3x^2 + 2x\).

Step 1: Write down the formula for the limit definition of a derivative

The formula for the limit definition of a derivative is \[\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\] We can start by plugging \(f(x) = x^3 + x^2\) into the formula.

Step 2: Calculate f(x + h)

We start by replacing \(x\) in the function with \(x + h\). So \(f(x + h) = (x + h)^3 + (x + h)^2 = x^3 + 3x^2h + 3xh^2 + h^3 + x^2 + 2xh + h^2 \)

Step 3: Substitute f(x + h) and f(x) into the difference quotient

Substitute \(f(x + h)\) and \(f(x)\) into the difference quotient: \[\lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3 + x^2 + 2xh + h^2) - (x^3 + x^2)}{h}\] Simplify this to \[\lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 + 2xh + h^2}{h}\]

Step 4: Simplify the Equation

Factor out an \(h\) from each term in the numerator and simplify the expression: \[\lim_{h \to 0} \left(3x^2+ 3xh + h^2 + 2x + h \right)\]

Step 5: Taking the limit

Now take the limit as \(h\) approaches 0. All the terms involving \(h\) will tend to zero, simplifying our expression to: \(3x^2 + 2x\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to non-negative integer powers and coefficients. These functions take the form:

• Linear: of the form \(f(x) = ax + b\)
• Quadratic: \(f(x) = ax^2 + bx + c\)
• Cubic: \(f(x) = ax^3 + bx^2 + cx + d\)
• And so on for higher degrees.

In the given exercise, the function \(f(x) = x^3 + x^2\) is a polynomial of degree 3, which is a cubic polynomial. Understanding how these functions behave informs us about their derivatives. Cubic functions, for instance, can have up to two turning points, which are revealed through their derivatives.
Analyzing these roots and behavior is key in calculus, helping to solve complex equations and understand curves better.

Calculus

Calculus is the branch of mathematics dealing with rates of change and accumulation of quantities. It has two main branches:

• Differential calculus: focuses on the concept of the derivative, which represents a rate of change.
• Integral calculus: concerns accumulation and the areas under or between curves.

In the limit definition of derivative, we focus on differential calculus. The derivative gives the slope of a function at any point. For polynomial functions, the derivative allows us to predict changes and behavior of the curve as you move along the x-axis.
This foundational concept in calculus simplifies understanding how a function behaves, enabling modeling in physics, economics, medicine, and more.

Differentiation

Differentiation is the process of finding a derivative, which reveals the rate of change of a function. In the problem given, we apply this by using the limit definition: \[\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]Steps involved include:

• Substituting the function \(f(x)\) into the formula.
• Expanding \(f(x+h)\) using algebra.
• Simplifying the difference quotient.
• Taking the limit as \(h\) approaches zero.

This mechanical process results in the derivative. For example, finding \(f'(x) = 3x^2 + 2x\) means at any \(x\) value, this polynomial function's slope can be determined.
This aspect is crucial in studying the behavior of materials, financial predictions, and engineering designs.

Limit Process

The limit process is a critical path to defining continuity and derivatives in calculus. Limits help understand behavior "as you approach" a certain point. Essentially, as the difference \(h\) becomes infinitesimally small, it shows how \(f(x+h)\) approximates \(f(x)\).To comprehend this better:

• A limit helps manage values that may become undefined if approached directly, like division by zero.
• In derivative calculations, this allows us to evaluate instantaneous rates of change.
• Using limits, we handle infinite processes in a clear and concise manner.

For the given problem, the derivatives' formula relies on letting \(h\) tend to zero, refining the difference quotient into the precise slope of the function at any \(x\) point. This fundamental limit approach acts as a bridge from algebraic expressions to the nuanced analysis in calculus.