Problem 23 - Calculus Early Transcendentals

Question

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x)=x^{3}-3 x+5\)

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 3x^2 - 3\), and the domain of both the function and its derivative is all real numbers, \(x \in \mathbb{R}\).

Step 1: Understand the Function

We start with the given function \(f(x) = x^3 - 3x + 5\). This is a polynomial function of degree 3.

Step 2: Apply the Definition of the Derivative

According to the definition of the derivative, \(f'(x)\) can be found using the limit: \[f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\]

Step 3: Substitute the Function into the Derivative Formula

First, calculate \(f(x+h)\):\[f(x+h) = (x+h)^3 - 3(x+h) + 5\]Expanding \((x+h)^3\) gives:\[x^3 + 3x^2h + 3xh^2 + h^3\]Thus,\[f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3 - 3x - 3h + 5\]

Step 4: Calculate f(x+h) - f(x)

Subtract \(f(x)\) from \(f(x+h)\):\[f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3 - 3x - 3h + 5) - (x^3 - 3x + 5)\]This simplifies to:\[3x^2h + 3xh^2 + h^3 - 3h\]

Step 5: Divide by h and Simplify

Divide the expression by \(h\):\[\frac{{3x^2h + 3xh^2 + h^3 - 3h}}{h} = 3x^2 + 3xh + h^2 - 3\]Cancel \(h\) to simplify the expression.

Step 6: Evaluate the Limit

Apply the limit as \(h\) approaches 0:\[f'(x) = \lim_{{h \to 0}}(3x^2 + 3xh + h^2 - 3) = 3x^2 - 3\]

Step 7: Determine the Domain of f(x)

The domain of \(f(x) = x^3 - 3x + 5\) is all real numbers, \(x \in \mathbb{R}\), because polynomials are defined for all real numbers.

Step 8: Determine the Domain of f'(x)

The derivative \(f'(x) = 3x^2 - 3\) is also a polynomial, so its domain is also all real numbers, \(x \in \mathbb{R}\).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's an equation of the form

• The typical form is: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]
• where \(n\) is a non-negative integer and \(a_n, a_{n-1}, \ldots, a_0\) are constants.
• The highest power of \(x\) determines the degree of the polynomial.

For example, in the function \(f(x) = x^3 - 3x + 5\), we have a third-degree polynomial. The leading term \(x^3\) indicates it is a cubic function, with the highest exponent being 3. This particular function includes both a cube term and a linear term, making it a combination of simple polynomial forms. Polynomial functions are very versatile and appear in numerous applications such as physics, economics, and engineering.

Limit Process

The limit process is a fundamental concept in calculus used to find instantaneous rates of change or slopes of curves. It involves evaluating the behavior of a function as it approaches a particular point. In derivative calculations, limits help us define the exact "slope" of a curve at a specific point.

• For derivatives, the limit is used in the definition:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]
• The \(h\) in the formula represents a very small change in \(x\).
• As \(h\) gets closer to 0, the fraction typically approaches a single value—this value is the derivative.

This process allows us to calculate the slope of the tangent line to the curve at any point \(x\). The limit is essential in obtaining precise values when finite differences alone are insufficient or impossible to compute accurately. Understanding limits deeply provides a foundation for advancing in calculus and analyzing functions deeply.

Domain of a Function

The domain of a function refers to all the possible input values (\(x\))- or arguments- that the function can accept without leading to any inconsistencies or undefined behaviors. In the context of polynomial functions, determining the domain is usually straightforward.

• In polynomial functions, like \(f(x) = x^3 - 3x + 5\), the domain is all real numbers.
• This means any real number can replace \(x\) without causing issues.
• Polynomials do not have restrictions commonly found in other functions like divisions by zero, or square roots of negative numbers.

The simplicity of calculating the domain for polynomials is due to their smooth and continuous nature. Unlike rational functions or radical functions, polynomial functions cover the entire set of real numbers without interruption, making them very approachable topics in early calculus studies.

Derivative Calculation

Derivative calculation is not only a critical concept but also a powerful tool in calculus used to determine how a function changes at any given point. The derivative of a function at a specific input value tells us the rate of change, or the slope of the tangent line to the function at that point. Calculations generally follow these steps:

• Begin with the definition of the derivative:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]
• Substitute \(f(x+h)\) and \(f(x)\) into the formula.
• Simplify the expression, factor and cancel terms as necessary.
• Finally, evaluate the limit by setting \(h \to 0\).

For our example function \(f(x) = x^3 - 3x + 5\), following these steps yields the derivative \(f'(x) = 3x^2 - 3\). This provides valuable information about the slope of the function at any point, and the derivative itself is also a polynomial with a domain of all real numbers. Calculating derivatives helps in various areas such as physics for analyzing motion, optimizing economic models, or even in biology to understand population changes.