Problem 69 - Applied Calculus

Question

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h} .\) (Assume \(\left.h \neq 0 .\right)\) (See instructions on previous page.) $$ \begin{array}{l} f(x)=x^{3} \\ \text { [Hint: Use } \left.(x+h)^{3}=x^{3}+3 x^{2} h+3 x h^{2}+h^{3} .\right] \end{array} $$

Step-by-Step Solution

Answer

The simplified expression is \(3x^2 + 3xh + h^2\).

Step 1: Apply the given hint to calculate f(x+h)

Using the hint, expand \((x+h)^3\) as follows: \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\). Substitute this into \(f(x+h)\):\[f(x+h) = (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3.\]

Step 2: Compute f(x+h) - f(x)

Now, calculate \(f(x+h) - f(x)\) by substituting \(f(x+h)\) and \(f(x)\): \[f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3) - x^3.\] Simplify by cancelling out \(x^3\): \[f(x+h) - f(x) = 3x^2h + 3xh^2 + h^3.\]

Step 3: Divide by h

Divide the expression \(3x^2h + 3xh^2 + h^3\) by \(h\): \[\frac{f(x+h) - f(x)}{h} = \frac{3x^2h + 3xh^2 + h^3}{h}.\] Simplify by canceling \(h\) in each term: \[\frac{f(x+h) - f(x)}{h} = 3x^2 + 3xh + h^2.\]

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to positive integer powers. The general form of a polynomial function is \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients and \( n \) is a non-negative integer indicating the degree of the polynomial. In this exercise, the function \( f(x) = x^3 \) is a simple cubic polynomial, as its highest power is 3.
Polynomials offer great flexibility in modeling various real-world phenomena due to their ability to easily represent curves and shapes. The operations on them are straightforward, involving addition, subtraction, multiplication, and division, as demonstrated during the expansion of \((x+h)^3\).
Understanding the behavior of polynomial functions is crucial, especially when dealing with their changes, as it directly connects to the derivative's concept.

Derivatives

Derivatives represent the rate at which a function changes concerning its input. Essentially, the derivative is a function's instantaneous rate of change or the slope of the tangent line to the function's graph at any given point.
The expression \( \frac{f(x+h) - f(x)}{h} \) is part of the difference quotient, which approximates the derivative of \( f(x) \) when \( h \) approaches zero. For polynomial functions such as \( f(x) = x^3 \), calculating the derivative known as \( f'(x) \) gives us an idea of its growth or decline at different points along its curve.
In this exercise, the process of finding the derivative of the polynomial \( x^3 \) simplifies into understanding how changes in \( x \) affect the function's value. The result \( 3x^2 \) reflects the derivative, indicating that for every unit increase in \( x \), the function's value increases by approximately \( 3x^2 \). This insight is crucial for analyzing motion, economics, and other fields requiring precise change measurements.

Limit Process

The limit process is a fundamental concept in calculus, allowing us to evaluate a function's behavior as its input approaches a particular value. In the context of derivatives, the limit process is crucial for finding a function's exact rate of change.
The expression \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \) defines the derivative rigorously. It implies that as \( h \), the small increment, approaches zero, the difference quotient turns into the precise derivative of the function \( f(x) \).
The limit process is what enables us to transition from the difference quotient to the derivative. In our example, simplifying \( \frac{3x^2h + 3xh^2 + h^3}{h} \) results in \( 3x^2 + 3xh + h^2 \). Taking the limit as \( h \to 0 \) yields \( 3x^2 \), the exact derivative. This step helps us precisely quantify change, such as the slope in geometrical terms, or growth rates in practical applications, making the limit process a foundational tool in calculus.