Problem 96 - Algebra and Trigonometry

Question

If \(f(x)=4 x^{2}-5 x-2,\) find $$ \frac{f(x+h)-f(x)}{h}, h \neq 0 $$ and simplify.

Step-by-Step Solution

Answer

The simplified difference quotient is \(8x + 4h - 5\).

Step 1: Replace \(x\) with \(x+h\) in \(f(x)\)

Substitute \(x\) with \(x+h\) in the function \(f(x)=4 x^{2}-5 x-2\) to find \(f(x+h) = 4 (x+h)^{2}-5 (x+h)-2\).

Step 2: Simplify \(f(x+h)\)

Expand and simplify \(f(x+h)\) to give \(f(x+h) = 4x^2 + 8hx + 4h^2 - 5x - 5h - 2\).

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

The formula for the difference quotient \(\frac{f(x+h)-f(x)}{h}\) is now substituted with \(f(x)=4 x^{2}-5 x-2\) and \(f(x+h) = 4x^2 + 8hx + 4h^2 - 5x - 5h - 2\) to form: \(\frac{4x^2 + 8hx + 4h^2 - 5x - 5h - 2 - (4x^2 - 5x - 2)}{h}\)

Step 4: Simplify the difference quotient

The equation simplifies to: \(\frac{8hx + 4h^2 - 5h}{h}\)

Step 5: Cancel common factors

Cancel the common factor of \(h\) in the numerator and the denominator to give the final simplified difference quotient: \(8x + 4h - 5\)

Key Concepts

Polynomial Functions

Polynomial functions are a type of mathematical expression that involves sums of powers of variables, each multiplied by coefficients. These functions often take the form:

• General Form: \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)
• Example: \( f(x) = 4x^2 - 5x - 2 \), where the terms \( 4x^2 \), \( -5x \), and \( -2 \) represent parts of the polynomial.

In our example, the polynomial involves quadratic and linear terms. The highest power, 2 here, is the degree of the polynomial, indicating its leading behavior as we move away from the origin.
This makes the function a quadratic polynomial, which can model various phenomena such as projectile motion or area calculations. Understanding these functions helps in analyzing the patterns or relationships expressed through their coefficients and terms.

Derivatives

Derivatives are foundational in calculus, representing the rate of change of a function with respect to its variable. In simpler terms, they show how a function changes as its input changes.

• The derivative of a constant function is 0.
• The derivative of \( x^n \) is \( nx^{n-1} \).

To find a function's derivative, especially with polynomial functions, we often use the difference quotient:\[\frac{f(x+h)-f(x)}{h}\]This measures the average rate of change of \( f(x) \) over a small interval \( h \), approaching zero. Our solution simplifies this to find the expression describing how quickly \( f(x) \) is changing. In the polynomial example, simplifying the difference quotient and taking its limit as \( h \to 0 \) reveals the derivative, guiding us through understanding instantaneous changes or slope at any point.

Function Notation

Function notation is a way to represent mathematical functions in a concise and readable form. It emphasizes the relationship between inputs and outputs in mathematical functions.
For example, if we have \( f(x) = 4x^2 - 5x - 2 \), \( f(x) \) denotes that the input \( x \) is applied through the function process to produce an output. This notation helps:

• Clearly convey what operations are happening to the input.
• Represent transformations of the input through specific expressions.

Using function notation also aids in understanding substitution and evaluation important for tasks like finding difference quotients or derivatives.
In operations like \( f(x+h) \), substituting \( x+h \) informs us about the function's behavior as the input shifts by a small amount \( h \), crucial for derivative concepts and analyzing changes in functions accurately.