Problem 63 - College Algebra

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=2 x^{2}+x-1$$

Step-by-Step Solution

Answer

The simplified difference quotient for the given function \(f(x)=2x^2+x-1\) is \(4x+1\).

Step 1: Substitute the Function into the Formula

Substitute \(f(x)=2x^2+x-1\) into the formula \(\frac{f(x+h)-f(x)}{h}\). This gives \[\frac{(2(x+h)^2+(x+h)-1) - (2x^2+x-1)}{h}\]

Step 2: Simplify the Numerator

Simplify the numerator by expanding the terms and combining like terms. This yields \[\frac{2x^2+4hx+2h^2+x+h-1 - (2x^2+x-1)}{h}\]which simplifies to \[\frac{4hx+2h^2+h}{h}\]

Step 3: Cancel Out the Common Factor

Cancel out \(h\) that is common to all terms in the numerator with the \(h\) in the denominator. This leaves \[4x+2h+1\]

Step 4: Evaluate the Limit as h Approaches Zero

Finally, note that in the difference quotient, \(h\) is supposed to approach zero. Therefore, setting \(h=0\), we get \[4x+1\]

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In our example, the function \( f(x) = 2x^2 + x - 1 \) is a polynomial because it consists of terms that are powers of \( x \). Each term follows this general format: \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer exponent. The characteristics of polynomial functions include:

• They can have constants, variables, and exponents combined using addition, subtraction, and multiplication.
• The exponents of the variables must be whole numbers.
• They are infinitely differentiable and continuous everywhere on the real number line.

Understanding polynomial functions is crucial for solving problems like the difference quotient, as they form the basis for the equations we manipulate. In this problem, each term of the polynomial \(2x^2\), \(x\), and \(-1\) contributes to how we compute \(f(x+h)\).

Limit Process

The limit process is a fundamental concept in calculus that allows us to analyze behaviors as something approaches a particular point. In the context of the difference quotient, the limit enables us to find the derivative of a function, which gives the rate of change.

In this problem, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) requires us to consider the behavior as \(h\) approaches zero. It helps determine the slope of the tangent line to the curve at any point \(x\). The steps we often follow are:

• Introduce a small change \(h\), typically along the x-axis.
• Evaluate the expression and keep terms involving \(h\).
• Finally, let \(h\) approach zero to "zero out" terms that become negligible and reveal the instantaneous rate of change.

When \(h\) approaches zero in our example, the expression simplifies to a linear form \(4x +1\), representing the derivative of \(f(x)\). This process is foundational for understanding how functions behave dynamically.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key skill in mathematics that involves combining and reducing terms to their simplest form. When dealing with the difference quotient, it's essential to simplify the expression to make it easier to handle the limit process.

In the given problem, simplifying involves several steps:

• Expand the polynomial \( (x+h)^2 \) to get \( x^2 + 2xh + h^2 \).
• Substitute back into the numerator, combine like terms, and cancel common factors like \( h \) from the numerator and denominator.
• After canceling \( h \), the expression simplifies to \(4x + 2h + 1\).

By systematically reducing the equation, we've prepared it for the limit process. Simplifying algebraic expressions is crucial not only in making calculations easier but also in ensuring the accuracy and clarity of results.