Problem 80 - College Algebra with Modeling and Visualization

Question

Complete the following for the given \(f(x)\) (a) Find \(f(x+h)\) (b) Find the difference quotient of \(f\) and simplify. $$ f(x)=x^{2}+3 x-2 $$

Step-by-Step Solution

Answer

(a) \(f(x+h) = x^2 + 2xh + h^2 + 3x + 3h - 2\); (b) Difference quotient is \(2x + h + 3\).

Step 1: Substitute x+h into f(x)

To find \(f(x+h)\), substitute \(x+h\) into the function \(f(x) = x^2 + 3x - 2\). This gives:\[f(x+h) = (x+h)^2 + 3(x+h) - 2\]

Step 2: Simplify (x+h)^2

Expand \((x+h)^2\) using the distributive property:\[(x+h)^2 = x^2 + 2xh + h^2\]

Step 3: Simplify 3(x+h)

Distribute \(3\) across \((x+h)\):\[3(x+h) = 3x + 3h\]

Step 4: Combine all expressions

Combine the expanded expressions from Steps 2 and 3, including the constant:\[f(x+h) = x^2 + 2xh + h^2 + 3x + 3h - 2\]

Step 5: State the simplified f(x+h)

Simplify the combined expression to write \(f(x+h)\):\[f(x+h) = x^2 + 2xh + h^2 + 3x + 3h - 2\]

Step 6: Find the difference quotient

The difference quotient is defined as \(\frac{f(x+h) - f(x)}{h}\). Subtract \(f(x)\) from \(f(x+h)\):\[f(x+h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h - 2) - (x^2 + 3x - 2)\]

Step 7: Cancel like terms

Cancel the \(x^2\) and \(3x\) terms, and simplify:\[f(x+h) - f(x) = 2xh + h^2 + 3h\]

Step 8: Divide by h to find difference quotient

Divide each term by \(h\):\[\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 3h}{h} = 2x + h + 3\]

Step 9: Simplify the difference quotient

Simplify the expression by canceling out \(h\):The final simplified difference quotient is \(2x + h + 3\). After quotient simplification, for most limits as \(h\to0\), it simplifies to \(2x+3\).

Key Concepts

Polynomials

Polynomials are mathematical expressions made up of variables and constants, connected by addition, subtraction, and multiplication. They are characterized by their degrees, which indicate the highest power of the variable present in the expression. In the given exercise, the function \( f(x) = x^2 + 3x - 2 \) is a polynomial. It includes terms like \( x^2 \) (quadratic term), \( 3x \) (linear term), and \( -2 \) (constant term).
Understanding how each term contributes is key to manipulating polynomial functions. For example, the quadratic term \( x^2 \) affects the curvature of the graph of the function. The linear term \( 3x \) impacts the slope, while the constant term shifts the graph up or down without changing its shape.
Polynomials are frequently used in mathematics due to their simplicity and ease of differentiation and integration. They often serve as approximations for more complex functions in various fields like physics and engineering.

Function Substitution

Function substitution involves replacing the variable in a function with another expression. This technique is crucial for solving various mathematical problems, especially when working with difference quotients. In the exercise, we find \( f(x+h) \) by substituting \( x+h \) into the polynomial \( f(x) = x^2 + 3x - 2 \).

• First, substitute \( x+h \) into each occurrence of \( x \) within the function. This results in \( (x+h)^2 + 3(x+h) - 2 \).
• Next, expand \((x+h)^2\) using the distributive property to get \(x^2 + 2xh + h^2 \).
• Similarly, expand \(3(x+h)\) to obtain \(3x + 3h \).

After substitution and expansion, combining all these expressions gives \( f(x+h) = x^2 + 2xh + h^2 + 3x + 3h - 2 \). This expression sets the stage for calculating the difference quotient.

Simplifying Expressions

Simplifying expressions is a method of reducing complex equations to their simplest forms by combining like terms and performing arithmetic operations. After finding \( f(x+h) \), simplification becomes necessary to ensure clarity and ease of further calculations.
In the given exercise, you start by subtracting \( f(x) \) from \( f(x+h) \) to find \( f(x+h) - f(x) \). This step results in the expression \( 2xh + h^2 + 3h \), after canceling similar terms like \( x^2 \) and \( 3x \).
The final piece involves dividing this expression by \( h \) when calculating the difference quotient:

• Divide each term by \( h \) to yield \( 2x + h + 3 \).

Through simplification, an otherwise cumbersome expression is transformed into a manageable form, making subsequent steps in the problem-solving process much more straightforward.

Limit of a Function

The limit of a function describes the behavior of that function as its input approaches a certain value. It is a fundamental concept in calculus, particularly in the definition and understanding of derivatives.
In the context of the difference quotient, the limit as \( h \to 0 \) provides the derivative of the original function. For the polynomial \( f(x) = x^2 + 3x - 2 \), after simplifying the difference quotient \( 2x + h + 3 \), we consider the limit as \( h \) approaches zero:

• The term \( h \) becomes infinitesimally small, effectively disappearing from the expression, resulting in the simplified derivative \( 2x + 3 \).

This result represents the slope of the tangent line to the curve defined by the polynomial function at any given point \( x \). Understanding limits is vital for evaluating derivatives, integrals, and overall function behavior near points of interest.