Problem 38 - College Algebra

Question

In Exercises \(33-44\), find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=2 x^{2}$$

Step-by-Step Solution

Answer

The difference quotient for the function \(f(x)=2x^2\) is \(4x + 2h\)

Step 1: Write Down the Difference Quotient

The difference quotient for a given function \(f(x)\) is defined as \(\frac{f(x+h)-f(x)}{h}\).

Step 2: Substitute the Function into the Difference Quotient

Substitute \(f(x) = 2x^2\) into the difference quotient, we get \(\frac{2(x+h)^2 - 2x^2}{h}\)

Step 3: Expand and Simplify the Numerator

Expand the numerator \((x+h)^2\) to \(x^2 + 2xh + h^2\), so we have \(\frac{2(x^2 + 2xh + h^2) - 2x^2}{h}\), which simplifies to \(\frac{2x^2 + 4xh + 2h^2 - 2x^2}{h}\)

Step 4: Cancel and Simplify the Fraction

Cancel the \(2x^2\) terms in the numerator and simplify the fraction to get \(4x + 2h\)

Key Concepts

Algebraic Functions

In mathematics, algebraic functions are expressions constructed using algebraic operations, such as addition, subtraction, multiplication, division, and taking roots of variables. A common representation of these functions is in the form of polynomials. For example, the function from the exercise, f(x) = 2x^2, is an algebraic function because it can be defined by simply squaring the variable x and multiplying by 2, which are both algebraic operations.

Algebraic functions play a vital role in various areas of mathematics, as they can model real-world situations like projectile motion in physics or represent the relationship between variables in economics. In the exercise provided, the calculation of the difference quotient of an algebraic function is an elementary step towards understanding more complex concepts such as derivatives in calculus.

When working with algebraic functions, it's crucial to follow the order of operations and to methodically manipulate the expressions to achieve the desired simplification or result, as demonstrated in the exercise. The simplification step is essential, as it paves the way for further analysis of the function's properties, such as evaluating limits or exploring its graph.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves rewriting expressions in a more concise and easily understandable form, without changing the value that the expression represents. This process often involves combining like terms, factoring, canceling terms, and reducing fractions to their simplest form.

In the context of finding the difference quotient, simplifying the expression makes it easier to identify patterns and changes as the function varies with the input. Here are some tips to effectively simplify expressions:

• Combine like terms, which are terms with the same variables raised to the same power.
• Expand the products of binomials or other polynomials using the distributive property.
• Factor expressions to find common factors that can be canceled out.
• Reduce fractions by dividing both the numerator and the denominator by their greatest common divisor.

In the given exercise, simplification involves expanding the squared term (x+h)^2, combining like terms, and canceling out terms that appear in both the numerator and denominator, which ultimately results in the simplified difference quotient of 4x + 2h.

Limits and Continuity

Limits and continuity are significant concepts in calculus, relating to the behavior of functions as inputs approach certain values. A limit evaluates what value a function approaches as the input gets infinitely close to a specific point, while continuity ensures that functions do not have any abrupt breaks or jumps.

The difference quotient plays a crucial role as it is the foundational building block for the definition of the derivative, which at its core is a limit of the difference quotient as h approaches zero. If the limit exists, we can say that the function is differentiable at that point, which also implies continuity.

For the algebraic function given in the exercise, understanding the behavior of the simplified difference quotient as h approaches zero would determine the slope of the tangent line at any point x on the function f(x). This crucial step is what connects algebraic manipulation of functions with the deeper analysis of their behavior through calculus. It highlights the importance of being able to confidently compute limits to fully understand the dynamics of a function around specific points.