Problem 101 - Applied Calculus

Question

True or False: If \(f(x)=x^{2},\) then \(f(x+h)=x^{2}+h^{2}\).

Step-by-Step Solution

Answer

False, because \(f(x+h) = x^2 + 2xh + h^2\), not \(x^2+h^2\).

Step 1: Identify the Function

The function given in the exercise is \(f(x) = x^2\). This is a basic quadratic function where the input \(x\) is squared.

Step 2: Substitute and Expand

We need to find \(f(x+h)\), which means we need to substitute \(x+h\) into the function \(f(x)\). The new expression becomes \(f(x+h) = (x+h)^2\).

Step 3: Apply the Binomial Formula

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

Step 4: Compare the Expressions

The expression for \(f(x+h) = (x+h)^2 = x^2 + 2xh + h^2\) does not equal to \(x^2 + h^2\). The term \(2xh\) is missing in the latter.

Step 5: Determine the Truth Value

Since \(f(x+h) = x^2 + 2xh + h^2\) is not the same as \(x^2 + h^2\), the statement in the exercise is false.

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra, particularly when dealing with expressions that are raised to successive powers. At its core, this theorem allows us to expand expressions of the form \( (a + b)^n \). For example, if we want to expand \( (x + h)^2 \), the binomial theorem helps us do so efficiently.

• The theorem states that \( (a + b)^n \) can be expanded into a sum involving terms of the form \( \binom{n}{k}a^{n-k}b^k \).
• For \( n = 2 \), this simplifies to \( a^2 + 2ab + b^2 \), which applies directly to our function.

In the exercise, the task was to find \( f(x+h) = (x+h)^2 \). Using the binomial theorem, the expansion yields \( x^2 + 2xh + h^2 \). This shows how critical it is to include the middle term \( 2xh \) in the correct expansion, which is a common mistake when students first encounter such problems. The error in the original statement of the exercise lies in omitting this middle term.

Function Substitution

Function substitution involves replacing a variable within a function with an expression or another variable. This is a fundamental concept in calculus and algebra, helping to evaluate and manipulate functions. When we talk about substitution in the context of functions, we typically refer to replacing the function's input with a different expression.

• In the given exercise, we started with \( f(x) = x^2 \).
• The goal was to determine what \( f(x+h) \) looks like after substitution, where \( x \) is replaced with \( x+h \).

By substituting \( x+h \) for \( x \) in \( f(x) \), we draft a new form of the function: \( f(x+h) = (x+h)^2 \). This substitution step is crucial because it lays the foundation for applying the binomial theorem or any other needed expansion techniques. It is this substitution that points out the errors in their misstatements or understanding of function behaviors.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. Understanding how to manipulate and expand these expressions is vital for solving a wide range of mathematics problems, especially in algebra.

• Consider the expression \( (x+h)^2 \), which is a binomial that can be expanded by knowing the rules for algebraic manipulation.
• The expanded form is \( x^2 + 2xh + h^2 \), clearly showing how each term is derived from the multiplication process.
• Knowing how to expand expressions allows students to simplify complex problems and find errors in incorrect algebraic steps.

This exercise demonstrates the importance of thorough knowledge of algebraic rules. Missteps often happen when parts of the expression are inadequately handled. Incorrect expansions or ignoring terms like \( 2xh \) can lead to false conclusions about the nature of the function. Therefore, mastering algebraic expressions is key to correctly interpreting and solving such problems.