Problem 59 - Brief Applied Calculus

Question

$$ \text { For each function, find and simplify } f(x+h) \text { . } $$ $$ f(x)=5 x^{2} $$

Step-by-Step Solution

Answer

\(f(x+h) = 5x^2 + 10xh + 5h^2\).

Step 1: Substitute \(x+h\) into the Function

Start by substituting \(x + h\) for \(x\) in the given function. So, for \(f(x) = 5x^2\), the expression becomes \(f(x+h) = 5(x+h)^2\).

Step 2: Expand the Binomial

Next, you need to expand the expression \((x+h)^2\). This can be done using the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Apply this to get: \((x+h)^2 = x^2 + 2xh + h^2\).

Step 3: Multiply by 5

Now, take the expanded expression and multiply each term by 5, since the original function is \(5x^2\). So, \(5(x^2 + 2xh + h^2) = 5x^2 + 10xh + 5h^2\).

Step 4: Combine and Simplify

Finally, gather all the terms to find the simplified expression for \(f(x+h)\). In this case, \(f(x+h) = 5x^2 + 10xh + 5h^2\). No further simplification is possible.

Key Concepts

Polynomial Functions

Polynomial functions are a cornerstone of algebra and calculus. They are expressions that consist of variables and coefficients, shaped by operations of addition, subtraction, multiplication, and exponentiation by a non-negative integer. For example, in the function \( f(x) = 5x^2 \), the term \( 5x^2 \) is a monomial, which is a single term that is itself a simple polynomial. Polynomials are defined by their degree, which is the highest power of the variable present in the expression. In \( f(x) = 5x^2 \), the degree is 2. Understanding the structure and behavior of polynomial functions is crucial for analyzing real-world phenomena, as they can be used to model diverse scenarios from physics to economics.

Function Evaluation

Evaluating a function involves substituting a given input value into the function to calculate the output. In the context of the exercise, we evaluate \( f(x+h) \) by substituting \( x+h \) into the function \( f(x) = 5x^2 \). By doing this, we are essentially determining the effect of a small change (\( h \)) on the variable \( x \). This process is vital in calculus, especially in concepts like limits and derivatives, where we study how functions behave as inputs change slightly. Here's how to evaluate:

• Identify the function and specify the new input.
• Replace the variable \( x \) with \( x+h \) in the function \( f(x) \).
• Perform algebraic operations to simplify the new expression.

In this way, function evaluation allows us to understand changes and predict outcomes in various fields.

Algebraic Manipulation

Algebraic manipulation is a set of techniques used to simplify or rearrange expressions and equations. In the exercise, we perform algebraic manipulation by expanding and simplifying \( (x+h)^2 \), and then multiplying the result by 5.This involves several steps:

• Apply the binomial expansion formula: \( (a+b)^2 = a^2 + 2ab + b^2 \).
• In our example: replace \( a \) with \( x \) and \( b \) with \( h \). This gives \( x^2 + 2xh + h^2 \).
• Multiply each term by the coefficient present in the original function, here it's 5, leading to \( 5x^2 + 10xh + 5h^2 \).

These steps help streamline problems, making complex expressions easier to understand and solve. Mastery of algebraic manipulation is essential for tackling polynomial functions, simplifying expressions, and solving equations efficiently.