Problem 60 - Applied Calculus

Question

For each function, find and simplify \(f(x+h)\). $$ f(x)=3 x^{2} $$

Step-by-Step Solution

Answer

\(f(x+h) = 3x^2 + 6xh + 3h^2\).

Step 1: Substitute into Function

Start by substituting \(x+h\) into the function \(f(x) = 3x^2\) for \(x\). This gives us \(f(x+h) = 3(x+h)^2\).

Step 2: Expand the Expression

Expand \((x+h)^2\) to get \(x^2 + 2xh + h^2\). The function now becomes \(f(x+h) = 3(x^2 + 2xh + h^2)\).

Step 3: Distribute

Distribute the 3 across \(x^2 + 2xh + h^2\) to get \(3x^2 + 6xh + 3h^2\). This is the simplified form of \(f(x+h)\).

Key Concepts

Functions and Their Role

A function in mathematics represents a relationship where each input (value from the domain) is assigned to exactly one output (value from the range). Think of it as a kind of machine. You feed it a number, and it gives you back another number, the output being determined by a specific rule.

• In our exercise, the function is given as \(f(x) = 3x^2\).
• This means for every number \(x\), multiply it by itself, and then multiply the result by 3.
• The goal was to substitute \(x+h\) into this function to explore how small changes (like adding \(h\) to \(x\)) affect the output.

Understanding functions helps in analyzing and predicting patterns, a crucial skill in calculus and various mathematical applications.

Concept of Expansion

Expansion in algebra involves taking an expression with one or more operators (like multiplication and addition) and rewriting it in an open form. For example, when we see something like \( (x+h)^2 \), it's a compact way of writing a sequence of operations.In our step-by-step solution, expanding \( (x+h)^2 \) is the second essential step.

• You rewrite \( (x+h)^2 \) as \( x^2 + 2xh + h^2 \).
• This involves using the distributive property to multiply each term thoroughly.
• Every term in the parentheses is accounted for, leading to a complete expression fully spread out into its separate parts.

By understanding expansion, you're able to transform complex expressions into forms that are easier to work with and comprehend.

Algebraic Manipulation Techniques

Algebraic manipulation is the art of rearranging, simplifying, and evaluating algebraic expressions. It's like solving a puzzle—putting pieces in the right place to reveal the bigger picture.

• In our example, after expanding \((x+h)^2\) as \(x^2 + 2xh + h^2\), the next step is distributing.
• This involves taking 3 and multiplying it with each term of \(x^2 + 2xh + h^2\), resulting in \(3x^2 + 6xh + 3h^2\).
• Such manipulations simplify expressions and make them manageable and solvable in calculus and other fields of mathematics.

Mastery of these manipulations is fundamental for pursuing advanced mathematical problems and understanding how different parts of an expression interact.