Problem 2 - Calculus Volume 3

Question

For the following exercises, evaluate each function at the indicated values. \(W(x, y)=4 x^{2}+y^{2}\). Find \(W(2+h, 3+h)\).

Step-by-Step Solution

Answer

The value of \(W(2+h, 3+h)\) is \(25 + 22h + 5h^2\).

Step 1: Understand the Function

The function given is \( W(x, y) = 4x^2 + y^2 \). This function takes two variables, \( x \) and \( y \), and computes a value based on those inputs.

Step 2: Substitute Variables

We are asked to find \( W(2+h, 3+h) \). Substitute \( x = 2+h \) and \( y = 3+h \) into the function: \( W(2+h, 3+h) = 4(2+h)^2 + (3+h)^2 \).

Step 3: Expand \((2+h)^2\)

Expand \((2+h)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Thus, \((2+h)^2 = 4 + 4h + h^2\).

Step 4: Expand \((3+h)^2\)

Expand \((3+h)^2\) using the same formula: \((3+h)^2 = 9 + 6h + h^2\).

Step 5: Plug Expanded Terms into the Function

Substitute the expanded expressions back into the function: \( W(2+h, 3+h) = 4(4 + 4h + h^2) + (9 + 6h + h^2) \).

Step 6: Distribute and Simplify

Distribute \( 4 \) across \( (4+4h+h^2) \): \( 16 + 16h + 4h^2 \). Now add the expressions: \( (16 + 16h + 4h^2) + (9 + 6h + h^2) = 25 + 22h + 5h^2 \).

Key Concepts

Function Evaluation

The process of function evaluation involves plugging specific values into a given mathematical function to uncover the output. It's a fundamental skill when dealing with multivariable calculus, particularly for understanding how a function behaves for different inputs. When evaluating the function \( W(x, y) = 4x^2 + y^2 \), we substituted \( x = 2+h \) and \( y = 3+h \) to find \( W(2+h, 3+h) \).

• Identify the Variables: First, identify the variables you need to replace. In our example, these are \( x \) and \( y \).
• Substitute Values: Next, substitute the identified variables with the provided values or expressions. For instance, \( x = 2 + h \) and \( y = 3 + h \) were substituted into the function \( W(x, y) \).
• Compute the Result: After substituting, you compute the value by carrying out algebraic operations on the resulting expression.

These steps are crucial, as they lay the groundwork for deeper operations like polynomial expansions and algebraic manipulation.

Polynomial Expansion

Polynomial expansion is a technique used to break down expressions that involve polynomials into simpler terms, making them easier to work with. This concept is vital when dealing with functions in calculus because it often simplifies subsequent operations.Use expansion formulas to transform expressions. For example, the expansion formula \( (a+b)^2 = a^2 + 2ab + b^2 \) was used to simplify \( (2+h)^2 \) and \( (3+h)^2 \).

• Square and Expand: Apply the expansion formula to quadratic terms. This involves squaring and then expanding binomial expressions such as \( (2 + h)^2 \) into \( 4 + 4h + h^2 \).
• Distribute and Simplify: After expanding, distribute any coefficients and combine like terms. For \( (3 + h)^2 \), expansion gives \( 9 + 6h + h^2 \).

By breaking down these expressions step-by-step, polynomial expansion allows us to simplify complex expressions significantly. This simplification eventually aids in accurately evaluating or solving functions.

Algebraic Manipulation

After evaluating and expanding functions, understanding algebraic manipulation is essential to simplifying expressions further. It's a core concept in calculus, providing a pathway to transform complicated expressions into simpler, more manageable ones.Here are key steps for algebraic manipulation:

• Distribution: When a polynomial or a term needs to be multiplied by each term in an expression, use distribution. For instance, in \( 4(4 + 4h + h^2) \), distributing 4 results in \( 16 + 16h + 4h^2 \).
• Combine Like Terms: Combine terms that have the same variable raised to the same power. The initially separate terms \( 16 + 9 \), \( 16h + 6h \), and \( 4h^2 + h^2 \) combine to yield \( 25 + 22h + 5h^2 \).

Practicing these methods will improve your ability to handle complex questions in multivariable calculus, as you can simplify intricate equations into easier forms to work with.