Problem 1 - Calculus 3

Question

For the following exercises, evaluate each function at the indicated values. $$ W(x, y)=4 x^{2}+y^{2} . \text { Find } W(2,-1), \quad W(-3,6) $$

Step-by-Step Solution

Answer

W(2, -1) = 17, W(-3, 6) = 72.

Step 1: Understand the Function

The given function is \( W(x, y) = 4x^2 + y^2 \). This is a two-variable function where \( x \) and \( y \) are the inputs. To evaluate it at specific values, we'll substitute those values into the function.

Step 2: Evaluate W(2, -1)

To find \( W(2, -1) \), substitute \( x = 2 \) and \( y = -1 \) into the function:\[W(2, -1) = 4(2)^2 + (-1)^2\]This becomes:\[W(2, -1) = 4 \times 4 + 1 = 16 + 1 = 17\]So, \( W(2, -1) = 17 \).

Step 3: Evaluate W(-3, 6)

To find \( W(-3, 6) \), substitute \( x = -3 \) and \( y = 6 \) into the function:\[W(-3, 6) = 4(-3)^2 + (6)^2\]This becomes:\[W(-3, 6) = 4 \times 9 + 36 = 36 + 36 = 72\]So, \( W(-3, 6) = 72 \).

Key Concepts

Understanding Two-Variable Functions

Two-variable functions are mathematical expressions that involve two different inputs, usually denoted as \( x \) and \( y \). These functions can be visualized as surfaces in three-dimensional space. Unlike single-variable functions, where a set of inputs produces a set of outputs along a line, two-variable functions generate outputs in a plane.
The function \( W(x, y) = 4x^2 + y^2 \) given in the exercise is a typical example. Here, \( x \) and \( y \) both influence the outcome of \( W(x, y) \). This means changes in either of these inputs will affect the result we get from the function.
To evaluate such functions, we often use specific pairs of \( x \) and \( y \) values, much like coordinates on a graph. These evaluations provide us with precise outputs for given input scenarios, helping us understand the function's behavior at different points.

Evaluating a Function

Function evaluation involves calculating the output of a function based on given inputs. In the context of two-variable functions, this means determining the value of the function at specific points defined by \( (x, y) \) pairs.
To evaluate \( W(2, -1) \), we replace \( x \) with \( 2 \) and \( y \) with \( -1 \) in the function \( W(x, y) = 4x^2 + y^2 \). The calculations are simple:

• Substitute \( x = 2 \) and \( y = -1 \).
• Solve \( 4(2)^2 + (-1)^2 \), which results in \( 16 + 1 = 17 \).

Similarly, for \( W(-3, 6) \), substitute \( x = -3 \) and \( y = 6 \) into the function:

• Compute \( 4(-3)^2 + 6^2 \), resulting in \( 36 + 36 = 72 \).

This method shows how the substitution of specific values allows us to accurately find the function's output at those points.

Mastering Mathematical Substitution

Mathematical substitution is the process of replacing variables with given values to simplify and evaluate expressions. It's a fundamental skill in solving mathematical problems, including functions of one or more variables.
When evaluating two-variable functions like \( W(x, y) = 4x^2 + y^2 \), substitution involves precisely swapping \( x \) and \( y \) with the numbers provided. This step ensures that each variable's part in the function is thoroughly understood and calculated:

• Identify the values to substitute — for instance, \((2, -1)\) and \((-3, 6)\) from the exercise.
• Substitute these into the function, maintaining the correct order and arithmetic.
• Solve the resulting expression to find the evaluated function value.

Substitution allows one to convert abstract expressions into concrete numbers, making it easier to interpret and apply mathematical functions in real-life scenarios.