Problem 8 - Calculus and its applications

Question

For \(f(x, y, z)=2^{x}+5 z y-x,\) find \(f(0,1,-3)\) and \(f(1,0,-3)\).

Step-by-Step Solution

Answer

\(f(0,1,-3) = -14\), \(f(1,0,-3) = 1\).

Step 1: Identifying the Variables

First, identify which values to substitute into the function for each problem. We are given two sets of variables: \((0, 1, -3)\) and \((1, 0, -3)\). The function is defined as \(f(x, y, z) = 2^x + 5zy - x\).

Step 2: Substitute First Set of Values

We have the first set of values \((x, y, z) = (0, 1, -3)\). Substitute these into the function: \(f(0, 1, -3) = 2^0 + 5(-3)(1) - 0\).

Step 3: Calculate Using First Set of Values

Simplify the expression: \(2^0 = 1\), hence the calculation becomes \(1 + 5(-3) - 0 = 1 - 15 = -14\).

Step 4: Substitute Second Set of Values

Now take the second set of values \((x, y, z) = (1, 0, -3)\) and substitute them into the function: \(f(1, 0, -3) = 2^1 + 5(-3)(0) - 1\).

Step 5: Calculate Using Second Set of Values

Simplify this expression: \(2^1 = 2\), hence the calculation becomes \(2 + 0 - 1 = 1\).

Step 6: Conclusion from Calculations

Based on the calculations, \(f(0, 1, -3) = -14\) and \(f(1, 0, -3) = 1\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a mathematical function for given input values. In our case, we have a multivariable function defined as \(f(x, y, z) = 2^x + 5zy - x\).
In function evaluation:

• We plug in specific values for each of the variables (\(x, y, z\)).
• The goal is to calculate the resulting value of the function.

Let's focus on evaluating the function with the provided sets of variables. For the set \((0, 1, -3)\), substituting these directly gives us the steps to calculate \(f(0, 1, -3)\). Similarly, for \((1, 0, -3)\), we substitute these into the function to find \(f(1, 0, -3)\).
Function evaluation is crucial as it helps us understand how a function behaves with different inputs.

Substitution Method

The substitution method involves directly replacing the variables in a function with given or specific values.
This allows for straightforward evaluation of a function.

• Identify the given function and its variables.
• Replace each variable in the function with the corresponding number from the provided ordered triplet.

In our exercise:- For \((x, y, z) = (0, 1, -3)\), substitute: \(f(0, 1, -3) = 2^0 + 5(-3)(1) - 0\). - Similarly, for \((x, y, z) = (1, 0, -3)\), substitute: \(f(1, 0, -3) = 2^1 + 5(-3)(0) - 1\).
This method helps simplify the evaluation by focusing only on the numbers representing the variables, eliminating the complexity of the expression when it is in variable form.

Simplifying Expressions

Simplifying mathematical expressions is an essential step to making calculations manageable and solving functions efficiently.
Here are some tips:

• Calculate any powers or exponents first, like \(2^0\) and \(2^1\).
• Follow the order of operations: parentheses, exponents, multiplication/division (from left to right), addition/subtraction (from left to right).

For our expressions:- With \(f(0, 1, -3) = 2^0 + 5(-3)(1) - 0\), solve \(2^0 = 1\), then \(5(-3)(1) = -15\) to find \(1 - 15 = -14\).- With \(f(1, 0, -3) = 2^1 + 5(-3)(0) - 1\), solve \(2^1 = 2\), then \(5(-3)(0) = 0\) to find \(2 - 1 = 1\).
Simplifying expressions avoids errors and reinforces understanding by breaking down the steps to an answer in a logical and simple manner.