Problem 9 - Calculus for Biology and Medicine

Question

Evaluate each function at the given point. \(f(x, y)=x^{2}+y\) at \((2,3)\)

Step-by-Step Solution

Answer

The value of the function at (2, 3) is 7.

Step 1: Understand the Function

The function given is \(f(x, y) = x^2 + y\). This is a function with two variables, \(x\) and \(y\), which means we need both an \(x\) and a \(y\) value to evaluate it.

Step 2: Identify the Given Point

We have been given the point \((2, 3)\). Here, \(x = 2\) and \(y = 3\). We will substitute these values into the function.

Step 3: Substitute the Values

Substitute \(x = 2\) and \(y = 3\) into the function: \(f(2, 3) = (2)^2 + 3\).

Step 4: Simplify the Expression

Now, calculate \((2)^2\), which is \(4\). Then, add the \(y\) value, which is \(3\). This gives \(4 + 3\).

Step 5: Calculate the Result

The final calculation is \(4 + 3 = 7\). Therefore, \(f(2, 3) = 7\).

Key Concepts

Function Evaluation

Evaluation of a function involves determining the value of a function for specific input values. Here, we are dealing with a multivariable function, which means it has more than one variable. In this exercise, the function is given as \( f(x, y) = x^2 + y \). Evaluating a function requires us to substitute particular values for the variables. Given the point \((2,3)\), we set \(x = 2\) and \(y = 3\) and substitute these values into the function. Evaluating multivariable functions can be seen as extending single-variable evaluations, where instead of a single input, we work with a pair (or more) of inputs corresponding to each variable in the function. This approach lets us find the function's value at a specific point on the plane, making it very useful in calculus.

Substitution Method

The substitution method is a powerful tool in function evaluation that lets us replace variables with specific numerical values. This process is simple and involves a few straightforward steps:

• Identify the variables in the given function. For example, our function has \(x\) and \(y\).
• Determine the specific values to substitute. For this problem, these are \(x = 2\) and \(y = 3\).
• Substitute the identified values into the function. This means replacing \(x\) with 2 and \(y\) with 3, transforming the function into \(f(2, 3) = (2)^2 + 3\).

When substituting, always double-check to ensure each variable is correctly replaced by its respective value. This careful substitution lays the foundation for correct evaluation and further simplification.

Simplification Steps

After substitution, simplifying the function is essential to arrive at a neat answer. Here are the simplification steps for the given function:

• First, focus on any arithmetic operations within the function. Our function involves squaring \(x\).
• Calculate \((2)^2\), which results in 4. Perform any similar operations within your function.
• Next, carry out any additional arithmetic, like addition or subtraction. For our function, add the result from the previous step to the \(y\) value, which is 3. So, we compute \(4 + 3\).
• The final result is 7. Ensure all computations are precise to prevent errors in the final evaluation.

Simplification not only makes the function's result cleaner but is also crucial in obtaining the correct result. Each step refines the function closer to its numerical value.