Problem 17 - Calculus for Biology and Medicine

Question

In Problems 17-24, find the indicated partial derivatives. $$ f(x, y)=3 x^{2}-y-2 y^{2} ; f_{x}(1,0) $$

Step-by-Step Solution

Answer

The partial derivative \(f_x(1, 0)\) is 6.

Step 1: Understanding Partial Derivatives

A partial derivative of a function of several variables is its derivative with respect to one of those variables, while keeping the other variables constant. Here, we are asked to find the partial derivative of the function \(f(x, y) = 3x^2 - y - 2y^2\) with respect to \(x\), and evaluate it at the point \((1, 0)\).

Step 2: Find the Partial Derivative with respect to x

To find \(f_x(x, y)\), the partial derivative of the function with respect to \(x\), take the derivative of each term that includes \(x\), treating \(y\) as a constant. The function is \(f(x, y) = 3x^2 - y - 2y^2\). The derivative of \(3x^2\) with respect to \(x\) is \(6x\). The terms \(-y\) and \(-2y^2\) do not contain \(x\), so they become 0 when differentiating with respect to \(x\). Thus, \(f_x(x, y) = 6x\).

Step 3: Evaluate the Partial Derivative at the Given Point

Substitute \((x, y) = (1, 0)\) into the expression for \(f_x(x, y)\). Thus, \(f_x(1, 0) = 6(1) = 6\).

Step 4: Verify the Result

Double check the calculation: \(f_x(x, y) = 6x\), substituting \(x = 1\) gives \(6 \times 1 = 6\). The evaluation is consistent. This confirms that \(f_x(1, 0) = 6\).

Key Concepts

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus. It deals with functions that have more than one variable. These functions can be more complex as they vary over two or more dimensions. Understanding multivariable calculus is crucial for many fields like physics, engineering, and economics.

• In this branch, you deal with functions such as \( f(x, y) \) or \( f(x, y, z) \), where there are multiple input variables.
• Operations include finding limits, derivatives, and integrals of these multivariable functions.
• These calculations often provide richer insights because they reveal how each variable influences the function's outcome independently and collectively.

For example, in the exercise we are dealing with \( f(x, y) = 3x^2 - y - 2y^2 \). This is a function of two variables that shows how both \(x\) and \(y\) contribute to its value. Multivariable calculus allows us to explore these contributions thoroughly.

Derivative with Respect to a Variable

A partial derivative is like a single-variable derivative but in a multivariable context. It shows how a function changes as one variable changes while keeping the other variables fixed.

• This is crucial when dealing with functions of more than one variable.
• For instance, if you have \( f(x, y) \), a partial derivative with respect to \( x \) treats \( y \) as a constant and focuses purely on how \( x \) influences \( f \).

In the given problem, we compute \( f_x(x, y) \), the partial derivative with respect to \( x \) of \( f(x, y) = 3x^2 - y - 2y^2 \). You find this by differentiating terms involving \( x \) and treating others as constants:

• The derivative of \(3x^2\) with respect to \(x\) is \(6x\).
• The terms \(-y\) and \(-2y^2\) do not affect the \(x\) derivative, leaving \(f_x(x, y) = 6x\).

This partial derivative tells us how the function's value changes with respect to small changes in \(x\), possessing practical applications in modeling and prediction.

Function Evaluation at a Point

Evaluating a function at a point involves plugging specific values into a function to determine its instantaneous rate or result at that particular input. This is often the final step after finding a derivative.

• It gives you a specific, often more relevant value for the function based on your situation.
• This is especially important for designing and analysis purposes in fields like physics and finance.

For the problem at hand, we evaluate the partial derivative at the point \((1, 0)\). This involves simply substituting \(x = 1\) and \(y = 0\) into \(f_x(x, y) = 6x\), and finding \(f_x(1, 0) = 6 \times 1 = 6\).
This value signifies the slope or rate of change of the function at the point \((1,0)\), which can be critical for understanding behavior at a specific spot.