Problem 110 - Calculus Volume 3

Question

Given \(\quad f(x, y)=x^{2}-4 y, \quad\) find \(\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}\).

Step-by-Step Solution

Answer

The limit is \( 2x \).

Step 1: Understanding the Problem

We are asked to find the limit that represents the derivative of the function \( f(x, y) = x^2 - 4y \) with respect to \( x \). This is essentially finding \( \frac{\partial f}{\partial x} \).

Step 2: Setting Up the Expression

First, substitute \( x+h \) into the function: \( f(x+h, y) = (x+h)^2 - 4y \). Then calculate \( f(x+h, y) - f(x, y) \).

Step 3: Expanding the Expression

Expand \( (x+h)^2 \):\[ (x+h)^2 = x^2 + 2xh + h^2 \]Replace back in the function: \[ f(x+h, y) = x^2 + 2xh + h^2 - 4y \].

Step 4: Calculating the Difference

Compute \( f(x+h, y) - f(x, y) \):\[ x^2 + 2xh + h^2 - 4y - (x^2 - 4y) = 2xh + h^2 \].

Step 5: Forming the Difference Quotient

Form the difference quotient:\[ \frac{f(x+h, y) - f(x, y)}{h} = \frac{2xh + h^2}{h} \].

Step 6: Simplifying the Expression

Simplify the quotient by canceling \( h \):\[ \frac{2xh + h^2}{h} = 2x + h \].

Step 7: Taking the Limit

Take the limit as \( h \to 0 \):\[ \lim_{h \to 0} (2x + h) = 2x \].

Step 8: Conclusion

The limit is \( 2x \), which represents the partial derivative of \( f(x, y) \) with respect to \( x \).

Key Concepts

Multivariable Calculus

Multivariable Calculus extends the concepts of single-variable calculus into higher dimensions. This means that functions can now depend on more than one variable, such as \( f(x, y) \), which has two inputs, \( x \) and \( y \). When dealing with such functions, it's essential to understand how each variable independently affects the output. This is why we study **partial derivatives**. Unlike ordinary derivatives, which consider changes in only one direction, partial derivatives focus on one variable at a time, treating all other variables as constants. For example, when calculating the partial derivative of \( f(x, y) = x^2 - 4y \) with respect to \( x \), we consider \( x \) as the changing variable and \( y \) as a constant. This involves calculating the derivative as if it were a single-variable problem with respect to \( x \).

• Partial derivatives provide insights into the slope and rate of change.
• They help in understanding multidimensional functions better.
• Essential in fields such as physics, engineering, and economics.

By mastering partial derivatives, one can analyze how changes in a specific direction impact the function, leading to applications like optimization and modeling real-world phenomena.

Limit Calculation

One of the fundamental concepts in calculus is **limit calculation**, which extends to multivariable contexts. When finding the derivative, either full or partial, the limit definition of a derivative is incredibly important. For instance, to find \( \frac{\partial f}{\partial x} \) of the given function \( f(x, y)\), we explore the behavior of the function as it approaches a specific point along the \( x \)-axis. This behavior is encapsulated in the limit expression \( \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h} \).
As \( h \) approaches zero, this expression describes the instantaneous rate of change or the slope of the tangent at any point on the function along this axis. Calculating this limit involves a few steps:

• Substitute \( x+h \) into the function to understand its behavior near \( x \).
• Expand and simplify the expression to separate out terms that disappear as \( h \) becomes negligibly small.
• Factor and reduce the quotient to reveal the underlying pattern or derivative.

The final step is to take the limit by allowing \( h \) to vanish, resulting in a simplified form that provides the derivative, like \( 2x \) in our example. This signifies a pivotal point where algebra converges into calculus, translating to a real-world rate of change.

Derivative Rules

In calculus, derivative rules are essential shortcuts for finding the derivatives of functions without resorting to first principles. When dealing with partial derivatives, these rules are extended to functions of multiple variables for swift calculation. For a given function such as \( f(x, y) = x^2 - 4y \), the derivative rules help us understand how to differentiate terms individually and then recombine them.
Key rules that apply include:

• **Power Rule**: If \( f(x) = x^n \), then \( \frac{d}{dx} x^n = nx^{n-1} \). This helps in instantly calculating derivatives of polynomial terms like \( x^2 \) resulting in \( 2x \).
• **Constant Rule**: The derivative of a constant is zero, and in partial derivatives, constant terms involving other variables remain unchanged.
• **Sum Rule**: The derivative of a sum of functions is the sum of the derivatives. This allows us to separate and individually handle each component, like \( x^2 \) and \(-4y \).

These rules simplify the computation remarkably. By applying them, the partial derivative of \( f(x, y) \) with respect to \( x \) was efficiently computed to be \( 2x \). Recognizing and applying these rules is critical for solving real-world problems involving dynamic changes across different dimensions.