Problem 207 - Calculus Volume 3

Question

Find the linear approximation of each function at the indicated point. \(\quad f(x, y)=x \sqrt{y}, \quad P(1,4)\)

Step-by-Step Solution

Answer

The linear approximation is \(L(x, y) = 2x + \frac{y}{4} - 1\).

Step 1: Find the Partial Derivatives

To find the linear approximation of the function \(f(x, y) = x \sqrt{y}\) at point \((1, 4)\), we start by calculating the partial derivatives with respect to \(x\) and \(y\). The partial derivative of \(f\) with respect to \(x\) is:\[ f_x(x, y) = \frac{\partial}{\partial x}(x\sqrt{y}) = \sqrt{y}. \]The partial derivative of \(f\) with respect to \(y\) is:\[ f_y(x, y) = \frac{\partial}{\partial y}(x\sqrt{y}) = x \cdot \frac{1}{2\sqrt{y}} = \frac{x}{2\sqrt{y}}. \]

Step 2: Evaluate Partial Derivatives at the Point

Next, evaluate the partial derivatives at the given point \((1, 4)\).Calculate \(f_x(1, 4)\):\[ f_x(1, 4) = \sqrt{4} = 2. \]Calculate \(f_y(1, 4)\):\[ f_y(1, 4) = \frac{1}{2\sqrt{4}} = \frac{1}{4}. \]

Step 3: Construct the Linear Approximation Formula

The linear approximation, \(L(x, y)\), of \(f\) at the point \((1, 4)\) can be written as:\[ L(x, y) = f(1, 4) + f_x(1, 4)(x - 1) + f_y(1, 4)(y - 4). \]From Step 2, substitute the values:\[ L(x, y) = 2 + 2(x - 1) + \frac{1}{4}(y - 4). \]

Step 4: Simplify the Linear Approximation Formula

To simplify, first calculate \(f(1, 4)\):\[ f(1, 4) = 1\sqrt{4} = 2. \]Substitute this into \(L(x, y)\):\[ L(x, y) = 2 + 2(x - 1) + \frac{1}{4}(y - 4). \]Expand and simplify:\[ L(x, y) = 2 + 2x - 2 + \frac{y}{4} - 1. \]Combine like terms:\[ L(x, y) = 2x + \frac{y}{4} - 1. \]

Key Concepts

Partial Derivatives

Partial derivatives are crucial in understanding how a multivariable function changes as one variable shifts while others are held constant. This concept extends from ordinary derivatives in single-variable calculus, giving us a window into how each individual dimension affects our function.
In the exercise, the function given is \( f(x, y) = x \sqrt{y} \), which is dependent on two variables, \( x \) and \( y \). To understand how changes occur near the point \((1, 4)\), we calculate the partial derivatives \( f_x \) and \( f_y \).

• The partial derivative with respect to \( x \), noted as \( f_x \), considers \( y \) as constant, resulting in \( \sqrt{y} \).
• Simultaneously, the partial derivative with respect to \( y \), labeled \( f_y \), looks at \( x \) as fixed and simplifies to \( \frac{x}{2\sqrt{y}} \).

These derivatives are foundational to forming the linear approximation, which predicts the function's behavior at slightly varied inputs from the point \((1, 4)\). Analyzing these derivatives helps in tackling more complex functions by breaking down their sensitivity to each variable.

Multivariable Calculus

Multivariable calculus extends the principles of differential calculus to functions involving several variables. Unlike single-variable calculus that examines the change of a function in one direction, multivariable calculus evaluates functions in a multi-dimensional space. This approach allows analyzing surfaces and how they change in every conceivable direction.
In the context of the exercise, the function \( f(x, y) = x \sqrt{y} \) relies on both \( x \) and \( y \), giving it a context in a two-dimensional plane. Finding linear approximations and partial derivatives are just a few of the tasks that multivariable calculus can handle.

• It enables exploration of gradients, which guide us to the steepest slope of any surface.
• It provides tools like Jacobians and Hessians to evaluate more complex systems.

For real-world applications, multivariable calculus can be used in optimizing functions where multiple factors are at play, such as in economic models or engineering simulations. Thus, mastering these techniques opens doors to solving a variety of realistic problems involving more than one variable.

Approximation Formula

The approximation formula is a strategy used to make functions easier to interpret. Linear approximation, specifically, estimates the value of a function near a chosen point by using the tangent plane in multivariable calculus, similar to using a tangent line in single-variable calculus.
For our exercise, the linear approximation \( L(x, y) \) at \((1, 4)\) starts formation after calculating \( f(1, 4) \), \( f_x(1, 4) \), and \( f_y(1, 4) \). With these values, the linear approximation is expressed as:\[L(x, y) = f(1, 4) + f_x(1, 4)(x - 1) + f_y(1, 4)(y - 4).\]By substituting the solved derivatives into the equation, we establish the linear behavior of \( f \) near our point of interest:\[L(x, y) = 2 + 2(x - 1) + \frac{1}{4}(y - 4).\]When expanded and simplified, the approximation becomes \( L(x, y) = 2x + \frac{y}{4} - 1 \). This form provides a simplified model of the function, only accurate near \((1, 4)\), but valuable for understanding local behavior trends.
Linear approximation is a powerful tool, harnessed across fields for making complex systems and functions more manageable, particularly near known values.