Problem 22 - Calculus for Biology and Medicine

Question

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right) .\) \(f(x, y)=e^{9 x+2 y} ;(1,2)\)

Step-by-Step Solution

Answer

The linearization is \(L(x, y) = e^{13}(9x + 2y - 12)\).

Step 1: Understand the Problem

We need to find the linear approximation of the function \(f(x, y) = e^{9x + 2y}\) at the point \((1, 2)\). This involves finding the equation of the tangent plane at that point.

Step 2: Recall the Linearization Formula

The linearization \(L(x, y)\) of a function \(f(x, y)\) at a point \((x_0, y_0)\) is given by:\[L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\]where \(f_x\) and \(f_y\) are the partial derivatives of \(f\) with respect to \(x\) and \(y\), respectively.

Step 3: Compute Partial Derivatives

Find \(f_x\) and \(f_y\):1. The partial derivative of \(f(x, y)\) with respect to \(x\) is:\[f_x(x, y) = 9e^{9x+2y}\]2. The partial derivative of \(f(x, y)\) with respect to \(y\) is:\[f_y(x, y) = 2e^{9x+2y}\]

Step 4: Evaluate Function and Derivatives at (1,2)

Calculate \(f(1,2)\), \(f_x(1,2)\), and \(f_y(1,2)\):1. \(f(1, 2) = e^{9 \cdot 1 + 2 \cdot 2} = e^{13}\)2. \(f_x(1, 2) = 9e^{13}\)3. \(f_y(1, 2) = 2e^{13}\)

Step 5: Write the Linearization Equation

Substitute the computed values into the linearization formula:\[L(x, y) = e^{13} + 9e^{13}(x - 1) + 2e^{13}(y - 2)\]This simplifies to:\[L(x, y) = e^{13} (1 + 9(x - 1) + 2(y - 2))\]Which can be rewritten as:\[L(x, y) = e^{13} (9x + 2y - 12)\]

Key Concepts

Partial Derivatives

Partial derivatives are an essential tool in understanding how multivariable functions change. When dealing with a function of multiple variables, such as \(f(x, y)\), we often want to know how the function changes as one specific variable changes while holding the other variable constant. This is where partial derivatives come into play.

• To find the partial derivative of \(f(x, y)\) with respect to \(x\), denoted as \(f_x(x, y)\), we treat \(y\) as a constant and differentiate \(f\) as if it were a function of \(x\) alone.
• Similarly, the partial derivative with respect to \(y\), denoted as \(f_y(x, y)\), involves holding \(x\) constant and differentiating the function as if it were solely dependent on \(y\).

Understanding partial derivatives helps us interpret the slopes of the surfaces represented by multivariable functions, which is crucial when constructing tangent planes and performing linear approximations.

Tangent Plane

The concept of the tangent plane extends the idea of a tangent line from one-dimensional calculus to functions of several variables. When we have a function \(f(x, y)\) and a specific point \((x_0, y_0)\) on its surface, the tangent plane provides the best linear approximation of the surface near this point.

• The tangent plane relies on knowing the point of tangency \((x_0, y_0)\) and the function's partial derivatives at this point: \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\).
• These partial derivatives give us the slopes in the direction of the \(x\)-axis and \(y\)-axis respectively.

By combining these slopes with the known value of the function at the point, we formulate the equation of the tangent plane, allowing us to estimate the function's behavior close to \((x_0, y_0)\). This is analogous to drawing a tangent line to a curve in single-variable calculus, providing a linear lens through which we can view changes in the function.

Linear Approximation

Linear approximation is a method used to approximate the value of a function using its tangent plane. For a multivariable function \(f(x, y)\), we use its partial derivatives to construct a linear function \(L(x, y)\) that approximates \(f\) near the point \((x_0, y_0)\).

• The formula for linear approximation is: \[L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\]
• This equation represents the tangent plane, providing a flat, simple model that estimates how \(f\) behaves nearby \((x_0, y_0)\).

The beauty of linear approximation is its simplicity. Despite the complexity of the original function, the tangent plane offers a straightforward way to estimate function values, giving us insights and predictions without the need to evaluate the full function.

Multivariable Function

A multivariable function is one that depends on more than one variable. In our case, \(f(x, y) = e^{9x + 2y}\) is a classic example of a function of two variables. Understanding such functions requires us to think beyond simple curves and look at surfaces or higher-dimensional objects.

• These functions can describe diverse phenomena, such as temperature over a geographical area (function of latitude and longitude), or profit depending on different factors like sales and cost.
• Visualization of multivariable functions typically involves 3D surface plots since they depend on multiple inputs to produce a single output.

Studying the behavior of these functions involves techniques like partial derivatives and linear approximations. These tools help us simplify and understand the complex changes and tendencies of multivariable functions. By mastering these core concepts, you unlock the potential to model and analyze systems across various fields effectively.