Problem 8 - Calculus for Biology and Medicine

Question

The tangent plane at the indicated point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=e^{x} \cos y ;(0,0,1) $$

Step-by-Step Solution

Answer

The equation of the tangent plane is \(z = x + 1\).

Step 1: Understand the Tangent Plane Formula

The equation of the tangent plane to the surface defined by \( z = f(x, y) \) at a point \((x_0, y_0, z_0)\) is given by: \( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \), where \(f_x\) and \(f_y\) are the partial derivatives of \(f\) with respect to \(x\) and \(y\).

Step 2: Calculate Partial Derivatives

Find the partial derivatives of \(f(x, y) = e^x \cos y\). First, the partial derivative with respect to \(x\) is \(f_x(x, y) = e^x \cos y\). Next, the partial derivative with respect to \(y\) is \(f_y(x, y) = -e^x \sin y\).

Step 3: Evaluate Derivatives at Given Point

Evaluate these derivatives at the point \((x_0, y_0) = (0, 0)\). Thus, \(f_x(0, 0) = e^0 \cos 0 = 1\) and \(f_y(0, 0) = -e^0 \sin 0 = 0\).

Step 4: Write the Tangent Plane Equation

Using the tangent plane formula and the derivatives evaluated, substitute into the equation: \( z - 1 = 1 \cdot (x - 0) + 0 \cdot (y - 0)\). Simplifying gives \( z - 1 = x\) or rearranged, \( z = x + 1\).

Key Concepts

Partial Derivatives

Partial derivatives are crucial in multivariable calculus because they allow us to see how a function changes as we tweak one variable at a time, keeping the others fixed.

• Think of partial derivatives as a way of "slicing" through a multidimensional surface to understand its directional change.
• For example, if you have a mountain landscape described by a function, the partial derivative with respect to one variable helps you see the incline when you walk in a straight line in one specific direction.
• In mathematical terms, for a function like \(f(x, y) = e^x \cos y\), the partial derivatives are \(f_x(x, y) = e^x \cos y\) and \(f_y(x, y) = -e^x \sin y\).

In the context of this exercise, these derivatives tell us how the function \(f(x, y) = e^x \cos y\) changes when we move either in the \(x\)-direction by the function itself and in the y-direction. These are key for formulating the tangent plane equation.

Tangent Plane Equation

The tangent plane provides a linear approximation to a surface at a given point, essentially acting like a flat surface that 'touches' the curve or surface at the specific point.

• This mathematical tool helps in making complex surfaces easier to work with, especially near the point of tangency.
• The general formula for the tangent plane to a surface \(z = f(x, y)\) at a point \((x_0, y_0, z_0)\) is: \(z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\).
• Here, \(f_x\) and \(f_y\) are partial derivatives, as we calculated earlier, and these help define the slope of the plane in both the x- and y-directions.

In our example, we apply this situation to point \(0, 0, 1\). After evaluating the derivatives at this point, we substitute into the tangent plane equation, reducing complex calculations to a manageable form: \(z = x + 1\). This linear equation represents the flat surface that grazes the point \((0,0,1)\) on the original function.

Multivariable Calculus

Multivariable calculus expands upon single-variable calculus, allowing us to explore functions of two or more variables. This includes thinking through problems that involve three-dimensional space and beyond.

• Imagine you're working with surfaces in three dimensions, such as mountains, wtih each point on the surface represented by a combination of x, y, and the function \(z=f(x,y)\).
• Concepts like partial derivatives are used to study how these variables interact and change together.
• One of the critical applications in multivariable calculus is the use of approximations like tangent planes to simplify analysis and computations.

Understanding these facets allows mathematicians and analysts to model real-world situations more precisely, encompassing tasks like predicting weather patterns, optimizing engineering designs, and even creating graphics in computer simulations. The tangent plane exemplifies using derivatives to approximate a surface, elucidating how changes in variables manifest within three-dimensional space.