Problem 1 - 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. $$ f(x, y)=2 x^{3}+y^{2} ;(1,2,6) $$

Step-by-Step Solution

Answer

The equation of the tangent plane is \( z = 6x + 4y - 8 \).

Step 1: Review the Problem Requirements

We are tasked with finding the equation of the tangent plane to the surface described by the function \( f(x, y) = 2x^3 + y^2 \) at the point \((1, 2, 6)\). We need to use the formula for the tangent plane to a function at a given point.

Step 2: Understand the Tangent Plane Equation

The equation of the tangent plane to a function \( z = f(x, y) \) 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 \( y \), respectively.

Step 3: Calculate Partial Derivatives

First, find the partial derivative of \( f(x, y) \) with respect to \( x \):\[ f_x(x, y) = \frac{\partial}{\partial x}(2x^3 + y^2) = 6x^2 \]Next, find the partial derivative of \( f(x, y) \) with respect to \( y \):\[ f_y(x, y) = \frac{\partial}{\partial y}(2x^3 + y^2) = 2y \]

Step 4: Evaluate Partial Derivatives at (1, 2)

Substitute \( x = 1 \) and \( y = 2 \) into the partial derivatives:\[ f_x(1, 2) = 6(1)^2 = 6 \]\[ f_y(1, 2) = 2(2) = 4 \]

Step 5: Construct the Tangent Plane Equation

Use the point \((1, 2, 6)\) and the partial derivative values to plug into the tangent plane equation:\[ z = 6 + 6(x - 1) + 4(y - 2) \]Simplify:\[ z = 6 + 6x - 6 + 4y - 8 \]\[ z = 6x + 4y - 8 \]

Key Concepts

Partial Derivatives

Partial derivatives are an important concept in calculus, specifically in dealing with multivariable functions. They represent the rate of change of a function as one variable changes while all other variables are kept constant.
For a function like \( f(x, y) \), which depends on two variables \( x \) and \( y \), you can find its partial derivatives with respect to each variable. This involves differentiating the function with respect to one variable while treating the other variable as a constant.

• To find the partial derivative \( f_x(x, y) \), differentiate \( f \) with respect to \( x \) and treat \( y \) as a constant.
• Similarly, to find \( f_y(x, y) \), differentiate \( f \) with respect to \( y \), keeping \( x \) constant.

In our exercise, the partial derivative \( f_x(x, y) = 6x^2 \) shows how the function changes with \( x \), and \( f_y(x, y) = 2y \) indicates how the function changes with \( y \). Calculating these derivatives accurately is crucial for solving problems like finding the equation of a tangent plane.

Calculus

Calculus is the branch of mathematics that studies continuous change. It is divided mainly into differential calculus and integral calculus. In differential calculus, we find the derivative, which measures how a function changes as its input changes.
For functions of more than one variable, like \( f(x, y) \), calculus is extended to partial derivatives. These allow us to analyze the change in the function with respect to each variable independently, while others remain fixed.

• Differential calculus helps to understand the behavior and innermost changes of curves and surfaces.
• It's essential for computing rates of change in various fields, like physics, engineering, and economics.

In the context of tangent planes, calculus provides the tools we need to find a plane that touches a surface at a given point and has the same instantaneous rate of change as the surface at that point.

Multivariable Functions

Multivariable functions are functions that have more than one input variable. For instance, \( f(x, y) \) is a multivariable function with two variables, \( x \) and \( y \). Such functions are used to model scenarios where outcomes depend on multiple factors.

• Understanding multivariable functions is crucial for sciences that deal with systems influenced by multiple changing conditions.
• They commonly appear in physics, economics, and engineering spaces where variables interact in complex ways.

When dealing with multivariable functions, we often explore their behavior using partial derivatives. This is particularly useful when finding tangent planes. The tangent plane to a function at a point provides an approximation of the surface near that point. It helps us understand the local properties of the function in contexts involving optimization, constraint problems, and surface analysis.