Question

Linear approximation a. Find the linear approximation to the function \(f\) at the given point. b. Use part (a) to estimate the given function value. $$f(x, y, z)=\frac{x+y}{x-z} ;(3,2,4) ; \text { estimate } f(2.95,2.05,4.02)$$

Short answer

Question: Use the linear approximation of the function \(f(x,y,z) = \frac{x+y}{x-z}\) at the point (3,2,4) to estimate the function value at the point (2.95, 2.05, 4.02). Answer: The estimated function value at the point (2.95, 2.05, 4.02) using the linear approximation is approximately -5.4.

Step-by-step solution

Compute the partial derivatives

We will compute the partial derivatives of the function \(f(x,y,z) = \frac{x+y}{x-z}\) with respect to x, y, and z. \(f_x = \frac{\partial}{\partial x}f(x,y,z)=\frac{(x-z)+(x+y)}{(x-z)^2}\) \(f_y = \frac{\partial}{\partial y}f(x,y,z)=\frac{1}{x-z}\) \(f_z = \frac{\partial}{\partial z}f(x,y,z)=\frac{-(x+y)}{(x-z)^2}\)

Evaluate the partial derivatives at the given point

Now, we will evaluate the partial derivatives at the given point (3,2,4). \(f_x(3,2,4)=\frac{(3-4)+(3+2)}{(3-4)^2}=5\) \(f_y(3,2,4)=\frac{1}{3-4}=-1\) \(f_z(3,2,4)=\frac{-(3+2)}{(3-4)^2}=-5\)

Find the linear approximation formula

The linear approximation formula for the function \(f(x,y,z)\) at the point (3,2,4) is given by: \(f_L(x,y,z) = f(3,2,4) + f_x(3,2,4)(x-3) + f_y(3,2,4)(y-2) + f_z(3,2,4)(z-4)\) We already know \(f_x(3,2,4)=5\), \(f_y(3,2,4)=-1\), and \(f_z(3,2,4)=-5\). Now we need to find the value of the function at the point (3,2,4): \(f(3,2,4)=\frac{3+2}{3-4}=-5\) So, the linear approximation formula is: \(f_L(x,y,z) = -5 + 5(x-3) - (y-2) - 5(z-4)\)

Estimate the function value at the point (2.95, 2.05, 4.02)

Using the linear approximation formula, we can estimate the function value at the given point (2.95, 2.05, 4.02): \(f_L(2.95,2.05,4.02) = -5 + 5(2.95-3) - (2.05-2) - 5(4.02-4)\) \(f_L(2.95,2.05,4.02) = -5 + 5(-0.05) - 0.05 - 5(0.02)\) \(f_L(2.95,2.05,4.02) = -5 - 0.25 - 0.05 - 0.1\) \(f_L(2.95,2.05,4.02) \approx -5.4\) Therefore, the estimated function value at the point (2.95, 2.05, 4.02) using the linear approximation is approximately -5.4.

Key concepts

Multivariable Calculus

Multivariable calculus is a branch of calculus that extends the concepts of single-variable calculus to more than one variable. It encompasses the study of functions that have multiple inputs, like the function in our exercise, which depends on three variables: x, y, and z. In these cases, understanding how a function changes in relation to each variable becomes crucial.

The heart of multivariable calculus lies in analyzing and understanding the behavior of these functions across various dimensions. For students beginning to explore this field, grasping the idea of functions with several inputs can be challenging, but it also opens up a fascinating world of possibilities, from predicting economic trends to modeling the flow of fluids.

Partial Derivatives

Partial derivatives are a component of multivariable calculus and are essential when you want to understand how a function changes as each variable is varied, while the others are held constant. Think of them as a tool that tells us the slope of the function in the direction of one of the axes in a multi-dimensional space.

In the context of our exercise, we calculated partial derivatives with respect to x, y, and z. The notation \( f_x \), \( f_y \), and \( f_z \) represent the rate at which the function changes in the direction of each respective variable. When these are calculated at a specific point, they provide local slope information which is a building block for approximating the function's behavior near that point.

Tangent Plane Approximation

When working with multivariable functions, such as in the exercise, a helpful way of approximating the function near a point is by using the tangent plane approximation. This is the multivariable equivalent to the tangent line in single-variable calculus. The tangent plane touches the surface created by the function at a given point and provides the 'best' linear approximation to the function at that vicinity.

In practical terms, the tangent plane approximation can be thought of as a flat, infinitely thin sheet that just grazes the surface of the function at a particular point. The linear approximation formula, derived using the function's value at a point along with its partial derivatives, gives us a simplified version of the function that can be used to estimate the function's value near the point of tangency, just like we have done in our step by step solution.