Question

Consider the function \(f(x, y)=5 x^{2} y\). (a) Evaluate this function and its first partial derivatives at the point \(A(2,3)\). (b) Suppose we consider point \(A\). Suppose small changes, \(\delta x, \delta y\), are made in the values of \(x\) and \(y\) so that we move to a nearby point \(B\). It is possible to show that the corresponding change in \(f\) is given approximately by \(\delta f \approx \frac{\partial f}{\partial x} \delta x+\frac{\partial f}{\partial y} \delta y\), where the partial derivatives are evaluated at the original point \(A\). Use this result to find the approximate change in the value of \(f\) if \(x\) is increased to \(2.1\) and \(y\) is increased to \(3.2\). (c) Compare your answer in (b) to the value of \(f\) at \((2.1,3.2)\)

Short answer

Question: Using the function \(f(x, y) = 5x^2y\), find the approximate change in the value of the function at point A(2,3) when \(x\) is increased to \(2.1\) and \(y\) is increased to \(3.2\) using the first partial derivatives. Compare the approximated change with the actual value of the function at point \((2.1, 3.2)\). Answer: The approximate change in the value of the function at point A(2,3) is \(10\), and our approximation results in a function value of \(70\) at point B(2.1, 3.2). The actual value of the function at point B is \(67.032\). There is a small difference between the approximation and the actual value, but the approximation is reasonably close.

Step-by-step solution

a) Find function value and partial derivatives at point A(2,3)

Firstly, we are given the function \(f(x, y) = 5x^2y\). To find the first partial derivatives with respect to \(x\) and \(y\), we differentiate the function partially. To differentiate with respect to \(x\): \(\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(5x^2y) = 10xy\) To differentiate with respect to \(y\): \(\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(5x^2y) = 5x^2\) Now, evaluate the function \(f(x, y)\) and its first partial derivatives at point A(2,3): \(f(2, 3) = 5(2)^2(3) = 60\) \(\frac{\partial f}{\partial x}(2, 3) = 10(2)(3) = 60\) \(\frac{\partial f}{\partial y}(2, 3) = 5(2)^2 = 20\)

b) Approximate change in the value of the function

To approximate the change in the value of the function using the given formula: \(\delta f \approx \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial y}\delta y\) We are given that \(x\) is increased to \(2.1\) and \(y\) is increased to \(3.2\). So, the small changes are: \(\delta x = 2.1 - 2 = 0.1\) \(\delta y = 3.2 - 3 = 0.2\) Now, using the partial derivatives at point A(2,3) that we found in part (a) and plugging in the values of \(\delta x\) and \(\delta y\): \(\delta f \approx 60(0.1) + 20(0.2) = 6 + 4 = 10\) Thus, the approximate change in the value of the function is \(10\).

c) Compare the approximated change to the actual value of \(f\) at \((2.1, 3.2)\)

Now, we will find the actual value of the function at \((2.1, 3.2)\): \(f(2.1, 3.2) = 5(2.1)^2(3.2) = 67.032\) To compare the answers, let's find the value of the function at the new point B with coordinates (2.1, 3.2) using the approximated change: \(f(2, 3) + \delta f = 60 + 10 = 70\) So, our approximation gives us a value of \(70\) for the function at point B (2.1, 3.2), while the actual value is \(67.032\). There is a small difference between the two values, but overall, the approximation is reasonably close.

Key concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus to functions of several variables. It involves the study of functions with more than one input, for example, a function like f(x, y) = 5x^2y from the original exercise, which depends on both x and y.

In this realm of calculus, we explore how these functions change as their inputs change, which is crucial for understanding phenomena in physics, engineering, and economics where multiple factors come into play. Key concepts in multivariable calculus include partial derivatives, multiple integrals, and vector calculus.

Function Approximation

In multivariable calculus, function approximation is a technique used to estimate the value of a function at a point based on the function's value and its derivatives at a nearby point. This concept is exemplified by the exercise, where we approximate the value of f at a new point B using the values of the function and its derivatives at point A.

Function approximation is especially useful when dealing with complex functions for which exact calculations may be difficult or impossible. It is commonly employed in fields such as physics, statistics, and machine learning, where predicting outputs based on inputs is a routine task. The tools of differential calculus, like derivatives, play a fundamental role in these approximations.

First Order Partial Derivatives

First order partial derivatives are derivatives of a function with respect to one variable while holding the other variables constant. These derivatives measure the rate at which the function changes as one particular input changes, and are essential in finding the slope of the function in the direction of that variable.

In the given exercise, we calculate the first order partial derivatives of f(x, y) with respect to both x and y to find how the function changes as either variable changes, while the other remains fixed. Understanding and calculating these derivatives is an important step towards mapping out the behavior of multivariable functions.

Differential Calculus

Differential calculus is a major branch of calculus focused on the study of rates at which quantities change. It's all about the concept of the derivative, which represents an instantaneous rate of change or the slope of a tangent to a curve at a point.

The exercise showcases the application of differential calculus to find the rate of change of a function of two variables. This is done by computing its first order partial derivatives. This application is incredibly powerful as it helps predict how small changes in input variables affect the output of a function, thereby enabling the approximation of function values at points that are not readily calculable.