Question

Find \(z=f(x, y)\) and use the total differential to approximate the quantity. \(\sin \left[(1.05)^{2}+(0.95)^{2}\right]-\sin \left(1^{2}+1^{2}\right)\)

Short answer

To approximate the quantity \(\sin \left[(1.05)^{2}+(0.95)^{2}\right]-\sin \left(1^{2}+1^{2}\right)\), apply the total differential to the function \(z=\sin \left(x^{2}+y^{2}\right)\). The approximate value can be obtained by calculating the total derivative at the point (1,1)

Step-by-step solution

Identify Function and Variables

In the given problem, \(z=\sin \left(x^{2}+y^{2}\right)\). Here, x and y are variable quantities while z is the dependent variable. The values of x and y are given as x=1 and y=1 with small increments 0.05 and -0.05 respectively.

Calculate Partial Derivatives

The next step is to calculate the partial derivative of z with respect to x and y. The result are \(\frac{\partial z}{\partial x} = 2x \cos (x^{2} + y^{2})\) and \(\frac{\partial z}{\partial y} = 2y \cos (x^{2} + y^{2})\). Calculate the derivatives at the point (1,1) using these formulas.

Apply the Total Differential

The total differential dz is given by \(dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy\). Substitute the obtained values from Step 2 into this equation alongside the small increments for x and y mentioned before.

Compute the Approximate Value

By adding \(\sin \left(1^{2}+1^{2}\right)\) to dz obtained in step 3, the approximate value of \(\sin \left[(1.05)^{2}+(0.95)^{2}\right]\) can be found.

Key concepts

Partial Derivatives

At the heart of multivariable calculus, partial derivatives represent how a function changes as each variable is altered separately. Imagine holding one variable constant and allowing another to vary; this is effectively what a partial derivative measures. For the function in our example, which is a function of both x and y, we denote partial derivatives with respect to x and y by \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\), respectively. When calculating these, we treat all other variables as constants and differentiate the function with respect to the variable of interest. This concept is crucial in determining the rate of change of a multivariable function in a specific direction and is a fundamental operation in both theoretical and applied mathematics.

To compute \(\frac{\partial z}{\partial x}\) for \(z = \sin (x^{2} + y^{2})\), we differentiate with respect to x, treating y as a constant, yielding \(2x \cos(x^{2} + y^{2})\). Similarly, for \(\frac{\partial z}{\partial y}\), we treat x as constant and differentiate with respect to y, obtaining \(2y \cos(x^{2} + y^{2})\). These expressions represent the instantaneous rates of change of the sine function with respect to x and y.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of more than one variable. While single-variable calculus asks questions about the slope of a curve at a point, multivariable calculus involves surfaces and contours in higher dimensions. This complexity allows us to solve real-world problems from various fields such as engineering, physics, and economics.

In our problem, we encounter a function \(z = \sin (x^{2} + y^{2})\), which is dependent on two variables, x and y. Understanding how z changes as both x and y vary simultaneously is what multivariable calculus equips us to do. Tasks such as finding the slope of a function at a point in a multidimensional space, maximizing or minimizing functions, and even approximating values all rely on concepts from this rich field of mathematics. The total differential, which is a linear approximation of how the function changes, is an excellent tool from multivariable calculus applied to find approximate values of functions near a known point.

Sin Function Approximation

Approximating the value of functions like the sine function can be particularly challenging when dealing with non-linear behavior. However, differential calculus provides tools for making these approximations both accessible and precise. When we consider the function \(z = \sin (x^{2} + y^{2})\), we are dealing with a composition of a trigonometric function and a polynomial. Approximating this function's value near a point involves understanding its local linear behavior, which is effectively captured by its first-order partial derivatives.

The sin function approximation made through the total differential takes into account the initial value at a known point and the rate of change at that point with respect to each variable. By working with a linear approximation using partial derivatives, we can find an approximate value for \(\sin ((1.05)^{2} + (0.95)^{2})\) that is accurate to a degree of certainty for small perturbations in the values of x and y from the point (1,1).

Differential Calculus

Differential calculus is the branch of mathematics that deals with the rate at which quantities change. It is the foundation for calculating slopes, rates of change, and approximating function values using differentials. In a nutshell, differential calculus focuses on the concept of the derivative, which represents an instantaneous rate of change.

In relation to our exercise, the derivative takes the form of partial derivatives when dealing with functions of multiple variables. The total differential \(dz\) is then constructed out of these partial derivatives to approximate the change in the function z. This is calculated as \(dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy\). By considering small changes in x and y (denoted by dx and dy), we can determine the approximate change in z. This approach is extremely useful for estimating values of complex functions at points where the exact calculation might be cumbersome or even impossible to determine analytically.