Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(f(x, y)=x^{2}+2 y^{2}\). a. Find \(f_{x}(2,1)\) and \(f_{y}(2,1)\). b. Interpret the numbers in part (a) as slopes. c. Interpret the numbers in part (a) as rates of change.

Step-by-Step Solution

Answer

The partial derivatives, \(f_x(x, y) = 2x\) and \(f_y(x, y) = 4y\), at the point (2,1) are \(f_x(2,1) = 4\) and \(f_y(2,1) = 4\). These values represent the slopes of the tangent lines to the level curves of the function at this point, indicating an increase of 4 units for each unit increase in both x and y directions. Interpreted as rates of change, this means that the function is equally sensitive to changes in x and y at (2,1), with any small change resulting in an increase of 4 times the change in the function value.

Step 1: Find the partial derivatives

To find the partial derivatives, we differentiate the function partially with respect to each variable (x and y). \[\frac{\partial f}{\partial x} = f_x(x, y) = 2x\] \[\frac{\partial f}{\partial y} = f_y(x, y) = 4y\]

Step 2: Calculate \(f_x( 2,1)\) and \(f_y(2,1)\)

Now let's find the \(f_x(2,1)\) and \(f_y(2,1)\) by plugging the values of x=2 and y=1 into the partial derivatives we found: \(f_x(2, 1) = 2(2) = 4\) \(f_y(2, 1) = 4(1) = 4\) So, the values of \(f_x(2,1)\) and \(f_y(2,1)\) are both 4.

Step 3: Interpret the partial derivatives as slopes

The partial derivatives represent the slopes of the tangent lines to the level curves of the function f(x,y) at the point (2,1). In other words, they indicate how much f(x,y) is changing in the x and y directions. \(f_x(2,1) = 4\) means that at the point (2,1), the slope of the tangent to the level curve of f(x,y) in the x direction is 4. This implies that the function f(x,y) increases at a rate of 4 units for each unit increase in x. Similarly, \(f_y(2,1) = 4\) means that at the point (2,1), the slope of the tangent to the level curve of f(x,y) in the y direction is 4. This implies that the function f(x,y) increases at a rate of 4 units for each unit increase in y.

Step 4: Interpret the partial derivatives as rates of change

The partial derivatives, as rates of change, indicate how sensitive the function f(x,y) is to changes in the variables x and y at the point (2,1). Both \(f_x(2,1) = 4\) and \(f_y(2,1) = 4\) imply that the function f(x,y) is equally sensitive to changes in both x and y at the point (2,1). Therefore, small changes in either x or y will result in an increase of 4 times the change in the function value at this point.

Key Concepts

Understanding Slopes of Tangent Lines

When you hear about slopes of tangent lines, think about how a hill slopes. Just like a hill can be steep or gentle, tangent lines can have different slopes. These slopes tell us how a function is behaving at a specific point.
Partial derivatives help us find the slopes of the tangent lines to the curves in a function of multiple variables. When we take the partial derivative with respect to one variable, we're finding the slope of the tangent line as we hold the other variables constant.
For example:

• The partial derivative \(f_x(2,1) = 4\) indicates the slope of the tangent line in the x-direction. It means at the point \((2,1)\), if we imagine slicing the surface along the x-axis, the increase is 4 units vertically for every 1 unit horizontally.
• Similarly, \(f_y(2,1) = 4\) tells us about the slope in the y-direction. At that same point, the function rises 4 units vertically per 1 unit horizontally in the y-direction.

This gives us a clear picture of how the function behaves locally, around a given point.

Interpreting Rates of Change

Rates of change are all about understanding how things are shifting or evolving. In the context of multivariable functions and partial derivatives, these rates tell us how a function's value changes as each variable changes.
Just like in single-variable calculus, where you learn that the derivative gives you a rate of change, partial derivatives perform the same role, but for each variable separately.
From our exercise:

• The derivative \(f_x(2,1) = 4\) means that at the point \((2,1)\), a change in \(x\) will cause a change in the function’s value at a rate of 4 units per unit change in \(x\). So, if \(x\) goes up by a little, the value of \(f(x, y)\) increases by 4 times that change.
• Similarly, \(f_y(2,1) = 4\) shows how the function's value responds to changes in \(y\). For every small increase in \(y\), the function value increases by 4 times that change.

These rates of change can help predict how the function value responds to small modifications in its input variables.

Exploring Multivariable Calculus

Multivariable calculus extends the ideas of calculus into functions that involve more than one input variable. This is super useful in many real-world situations where you have more than one factor influencing an outcome.
In multivariable calculus, we study how these functions behave using concepts like partial derivatives, gradients, and level surfaces.

• Partial derivatives are foundational because they allow us to analyze how each individual variable affects the function while ignoring others.
• Level curves or surfaces represent sets of points where the function has the same value, kind of like elevation lines on a map.

The function in our exercise, \(f(x, y) = x^2 + 2y^2\), is a simple case. It lets us explore these concepts by calculating how steep the function's surface is at any point. The slopes and rates of change we computed show us how flexible multivariable calculus can be in giving insights into a function’s behavior in every possible direction.