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Chapter 15: Problem 5

Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=3 x^{2} y+2\).

Short Answer

Expert verified
Question: Find the first-order partial derivatives of the function \(f(x, y) = 3x^2y + 2\) with respect to \(x\) and \(y\). Answer: The first-order partial derivatives of the given function are \(f_x = 6xy\) and \(f_y = 3x^2\).

Step by step solution

01

Calculate the partial derivative with respect to x (\(f_{x}\))

To find the partial derivative of \(f(x, y)\) with respect to \(x\), we'll take the derivative of the function while treating \(y\) as a constant. So, \(f_x = \frac{\partial}{\partial x}(3x^2y+2)\). The derivative of \(3x^2y\) with respect to \(x\) is \(6xy\), and the derivative of a constant (like \(2\)) is \(0\). Therefore, \(f_x = 6xy\).
02

Calculate the partial derivative with respect to y (\(f_{y}\))

Now, we'll find the partial derivative of \(f(x, y)\) with respect to \(y\), treating \(x\) as constant. So, \(f_y=\frac{\partial}{\partial y}(3x^2y+2)\). The derivative of \(3x^2y\) with respect to \(y\) is \(3x^2\), and the derivative of \(2\) with respect to \(y\) is \(0\). Therefore, \(f_y = 3x^2\).
03

Final Result

Given the function \(f(x, y) = 3x^2y + 2\), the first-order partial derivatives are \(f_x = 6xy\) and \(f_y = 3x^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multivariable Calculus
Multivariable calculus is an extension of calculus that involves more than one variable. This branch of mathematics is incredibly powerful as it lets us model and analyze systems in multiple dimensions.
Rather than dealing with functions of a single variable, we work with functions of two or more variables, which can describe surfaces and volumes rather than just lines and areas. Some essential topics in multivariable calculus include:
  • Partial derivatives: the core part of the problem we're solving, which involves differentiating with respect to one variable while holding the others constant.
  • Gradient vectors: combining partial derivatives to describe the slope of a multivariable function.
  • Multiple integration: extending concepts like the integral to functions with more than one variable.
Understanding multivariable calculus is crucial not only for mathematics but also for engineering, physics, and a variety of other fields that use mathematical modeling.
Differentiation
Differentiation, at its core, is about finding the rate at which a quantity changes. For a function of a single variable, differentiation gives us the slope of the function at any point. In simple terms, it's like finding how steep a hill is at a particular spot.
When we move to functions with several variables, differentiation takes the form of partial derivatives. These allow us to see how the function changes as each individual variable changes:
  • The partial derivative with respect to a variable measures how the function changes as that specific variable changes, with all other variables held constant.
  • It provides insight into the behavior of complex systems affected by multiple factors.
Here, we found two partial derivatives for the function: one with respect to 'x' and one with respect to 'y', which paint a picture of how the function behaves in different directions.
Functions of Several Variables
Functions of several variables are fundamental in describing situations where more than one input determines the output. They take the form \( f(x, y) \), meaning the output depends on both \( x \) and \( y \).
Such functions often arise in real-world scenarios where multiple parameters interact. For example:
  • A function describing the temperature at any point on a Earth's surface would depend on the geographical coordinates, thus requiring two variables.
  • An economic model predicting profit might depend on both demand and cost input variables.
In this problem, the function \( f(x, y) = 3x^2y + 2 \) uses two variables, \( x \) and \( y \). By calculating partial derivatives, we determined how changes in these variables individually affect the function's output.

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Most popular questions from this chapter

Heron's formula The area of a triangle with sides of length \(a, b\) and \(c\) is given by a formula from antiquity called Heron's formula: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where \(s=\frac{1}{2}(a+b+c)\) is the semiperimeter of the triangle. a. Find the partial derivatives \(A_{\sigma}, A_{b},\) and \(A_{c}\) b. A triangle has sides of length \(a=2, b=4, c=5 .\) Estimate the change in the area when \(a\) increases by \(0.03, b\) decreases by \(0.08,\) and \(c\) increases by 0.6 c. For an equilateral triangle with \(a=b=c,\) estimate the percent change in the area when all sides increase in length by \(p \% .\)

The electric potential function for two positive charges, one at (0,1) with twice the strength of the charge at \((0,-1),\) is given by $$\varphi(x, y)=\frac{2}{\sqrt{x^{2}+(y-1)^{2}}}+\frac{1}{\sqrt{x^{2}+(y+1)^{2}}}$$ a. Graph the electric potential using the window $$[-5,5] \times[-5,5] \times[0,10]$$ b. For what values of \(x\) and \(y\) is the potential \(\varphi\) defined? c. Is the electric potential greater at (3,2) or (2,3)\(?\) d. Describe how the electric potential varies along the line \(y=x\)

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. $$w=f(u, x, y, z)=\frac{u+x}{y+z}$$

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). $$Q(x, y, z)=\frac{10}{1+x^{2}+y^{2}+4 z^{2}}$$

Suppose you make a one-time deposit of \(P\) dollars into a savings account that earns interest at an annual rate of \(p \%\) compounded continuously. The balance in the account after \(t\) years is \(B(P, r, t)=P e^{r^{n}},\) where \(r=p / 100\) (for example, if the annual interest rate is \(4 \%,\) then \(r=0.04\) ). Let the interest rate be fixed at \(r=0.04\) a. With a target balance of \(\$ 2000\), find the set of all points \((P, t)\) that satisfy \(B=2000 .\) This curve gives all deposits \(P\) and times \(t\) that result in a balance of \(\$ 2000\). b. Repeat part (a) with \(B=\$ 500, \$ 1000, \$ 1500,\) and \(\$ 2500,\) and draw the resulting level curves of the balance function. c. In general, on one level curve, if \(t\) increases, does \(P\) increase or decrease?
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