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Chapter 14: Problem 50

Are the statements true or false? Give reasons for your answer. If \(f(x, y)\) is a function of two variables and \(f_{x}(10,20)\) is defined, then \(f_{x}(10,20)\) is a scalar.

Short Answer

Expert verified
True; \(f_x(10, 20)\) is a scalar.

Step by step solution

01

Understanding the Notation

The notation \(f_x(10, 20)\) represents the partial derivative of function \(f\) with respect to variable \(x\) evaluated at the point \((10, 20)\). This means we are finding the rate of change of \(f\) in the \(x\) direction at this specific point.
02

Analyzing Result Type of a Partial Derivative

A partial derivative like \(f_x(10, 20)\) quantifies how much the function's output changes as we vary \(x\), while holding \(y\) constant at 20. The result of a partial derivative is a single number (a scalar) that indicates this rate of change at the specified point.
03

Conclusion About the Statement

Since \(f_x(10, 20)\) indeed yields a scalar value as a result of the partial derivative calculation, the statement given in the problem is correct.

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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 single-variable calculus to functions of two or more variables. Imagine you are dealing with a surface in three-dimensional space. Here, instead of just having a curve like in single-variable calculus, you can have a whole surface or a multi-faceted structure. In multivariable calculus, you deal with functions like \(f(x, y)\), where the function has more than one input. This allows us to analyze phenomena where there are multiple influencing factors.
  • In multivariable calculus, you can find partial derivatives, which tell you how the function changes as one variable changes, while the other variable(s) are kept constant.
  • This is especially useful in optimizing a function or understanding how a function varies in different directions.
Scalar
A Scalar is a single number, unlike vectors or matrices that contain multiple numbers. In the context of partial derivatives, when we evaluate something like \(f_x(10, 20)\), we are looking for a scalar. The partial derivative result is a scalar because it provides a specific value describing the rate of change. Think of it as asking, "How steep is the hill if I walk in this particular direction starting at point \((10, 20)\)?" The answer is a single number, not a complicated array of values.
  • Scalars are fundamental in understanding measurements and dimensions in calculus.
  • They provide clear and concise meaning to derivatives and their effects.
  • This simplicity is what's desired when working with rates of change across multiple dimensions.
Function of Two Variables
A Function of Two Variables, such as \(f(x, y)\), involves a relationship wherein one quantity depends on two inputs. This is an extension from a regular single-variable function \(f(x)\) but offers much richer information and interaction possibilities by including a second influence.
  • In a function of two variables, output changes depending on the variations in any of the two input variables.
  • Partial derivatives come into play to examine these variations more closely. For example, \(f_x(x, y)\) examines the function's sensitivity to changes in \(x\) while fixing \(y\).
  • This relationship is useful in practical applications where multiple factors influence an outcome, such as economics (supply and demand), physics (temperature and pressure), or engineering (stress and strain).
Being able to understand and interpret these functions is crucial for solving dynamic problems where more than one variable significantly influences the output.

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Most popular questions from this chapter

The tastiness, \(T\), of a soup depends on the volume, \(V\) of the soup in the pot and the quantity, \(S,\) of salt in the soup. If you have more soup, you need more salt to make it taste good. Match the three stories (a)-(c) to the three statements (I)-(III) about partial derivatives. (a) I started adding salt to the soup in the pot. At first the taste improved, but eventually the soup became too salty and continuing to add more salt made it worse. (b) The soup was too salty, so I started adding unsalted soup. This improved the taste at first, but eventually there was too much soup for the salt, and continuing to add unsalted soup just made it worse. (c) The soup was too salty, so adding more salt would have made it taste worse. I added a quart of unsalted soup instead. Now it is not salty enough, but I can improve the taste by adding salt. (I) \(\partial^{2} T / \partial V^{2} < 0\) (II) \(\partial^{2} T / \partial S^{2} < 0\) (III) \(\partial^{2} T / \partial V \partial S > 0\)

Find the gradient of the function. Assume the variables are restricted to a domain on which the function s defined. $$z=x \frac{e^{y}}{x+y}$$

Find the gradient of the function. Assume the variables are restricted to a domain on which the function s defined. $$f(x, y)=\ln \left(x^{2}+y^{2}\right)$$

In a room, the temperature is given by \(T=f(x, t)\) degrees Celsius, where \(x\) is the distance from a heater (in meters) and \(t\) is the elapsed time (in minutes) since the heat has been turned on. A person standing 3 meters from the heater 5 minutes after it has been turned on observes the following: (1) The temperature is increasing by \(1.2^{\circ} \mathrm{C}\) per minute, and (2) As the person walks away from the heater, the temperature decreases by \(2^{\circ} \mathrm{C}\) per meter as time is held constant. Estimate how much cooler or warmer it would be 2.5 meters from the heater after 6 minutes.

Find the differential \(d f\) from the gradient. $$\text { grad } f=\left(2 x+3 e^{y}\right) \vec{i}+3 x e^{y} \vec{j}$$
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