Calculus Michael Spivak 3th ed. Pag 46: Function as a rule

In summary, on page 46, the author begins to question the definition of a function, realizing that it may not simply be a rule. He provides two examples, f(x) = x² and f(x) = x² + 3x + 3 - 3(x+1), and notes that they are different. However, he then rewrites the second example as f(x) = x² + 3x + 3 - c(x+1), realizing that the initial version may have been a small mistake. This leads to a discussion among colleagues about typos in books and the potential for newer editions to have errors.
  • #1
xorg
15
0
Pag 46 says:
NGw7QSt.png

He begins to do a rethink about what is a function.
Function is a rule? If it is a rule then:
f(x) = x²
and
f(x) = x² +3x + 3 -3(x+1)
are different. But that was not what we wanted.

But he rewrites, this way:
f(x) = x²
and
f(x) = x² +3x +3 -c(x+1)

Apparently this was a small mistake.

I would like to hear the opinion of colleagues. Thank you.
 
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  • #2
Looks like a typo yes. Every book has a few, you should just correct it as you did and carry on.
 
  • Like
Likes xorg
  • #3
I have the first edition, which doesn't have the typo. Sometimes books get worse instead of better when new editions are published. :frown:
 
Last edited:
  • #4
Ah, entropy...
 

Related to Calculus Michael Spivak 3th ed. Pag 46: Function as a rule

1. What is a function as a rule in Calculus?

A function as a rule is a way of describing how one quantity (the input or independent variable) is related to another quantity (the output or dependent variable). It is represented by an equation or set of rules that determines the output value based on the input value.

2. How is a function as a rule different from an ordinary function?

A function as a rule is a more general concept than an ordinary function. While an ordinary function maps a specific input value to a specific output value, a function as a rule can map a range of input values to a range of output values. It allows for a more versatile representation of relationships between quantities.

3. Can a function as a rule have more than one input variable?

Yes, a function as a rule can have multiple input variables. In this case, the output value will depend on the values of all the input variables. This type of function is known as a multivariable function and is commonly used in advanced calculus and physics.

4. How is a function as a rule graphically represented?

A function as a rule is represented by a graph that shows the relationship between the input and output values. The input values are usually plotted on the x-axis, while the output values are plotted on the y-axis. The shape of the graph can provide information about the behavior of the function, such as its domain, range, and whether it is increasing or decreasing.

5. What is the purpose of using a function as a rule in calculus?

A function as a rule is a fundamental concept in calculus that is used to model and analyze real-world phenomena. It allows us to understand and predict how quantities are related to each other, which is essential in many fields such as physics, economics, and engineering. It also serves as the basis for more advanced calculus concepts, such as derivatives and integrals.

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