Problem 1,v, Ch.1, Spivaks 4th Ed Calculus

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In summary, the user is struggling with Spivak's solution to Problem 1, v, and is trying to understand how to get rid of the middle section in the equation x^n - y^n = (x-y)(x^n-1 + x^n-2y + ... + xy^n-2 + y^n-1). The solution involves multiplying the two factors on the right and noticing that all but the first and last terms will drop out, resulting in xn - yn.
  • #1
linwoodc3
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Hi all:

Getting back into Physics so doing self study. Having an issue with Spivak's solution to Problem 1, v.
x^n - y^n= (x-y)(x^n-1+x^n-2y+...+xy^n-2+y^n-1

Got all the way to:
x^n+x^2y^n-2-x^n-2y^2-y^n

How do i get rid of the middle section?
 

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  • #2
linwoodc3 said:
Hi all:

Getting back into Physics so doing self study. Having an issue with Spivak's solution to Problem 1, v.
x^n - y^n= (x-y)(x^n-1+x^n-2y+...+xy^n-2+y^n-1

Got all the way to:
x^n+x^2y^n-2-x^n-2y^2-y^n

How do i get rid of the middle section?

Your attachment is so fuzzy it's difficult to read, and the lack of parentheses above makes it harder to understand what you have written than it needs to be.

Are you trying to show that if you multiply the two factors in your first equation, you get xn - yn?

If so, multiplying the factors on the right gives
xn + xn - 1y - xn - 1y + xn - 2y2 - xn - 2y2 + ... + xyn - 1 - xyn - 1 - yn. Notice that instead of multiplying all of the terms in the larger factor by x, and then repeating the process by multiplying all of the terms by -y, I have instead alternated to get partial products where the exponents add up to the same value.

The first partial product is xn. The next two products come from multiplying x by xn -2y, resulting in xn-1y, and by multiplying -y by xn-1, resulting in -xn - 1y. These two terms add to zero, as do the following pairs of terms.

Doing the multiplication this way makes it clear that all but the first and last terms will drop out.
 

1. What is Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus?

Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus is a math problem that involves finding the derivative of a function using the limit definition of a derivative. It is a common problem used to introduce students to the concept of derivatives and their calculation.

2. How do you solve Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus?

To solve Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus, you must use the limit definition of a derivative. This involves finding the limit of a function as the change in the independent variable approaches 0. With this limit, you can then use algebraic manipulation to find the derivative of the given function.

3. Why is Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus important?

Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus is important because it introduces the fundamental concept of derivatives in calculus. It helps students understand the relationship between the slope of a curve and the derivative of a function, which is essential in many applications of calculus.

4. What are the key concepts needed to solve Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus?

The key concepts needed to solve Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus are the limit definition of a derivative, algebraic manipulation, and understanding the relationship between the slope of a curve and the derivative of a function.

5. Are there any tips for solving Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus?

Some tips for solving Problem 1,v in Chapter 1 of Spivak's 4th Edition Calculus include understanding the limit definition of a derivative, practicing algebraic manipulation, and checking your answer by using the derivative rules. It can also be helpful to break the problem down into smaller, manageable steps.

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