Question

18.01 @20:00 How do I factor (1/delta x) out? Why do the terms (xsubzero+deltax) and (xsubzero) from the original 1/(xsubzero+deltax) - 1/(xsubzero)change signs when it's rewritten as (1/deltax)((xsubzero-xsubzero+deltax)/((xsubzero+deltax)(xsubzero)))

Answer 1

For those who are unclear about where we are, this is from lecture 1 of 18.01 (not the Scholar version). Prof. Jerison is working with this somewhat unwieldy expression:\[\frac{ \frac{ 1 }{ x_0+\Delta x }-\frac{ 1 }{ x_0 } }{ \Delta x }\]He performs two operations at the same time, which may be why it is difficult to see what he's doing. One operation is just rearranging the overall fraction to make it more manageable. Just as we can do this\[\frac{ a-b }{ c }=\frac{ 1 }{ c }(a-b)\]we can perform the same operation on that messier fraction:\[\frac{ \frac{ 1 }{ x_0+\Delta x }-\frac{ 1 }{ x_0 } }{ \Delta x }=\frac{ 1 }{ \Delta x }\left( \frac{ 1 }{ x_0+\Delta x }-\frac{ 1 }{ x_0 } \right)\]All we've done here is rearrange the fraction to make it easier to work with the numerator. As you work through calculus you'll often find it convenient to do this because many of the calculations produce top-heavy fractions.

The second operation is to create a common denominator for the two fractions between the parentheses. Here again Prof. Jerison combines steps, making it a little difficult to see what he's doing. Here's an expanded version:\[\frac{ 1 }{ \Delta x }\left( \frac{ 1 }{ x_0+\Delta x }-\frac{ 1 }{ x_0 } \right)=\frac{ 1 }{ \Delta x }\left( \frac{ 1 }{ x_0+\Delta x }\frac{ x_0 }{ x_0 }-\frac{ 1 }{ x_0 } \frac{ x_0+\Delta x }{ x_0+\Delta x }\right)\]\[=\frac{ 1 }{ \Delta x }\left( \frac{ x_0 }{x_0( x_0+\Delta x) }-\frac{ x_0+\Delta x }{ x_0(x_0+\Delta x) }\right)\]\[=\frac{ 1 }{ \Delta x }\left( \frac{ x_0-(x_0+\Delta x) }{ x_0(x_0+\Delta x)}\right)\]In the first step we're simply multiplying each fraction by something that's equal to 1, but doing it in a way that produces a common denominator. This is called cross multiplying because we multiply one fraction by the denominator of the other. The end result of this particular operation is the appearance that something switched signs, but that's not what happened.