Difference Quotients

[ktkeiper]

There's one example in my book, but it doesn't explain anything. It just goes through the steps. My teachers didn't teach us this section in class, but it's assigned for homework that we have to turn in. I just don't know where to plug all the numbers into the formula. Any help/explanation would be greatly appreciated!

f(x)=5x-x^2, [f(5+h)-f(5)]/h

 

[ktkeiper]

Oh wow, I feel like an idiot now that I finally figured it out... Thank you!

 

[soroban]

Hello, ktkeiper!

My teachers didn't teach us this section in class, but it's assigned for homework that we have to turn in.
Yeah, I don't know why teachers like to do that!

\(\displaystyle \text{Given: }\:f(x)\:=\:5x-x^2\)

\(\displaystyle \text{Find: }\:\dfrac{f(5+h)-f(5)}{h}\)

I like to separate the Difference Quotient into three steps:

. . [1] Find \(\displaystyle f(5+h)\) . . . Replace \(\displaystyle x\) with \(\displaystyle 5+h\) ... and simplify.

. . [2] Subtract \(\displaystyle f(5)\) . . . Subtract \(\displaystyle f(5)\) ... and simply.

. . [3] Divide by \(\displaystyle h\) . . . Factor and reduce.


Here we go . . .


\(\displaystyle [1]\;f(5+h) \:=\:5(5+h) - (5+h)^2\)

. . . . . . . . . .\(\displaystyle =\;25 + 5h - (25 + 10h + h^2) \)

. . . . . . . . . .\(\displaystyle =\;25 + 5h - 25 - 10h - h^2\)

. . . . . . . . . .\(\displaystyle =\;-5h - h^2\)


\(\displaystyle [2]\;f(5+h) - f(5) \;=\;(-5h - h^2) - (5\!\cdot\!5 - 5^2) \)

. . . . . . . . . . . . . . . \(\displaystyle =\;-5h - h^2 - 0\)

. . . . . . . . . . . . . . . \(\displaystyle =\; -5h - h^2\)


\(\displaystyle [3]\;\dfrac{f(5+h)-f(5)}{h} \;=\;\dfrac{-5h - h^2}{h}\)

. . . . . . . . . . . . . . . \(\displaystyle =\;\dfrac{-h(5 + h)}{h}\)

. . . . . . . . . . . . . . . \(\displaystyle =\;-(5 + h)\)