Differential equation for slope

3B1B

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Apr 11, 2021
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I have this slope equation where it wants me to find the slope of two points from a function:

What is the slope of the secant line that intersects the graph of f(x) = 32x at x=0, and x=2?

The first thing I don't get is the power to 2x, what does that mean? does it mean power by 2, then by x or all in one go?

other than that, I believe I do know how to do the rest, by finding the difference and then just plugging the values in the slope formula.


Any help is appreciated.
 
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skeeter

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\(\displaystyle m = \frac{f(2)-f(0)}{2-0}\)
 

AmandasMathHelp

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Feb 9, 2021
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You should multiply x by 2 before you take the 3 to a power. The 2x is the exponent on 3.

For example when x=5 you would do \(\displaystyle 3^{2*5}= 3^{10}\)

It is notable that you can actually rewrite this to do the power of 2 first then x, like you said. But that would look more like this. You will get the same answers either way.

\(\displaystyle 3^{2*x} = (3^{2})^x = 9^x \)
 

3B1B

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Apr 11, 2021
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You should multiply x by 2 before you take the 3 to a power. The 2x is the exponent on 3.

For example when x=5 you would do \(\displaystyle 3^{2*5}= 3^{10}\)

It is notable that you can actually rewrite this to do the power of 2 first then x, like you said. But that would look more like this. You will get the same answers either way.

\(\displaystyle 3^{2*x} = (3^{2})^x = 9^x \)
Thanks for your help!
I understand now.
 

HallsofIvy

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Jan 27, 2012
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As AmandasMathHelp said, \(\displaystyle 3^{2x}= (3^2)^x= 9^x\). For x= 2 that is \(\displaystyle 9^2= 81\).

You could also do this as \(\displaystyle 3^{2(2)}= 3^4= 3(3)(3)(3)= 9(3)(3)= 27(3)= 81\).
 
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