SanjanaP
Help with Increasing and Decreasing Derivatives Please!!
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myininaya
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what is the question exactly?
myininaya
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If f'>0, then f is increasing
If f'<0, then f is decreasing
SanjanaP
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Oh...sorry I thought I had the attachment on
SanjanaP
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I neeed help with C and D
SanjanaP
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@myininaya
myininaya
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so are you having problems with the part highlighted in yellow?
SanjanaP
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Basically...
myininaya
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ok then so you know that g' will tell us if g is decreasing or increasing right?
SanjanaP
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Yeah
myininaya
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so find g' given that:
\[g(x)=\int\limits_{0}^{x} f(t) dt \]
don't look at the picture yet
just tell what g' is
SanjanaP
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f(x)+0.5
SanjanaP
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F(x)+c right?
myininaya
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no i'm sorry that isn't correct
ok let's look at this...
I will rewrite it a little...
\[g(x)=F(x)-F(0)\]
where F'=f
Now differentiate to find g'
SanjanaP
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I'm sorry..so what is g'?
myininaya
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you have to differentiate g(x)=F(x)-F(0) to find g'
can you do that?
SanjanaP
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I'm not sure what differentiate means anymore?
myininaya
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to find derivative
SanjanaP
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It's f'(x)-f'(0)
SanjanaP
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f'(x)(x) right?
myininaya
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I will help you out some more.
derivative of g is g'
derivative of F is f (this was given above when I said F'=f)
derivative of a constant is 0
everything i said in this little post right here will need to be used
SanjanaP
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oh....so the derivative of F(x) is f(x). Is that what you mean?
myininaya
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That is what I said
myininaya
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so do you know F(0) is a constant?
SanjanaP
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its 0.5
myininaya
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I think you are thinking of f(0) not F(0)
f is given not F
SanjanaP
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yea?
myininaya
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Anyways F(0) is a constant.
Because F is a just a function of x
any if you plug in a number for x then you will definitely receive a constant
----------------------------------------------------
example:
Say F(x)=cos(x)
well F(0) is definitely a constant because F(0) is 1 and 1 never changes (it is and will always remain 1)
myininaya
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So going back to \[g(x)=\int\limits_{0}^{x} f(t) dt \\ g(x)=F(x)-F(0)\]
can you differentiate g now?
myininaya
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what that means is you will have to differentiate both sides (not just one side)
SanjanaP
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Can you just tell me where it is increasing and concave up...so I'll try to figure it out?
myininaya
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Try to use what I said earlier...
derivative of g is g'
derivative of F' is f
derivative of a constant is 0
you can do this
SanjanaP
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I would prefer the answer because I have to write the explanation
anyway.
SanjanaP
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Okay...so you g(x)= F(x)-F(0)
SanjanaP
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that means g'(x) = f(x)-0?
myininaya
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I'm not going to give just the answer. Sorry.
But right g'=f
so that means the picture given is g'
myininaya
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and you know if g'>0, then g is increasing
and you know if g'<0, then g is decreasing
myininaya
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where on the picture that is given is g' above the x axis (because that is where g is increasing)
and when g' is below the x-axis that is where g is decreasing
SanjanaP
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what about concave up?
SanjanaP
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