Hmm, your explanation is great! I am yet to confidently say I get it tho!
My teacher has given me some sample problems from Schaum's Outlines to go through which start easy and get harder.
I am hoping to get through them before the exam and also some practice at $related rates$ word problems...
Thanks Mark for the extra explanation.
What happens to the $"+c"$ after the anti-derivative?
Is it a case of $+c$ on the first integral and $-c$ on the other?
Cheers Nemo
Hi Mark,
So after applying the FTOC:
\d{}{x}\int_{a}^{x} f(t) \,dt = f(x)h(t)=-k\cdot t+c
Which set to zero:
h(0)=-k\cdot 0+c
We get:
c=0Which makes sense as when the height is zero the surface area is also zero.
Does my explanation make sense?
Cheers Nemo
Hi Mark, Ok, so I have been working on this one for a while now, and think I have solved part a from your help.
I feel like it is simple algebraic manipulation but wish I fully understood the concepts.
So if we have:
V=\int_{0}^{h}A(x) \,dx
Then setup the derivative with respect to $t$...
For Part C:
Using the formula for volume around an axis:
\pi\int_{a}^{b}r^2 \,dx
I can set up my integral as:
\pi\int_{0}^{1}(\frac{e^{x}+e^{-x}}{2})^2 \,dx
Simplifying and taking the indefinite integral (Take too long to type in my working out):
\pi \int \frac{1}{4}(e^{x}+e^{-x})^2 \,dx =...
Hi Mark, to find the length,
I think its the following,
Using the formula:
Length=\int_{a}^{b}\sqrt{1+f'(x)^2} \,dx
Knowing that the derivative of \frac{e^{x}+e^{-x}}{2} = \frac{1}{2}(e^{-x}+e^{x})
We can plug this into get:
Length=\int_{a}^{b}\sqrt{1+(\frac{1}{2}(e^{-x}+e^{x}))^2} \,dx...
I am still confused even with your explanation, sorry.
How do you differentiate a function or apply the chain rule when it is only $A(x)$.
Totally lost.
Cheers Nemo
\frac{e^{2x}+2+e^{-2x}}{4}
Can become:
\frac{e^{2x}+2(e^xe^{-x})+e^{-2x}}{4}
Which is in the format of:
(a+b)^2=A^2+2ab+b^2
Which then gives us:
\frac{(e^{x}+e^{-x})^2}{4}
Which is finally equal to:
(\frac{e^{x}+e^{-x}}{2})^2
which is the function we are trying to get.
Wow! I would not...
Hi Mark,
I am completely lost now,
When I read: \d{V}{t}=\frac{d}{dt}\int_0^h A(x)\,dx=A(h)\d{h}{t}
I am thinking that the derivative of the volume with respect to time is equal to the derivative of time from $0$ to $h$ of $A(x)dx$
If $A(x)dx$ is the area of the surface, the volume...
So I have been working on this and must admit I am overwhelmed!
I have found that we are missing a $1+$ to the front of our formula to be equal to the original 1+({\frac{e^{x}-e^{-x}}{2}})^2
By using the exponent rule a^b\cdot a^c=a^{b+c}
I can simplify the centre to...
Hi Mark,
When I work it out I get:
V=\int_0^h A(x)\,dx = \frac{1}{2}Ah^2
By using the Power rule and the F.T.O.C, this seems very abstract for me as I am struggling to understand what the dx truly means in this context. At a basic level, I know that dx is the same as \Delta x and visually I...
Hi Community,
I have the following problem and I am completely stuck. I really struggle to get my head around how to break down these questions into chunks that I can then apply the math to.
From what I can see so far, I have a to be able calculate the surface area at any height to get the...
Hi Community,
I have the following problem and I would like some help in understanding part a.
So far I far I have been able to show that:
1+\frac{(e^x-e^{-x})^2}{4} = \frac{(e^x)^2-2(e^x-e^{-x})+(e^{-x})2}{4}+1
But I am unsure of how to proceed.
Also any pointers on how to look at the...