The question I'm stuck on is this:
Let f(x,y) = (x+1)^2 + y^2
I'm asked to find the partial derivatives and then evaluate them at (1,2). From there, find L(x,y), the linear approximation to f(x,y) at (1,2).
That part I'm ok with. I got the following:
f_x(x,y) = 2(x+1)
f_y(x,y) = 2y...
Thanks for the replies. It's helped me get my head around these problems better.
I like jbunniii's solution. very clever and succinct.
For this question:
\lim_{n \to \infty} \frac {n^p}{a^n} = 0
my solution is:
Apply L'Hopital's Rule:
\frac{p}{ln(a)} [\lim_{n \to \infty} \frac...
I've got a couple of problems I'm stuck on. Any help gratefully received!
Test for convergence/divergence:
\sum_{n=1}^{\infty} \frac {(n+1)}{n^3 ln(n+2)}
What test should I do here? Can I rearrange the equation to be:
\frac{(\frac{1}{n^2} + \frac{1}{n^3})}{ln(n+2)}
and then use...
I've got this question to do:
A bank is attempting to determine where its assets should be invested during the current
year. At present $500 million is available for investment in bonds, home loans, car loans,
and personal loans. The annual rate of return on each type of investment is known...
sorry I should have stated the problem a bit more in detail.
Once I've got the eigenvalues, I'm to put them into the equation
\lambda1x^2 + \lambda2y^2 = 9
so knowing which is which is important as swapping produces vastly different graphs with either:
9x^2 - y^2 = 9 giving x^2 - y^2/9 = 1...
I have a problem with determining eigenvalues. This is what I've got thus far:
Homework Statement
Identify and sketch the graph of the quadratic equation
4x² + 10xy + 4y² = 9
The Attempt at a Solution
We put it in the matrix form:
\begin{pmatrix} 4 & 5 \\
5 & 4 \\
\end{pmatrix}
Now we find...
Question One:
Train B travels 10km/hr faster than train A. Thus it's speed, relative to A, is 10km/hr.
Train A leaves 1/2 hour earlier meaning, at 25km/hr, it's 12.5km ahead.
So how long will it take train B, moving at a relative speed of 10km/hr, to travel 12.5km?
think about it a second:
you've already found that x^{2} - 8x + 11
Now you need to find y - 8y^{1/2} + 11
If you substituted x^{2} = y,
you would have x^{2} - 8x + 11
which you already have the answer to. If x = 4+/-\sqrt{5},
what is x^{2} (i.e. y) going to equal?
yay, got it right! Off to the post office I scurry.
And that other bit just came out poorly due to bad formating. It looks better in my assignment :wink:
Okay, I'm 99% sure I've got the right answer here, but I just wanted to make certain before I send my assignment in. It's the last question and has been bugging me for the last few days until I had an eureka moment just a few minutes back.
(In case you're wondering, I'm doing my studies by...
Sometimes it's easier to stick with whole numbers and fractions:
length of side = \sqrt{3}
length of QT = \frac{1}{2}\sqrt{3}
area of triangle QTS = \frac{1}{2}*\frac{\sqrt{3}}{2}*\sqrt{3}
=\frac{\sqrt{3}}{2}*\frac{\sqrt{3}}{2}
and what does \sqrt{3}*\sqrt{3} = ?
divide that by 4 (1/2 *...
Just a little. It wasn't helped by HoI using the wrong number! (it should read 10/3 not 4/3 - the 4/3 is correct but HoI skipped a line). Also you're only / by H in your equation whereas you should be / by W + H.
Here's how I'd do it:
Let's call the time taken to drive home x.
Time taken...
Okay. Here's what I got for the box question:
base of box = x, height = h.
Volume of box = x²h = 2000; therefore h = 2000/x²
Surface area of box = 2x² + 4xh
Assume the sides cost $a p/cm².
Then the total cost of the sides will be 4axh. Since the top and bottom cost twice as much (ie...
[SOLVED] Maximum area/volume problems
I have two here. One I feel I've solved but am just looking for reassurance (I'm needy that way!), The other I've got an answer but I know I've missed something.
Homework Statement
1. A rectangle is inscribed in a 6/8/10 cm right-angle triangle where two...
By that I assume you mean this: 72x^3-16x+45) = 4x^2(18x^3-12x+15)
If so, expand back out. the two sides don't match up.
That would indicate a flat spot then, not a max/min point. You can find that out by checking to see if the signs on f''(x) change at that point. If they don't then it isn't...
You really need to recheck your work there. Your f'(x) is correct as far as I can see, but not when you reduce it to x^2(18x^3-12x+15). Expand it out and you get a much different answer to your first one of f'(x).
Also, your f''(x) is wrong. 72x^5 differentiated doesn't equal 288x^3 Nor does...
third derivative is 4x^{3}-8x
solving for f'''(x) = 0 gives us
x = +/-\sqrt{2} and x = 0.
minima points of f''(x) are at x = (-\sqrt{2}, -4) & (\sqrt{2}, -4)
maxima point of f''(x) is (0, 0).
I still confused here! How does this help me find the y co-ordinates for inflection points of...
Homework Statement
Suppose the graph f''(x) of a function is given by:
(see attachment)
(a) Find all points of inflection of f(x)
The Attempt at a Solution
First I figured, by looking at the graph and seeing the intercept points [(-2,0),(0,0),(2,0)] that f''(x) = x^{4}-4x^{2}
solving for...
Right, I think I'm getting on okay with these. Thanks for all the help.
I'm still having issues with c though.
I tried doing it a different way:
Now correct me if I'm wrong but:
cos^{4}(t^{2} + e^{2t})
can be written as
(cos(t^{2} + e^{2t}))^{4}
right? (please say it is!)
And then...
I think the problem here is that Francis is getting confused between dependent vs independent probability. (sorry there's prob a better way to say that but that's the best I can come up with!)
By this I mean that he (assuming Francis is a he) is correct in saying the probability of the next...
Maybe drawing it 1st will give you an idea of what the equation should be.
A line of inlcination 180\cdot going through (0,2) would have the equation: y = 2
Well I did say I was very rusty!
okay, looking at them again I think where I'm going wrong is that I just differentiating parts seperately, so not using the Product rule correctly. Is that fair to say?
For a), Is this closer to the answer:
(x-2)^{3}.-2e^{-2x} + 3(x-2)^{2}.e^{-2x}
and then...
Help please on these. I'm extremely rusty with differentiation and want to know if I've got the right answers here
Homework Statement
differentiate the following functions (you do not have to simplify):
a. f(x)=(x-2)^{3}e^{-2x}
b. f(x) = cos x / ln (x^{2} + x)
c. g(t) = cos^{4}(t^{2} +...
solving the second equation is easy enough - just make y the subject.
As for the first, I find it's easier to rewrite it without using negative powers.
x^{-2/3} = \frac{1}{x^{2/3}}
From there, it's just a matter of rearranging to make y the subject again.
From there, you can find x...
A. Think about it for a second:
If each successive swing is less than the one prior, then how could the 10th swing be 11 feet more than the 1st?
Looks to me like you've got the total sum of the ten swings, not the length of the 10th.
B. We know that the 1st swing is 2 foot. Each...
minor quibble, but you should really write that as ln(1.08), not ln1+0.08, as it does make it look like you're wanting to multiply t by the ln of 1 and then add 0.08.
If you did this, you're answer would end up being out by a little over 8%.