Thread: MATH2111 Higher Several Variable Calculus

1. Re: MATH2111 Higher Several Variable Calculus

$f:[-\pi, \pi]\to \mathbb{R}\text{ satisfied }f(-x)=-f(x), \, f(\pi)=0, \, f^{\prime\prime\prime}(x) = -6$

$\text{Proven in b): The relevant Fourier series is}\\ Sf(x) = \sum_{k=1}^\infty \frac{12(-1)^k}{k^3}\sin(kx)$

c) Give a simple formula for f(x)

$\text{Proven in a): } f^{\prime\prime}(x)=-6x\\ \text{so I wanted to keep integrating up to get }f(x) = -x^3\\ \text{But then what? I still have to satisfy }f(\pm \pi) = 0\\ \text{Do I just define a piecewise function to get out of this issue?}$

2. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
$f:[-\pi, \pi]\to \mathbb{R}\text{ satisfied }f(-x)=-f(x), \, f(\pi)=0, \, f^{\prime\prime\prime}(x) = -6$

$\text{Proven in b): The relevant Fourier series is}\\ Sf(x) = \sum_{k=1}^\infty \frac{12(-1)^k}{k^3}\sin(kx)$

c) Give a simple formula for f(x)

$\text{Proven in a): } f^{\prime\prime}(x)=-6x\\ \text{so I wanted to keep integrating up to get }f(x) = -x^3\\ \text{But then what? I still have to satisfy }f(\pm \pi) = 0\\ \text{Do I just define a piecewise function to get out of this issue?}$
It won't be -x^3. It'll be a different cubic. It will be an odd cubic that satisfies f(pi) = 0.

3. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by InteGrand
It won't be -x^3. It'll be a different cubic. It will be an odd cubic that satisfies f(pi) = 0.
Oh, now that I think about it could it potentially be x(x^2-pi^2)?

4. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
Oh, now that I think about it could it potentially be x(x^2-pi^2)?
Yeah but the third derivative was negative, so it'd be the negative of what you wrote there.

5. Re: MATH2111 Higher Several Variable Calculus

(Stats is done for this sem)

$\\\text{Let }\textbf{f}:\mathbb{R}^n\to \mathbb{R}^n\text{ and }g:\mathbb{R}^n\to \mathbb{R}\text{ be differentiable, and define}\\ h = g\circ \textbf{f}$

$\text{If }\textbf{a}\in \mathbb{R}^n\text{ is a stationary point of }h\text{, what can you say about }\textbf{f}\text{ and }g?$

6. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
(Stats is done for this sem)

$\\\text{Let }\textbf{f}:\mathbb{R}^n\to \mathbb{R}^n\text{ and }g:\mathbb{R}^n\to \mathbb{R}\text{ be differentiable, and define}\\ h = g\circ \textbf{f}$

$\text{If }\textbf{a}\in \mathbb{R}^n\text{ is a stationary point of }h\text{, what can you say about }\textbf{f}\text{ and }g?$
$\noindent Have you tried using the chain rule?$

7. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by InteGrand
$\noindent Have you tried using the chain rule?$
Chain rule was what was on my mind, but I was getting lost at how to use it

$\text{I thought of using Jacobians and saying }Jh(\textbf{a}) = Jg(\textbf{f}(a))Jf(\textbf{a})\\ \text{But then I got lost from there}$

I also started doubting myself, questioning if it's meant to be a Jacobian or just h-dash.

Unless, the answer is just that Jh(a) = 0 so RHS = 0 as well

Edit: Uh, nvm, found a gap in my learning.

8. Re: MATH2111 Higher Several Variable Calculus

Found it really hard to work around the wording for this question. Not sure how to properly use the implicit function theorem.

Not too sure what exactly they want for my matrices (A|B) either

9. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
Edit: Uh, nvm, found a gap in my learning.

10. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
(Stats is done for this sem)

$\\\text{Let }\textbf{f}:\mathbb{R}^n\to \mathbb{R}^n\text{ and }g:\mathbb{R}^n\to \mathbb{R}\text{ be differentiable, and define}\\ h = g\circ \textbf{f}$

$\text{If }\textbf{a}\in \mathbb{R}^n\text{ is a stationary point of }h\text{, what can you say about }\textbf{f}\text{ and }g?$
lel gl with calculus, most aids course

11. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by seanieg89
I thought I had it but I probably didn't. Not sure what I could possibly do next after $Jg(f(\textbf{a})J\textbf{f}(a)=\textbf0$

I really don't feel like that should be the final answer either. Any further guidance?

lel gl with calculus, most aids course
dying

12. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
I thought I had it but I probably didn't. Not sure what I could possibly do next after $Jg(f(\textbf{a})J\textbf{f}(a)=\textbf0$

I really don't feel like that should be the final answer either. Any further guidance?
Idk what they want you to say with that wording and those assumptions really. Basically it just means that the image of f'(a) is a subspace of the kernel of g'(f(a)), or equivalently that the image of f'(a) is orthogonal to the gradient vector of g at f(a).

13. Re: MATH2111 Higher Several Variable Calculus

Figures, should've thought in that direction
_______________________

$\text{Suppose }\textbf{a}\text{ is an interior point of }U\text{ and }\textbf{f}\text{ is differentiable at }\textbf{a}$

$\\\text{Suppose }\det(J_\textbf{a}\textbf{f})\neq 0.\, \text{Can we conclude that }\\\textbf{f}(\textbf{a})\text{ is an interior point of }\textbf{f}(U)?$

14. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
Figures, should've thought in that direction
_______________________

$\text{Suppose }\textbf{a}\text{ is an interior point of }U\text{ and }\textbf{f}\text{ is differentiable at }\textbf{a}$

$\\\text{Suppose }\det(J_\textbf{a}\textbf{f})\neq 0.\, \text{Can we conclude that }\\\textbf{f}(\textbf{a})\text{ is an interior point of }\textbf{f}(U)?$
What have you tried so far? Did you try maybe considering the inverse function theorem?

15. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by InteGrand
What have you tried so far? Did you try maybe considering the inverse function theorem?
That was definitely the starting point, but that just asserted that there exists an open set $V\subseteq U$ that is open, not $\textbf{f}(V)\subseteq \textbf{f}(U)$ right?

Image of an open set under a continuous function might not be open

16. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
Image of an open set under a continuous function might not be open
Generally this is correct, but consider the implications of f being an invertible (locally) continuous map.

17. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
That was definitely the starting point, but that just asserted that there exists an open set $V\subseteq U$ that is open, not $\textbf{f}(V)\subseteq \textbf{f}(U)$ right?
Also, if V is a subset of U, then f(V) is automatically a subset of f(U) (true for any function and sets (that make sense)).

(Or maybe you meant that it doesn't state that f(V) is open.)

18. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by InteGrand
Also, if V is a subset of U, then f(V) is automatically a subset of f(U) (true for any function and sets (that make sense)).

(Or maybe you meant that it doesn't state that f(V) is open.)
Yeah the latter lol
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Just a yes or no answer please because I can't just tell what to use immediately. I know that it converges pointwise to the zero function.

$\text{Does }f_k(x)=\begin{cases}k & \text{if }0 < x \le k^{-1}\\ 0 & \text{otherwise}\end{cases}\text{ converge to 0 uniformly?}$

Because seeing as though it converges pointwise to something continuous I doubt I can use that, and the Weierstrass M-test doesn't even seem relevant.

19. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
Yeah the latter lol
________________________________

Just a yes or no answer please because I can't just tell what to use immediately. I know that it converges pointwise to the zero function.

$\text{Does }f_k(x)=\begin{cases}k & \text{if }0 < x \le k^{-1}\\ 0 & \text{otherwise}\end{cases}\text{ converge to 0 uniformly?}$

Because seeing as though it converges pointwise to something continuous I doubt I can use that, and the Weierstrass M-test doesn't even seem relevant.
No

20. Re: MATH2111 Higher Several Variable Calculus

$\frac{d}{dx}\int_0^{u(x)}f(v(x),y)dy = u^\prime(x) f(v(x),y) + \underbrace{\int_0^{u(x)}\frac{\partial f}{\partial x}(v(x),y)dy}_{\text{how to simplify this further?}}$

21. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
$\frac{d}{dx}\int_0^{u(x)}f(v(x),y)dy = u^\prime(x) f(v(x),y) + \underbrace{\int_0^{u(x)}\frac{\partial f}{\partial x}(v(x),y)dy}_{\text{how to simplify this further?}}$
Also the first term on your RHS has a mistake/typo (shouldn't have y).

22. Re: MATH2111 Higher Several Variable Calculus

Originally Posted by leehuan
$\frac{d}{dx}\int_0^{u(x)}f(v(x),y)dy = u^\prime(x) f(v(x),y) + \underbrace{\int_0^{u(x)}\frac{\partial f}{\partial x}(v(x),y)dy}_{\text{how to simplify this further?}}$
chain rule.

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