I have the following function
$$f^{(0)}\left(x\right)=f\left(x\right)=e^{x}$$
And want to approximate it using Taylor at the point ##\frac{1}{\sqrt e} ##
I also want to decide (without calculator)whether the error in the approximation is smaller than ##\frac{1}{25} ##
The Taylor polynomial is...
Hi there.
I have the following function:
$$f(x)=\ln|\sin(x)|$$
I've caculated the derivative to:
$$f'(x)=\frac{\cos(x)}{\sin(x)}$$
And the domain of f(x) to: $$(2\pi n, \pi+2\pi n ) \cup (-\pi + 2\pi n, 2\pi n)$$
And the domain of f'(x) to: $$(\pi n, \pi+\pi n )$$
I want to determine for...
Hi there.
I have the following function:
$$f(x)=arcsin(\sqrt x)$$
I've caculated the derivative to:
$$f'(x)=\frac{1}{2\sqrt x\sqrt{ (1-x}}$$
And the domain of f(x) to: $$[0, 1]$$
And the domain of f'(x) to: $$(0, 1)$$
I want to determine for which x the derivative exists but I'm not...
Hi there.
I have the following function:
$$f(x)=x+\frac{1}{(x+1)}$$
I've caculated the derivative to:
$$f'(x)=1-\frac{1}{(1+x)^2}$$
And the domain to: $$(-\infty, -1)\cup(-1, \infty)$$
I've also found two extreme point: $$x=0, x=-2$$
I know that a function is strictly increasing if...
The vector field F which is given by $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$
And the line integral $$ \int_{C} F \cdot dr $$
C is the path of $$\dfrac{\ (\cos (t), \sin (t))}{ 1+ e^t}$$ , and $$0 ≤ t < \infty $$
How do I calculate this? Anyone got a tip/hint? many thanks
Hahaha.. I finally got it! the mixed partial cancels and if I use the pythagorean trigonometric identity on what's left I get the answer!
I can't explain how much you've made my day! <3
then I get:
$$\dfrac{\partial^2}{\partial y^2} = \sin^2 \theta \dfrac{\partial^2}{\partial u^2} - 2sin \theta\cos \theta\dfrac{\partial}{\partial u\partial v}+ \cos^2 \theta \dfrac{\partial^2}{\partial v^2}$$
Can I somewhat use this to calculate $$\dfrac{\partial^2}{\partial u^2}$$ ?
Thank...
$$\dfrac{\partial^2}{\partial x^2} = \cos^2 \theta \dfrac{\partial^2}{\partial u^2} + 2sin \theta\cos \theta\dfrac{\partial}{\partial u\partial v}+ \sin^2 \theta \dfrac{\partial^2}{\partial v^2}$$
Is that correct? :)
Can you give me any tips on how to calculate $$\dfrac{\partial^2}{\partial...
Thank you so much for helping me.. should the first line be:
$$\dfrac{\partial}{\partial x} = \dfrac{\partial u}{\partial x}\dfrac{\partial}{\partial u} + \dfrac{\partial v}{\partial x}\dfrac{\partial}{\partial v} = \cos \theta \dfrac{\partial}{\partial u} + \sin \theta\dfrac{\partial}{\partial...
Mentor note: Fixed the LaTeX in the following
I have the following statement:
\begin{cases} u=x \cos \theta - y\sin \theta \\ v=x\sin \theta + y\cos \theta \end{cases}
I wan't to calculate:
$$\dfrac{\partial^2}{\partial x^2}$$
My solution for ##\dfrac{\partial^2}{\partial x^2}##...