# Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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### Finding the derivative of a function.

Differentiate $$f(x) = \sin(\ln(\cos(x^2+1)))$$ My work: $u = \ln(\cos(x^2+1))$ so $f(x) = \sin u$ , $f'(x) = \cos u = \cos(\ln(\cos(x^2+1)))$. I keep getting this answer, but where am I going wrong?...
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### Proving uniform continuity of function of two variables.

Proving uniform continuity of function:$$f(x,y)=\begin{cases} \frac{x^3-xy}{x^2+y^2}, & (x,y)\neq (0,0) \\ 0, & (x,y)=(0,0) \end{cases}$$ This is supposedly solve, but I don't understand the ...
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### showing an inequality not using stirling formula

I don't know how to show that $$\frac{k^k}{k!}\leq e^{k}$$ without using Stirling's approximation. I want to show it directly. I guess I need some inequality to achieve this but I don't know.
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### To evaluate $\int_{0}^\pi \frac {\sin^2 x}{a^2-2ab \cos x + b^2}dx$? [duplicate]

How do we evaluate $\int_{0}^\pi \dfrac {\sin^2 x}{a^2-2ab \cos x + b^2}dx$ ? I tried substitution and some other methods , but its not working ; please help .
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### Solving a double integral using change of variables.

$$\int ^{1}_{0} \int^{1}_{y}e^{-x^{2}}\,dx\,dy$$ To solve this I know one must use change of variables, but the problem is that I do not know how to approach the actual change. Just thinking out ...
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### Find the nodes and coefficients of Gauss-Lobatto Quadrature with $n=4$

I am stucked at this problem: Gauss-Lobatto quadrature is defined as: $\int_{-1}^1 f(x)dx\approx w_1 f(-1)+w_n f(1) + \Sigma_{k=2}^{n-1}w_k f(x_k)$ ($2\leq n\in\Bbb{N}$) Where the nodes $x_k$ ...
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### A simple looking integration : $\left(\frac{x^3}{1+x^5}\right)$

One of my friends gave me this problem about a week back and since then, I have been toiling to get a solution to this problem, but I just get stuck at some step. Can someone please tell me the steps ...
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### $\displaystyle\int_{0}^{\pi} \frac{\sin^2 x}{a^2 - 2ab\cos x +b^2} \,dx$ [duplicate]

An integration: $$\int_{0}^{\pi} \frac{\sin^2 x}{a^2 - 2ab\cos x +b^2} \,dx$$ I am stuck with this definite integral. Will putting $\,\cos x = z\,$ help out here?
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### Are Riemann integrals necessarily convergent sums?

For example, does the Riemann sum of $\sum_2^n$ $\frac{1}{k} \frac{1}{n}$ converge to the integral $\int_2^n \frac{1}{x}dx$? Does the 1/n factor help the sum of 1/k converge? It doesn't look obvious,...
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### Calculating the volume of a surfboard

I'm building a website for a client in which customers can customise the shape of their board (curvature, length, width, thickness, and so forth) and the client has asked if we can calculate the ...
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### “multiplicative inverse in the modulo of the larger number” what does that mean?

while I was reading this artical I have read the following paragraph: The interesting thing is that if two numbers have a $\gcd$ of $1$, then the smaller of the two numbers has a multiplicative ...
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### Proving a function has real roots

I am not interested in finding roots but interested in proving that the function has real roots. Suppose a function $f(x) = x^2 - 1$ This function obviously has real roots. $x = {-1, 1}$ How could ...
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### How would I find the second derivative of the bilinear $B(x,y)=Ax \times y$ where $A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$

$$B(x,y)=Ax \times y \text{ where } A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$$ Second derivative is obviously the first derivative of the first ...
### Integrating $\frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\infty} e^{-x^2} \cos (2x) \, dx}}$
I need help calculating the following integrals. For the top integral we can use the jacobin, right? But how do I calculate the bottom one?:  \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\...
### If $f(x)$ is diferentiable in $[0,1]$ and $f(0)=f(1)=0$, then find $\alpha$ such that $f'(c)+\alpha f(c)=0$
Attempt: Let $g(x)=e^{\alpha x}f(x)$ $g'(x)=\alpha e^{\alpha x}f(x)+e^{\alpha x}f'(x)$ $g'(c)=\alpha e^{\alpha c}f(c)+e^{\alpha c}f'(c)$ Now, $g(0)=f(0)$ and $g(1)=e^{\alpha }f(1)=e^{\alpha }g(0)$...