For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.
0
votes
1answer
23 views
Find the area of region
Find the area of the region bounded by the curves $y=2x^2-6x+5$ and $y=x^2+6x-15$.
I found the critical points $2$ and $10$. My trouble is making the integral work. My teacher's answer key says that ...
3
votes
3answers
43 views
Trying to understand the meaning of symmetry
The picture below is the solution to the following problem as presented in my book:
Find the area of the region that lies inside both curves $$r = 8 + \cos \theta \\r = 8 − \cos θ$$
According to ...
0
votes
0answers
20 views
Is Risch's algorithm powerful enough to determine any integral of a function have a closed form or not?
Is Risch's algorithm powerful enough to determine any integral of a function have a closed form or not?
Is there a historic piece of reference that support your answer?
...
0
votes
2answers
23 views
Finding derivative of log and trigo function
How to find the derivative of $y =(\log_{\cos x}\sin x)(\log_{\sin x}\cos x)+\sin^{-1}\frac{2x}{1+x^2}$
Please guide...thanks..
0
votes
0answers
14 views
How to determine if a integral has not a closed form solution in terms of elementary functions [duplicate]
How to determine if a integral has not a closed form solution in terms of elementary functions?
I guess my professor will not teach me this!
0
votes
2answers
95 views
How did Euler and Bernoulli prove this limit?
Prove that the lim as x approaches infinity of $(1+1/x)^x$ exists, and prove this without assuming any prior knowledge of $e$.
1
vote
0answers
31 views
Solving for $x$ in this simple differential equation?
$\dfrac{dx}{dt}=2\dfrac{\sqrt{2g(\sin c- \sin x)}}{\sqrt{l}}$. $g$, $c$, and $l$ are all constants. Is there a way to solve for $x$ in terms of $t$ here? Once I did separation of variables and plugged ...
0
votes
2answers
54 views
How to Evaluate this integral in the simplest way?
Find $$\int 5^{(x^2)}\; dx.$$
Show me the step to find this please!
1
vote
1answer
23 views
Points around which one expands and the radiuses of convergence
I'm trying to make sense of the following passage:
Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
2
votes
3answers
39 views
Series evaluated to $m$ terms, approximating the error
Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms?
$$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$
...
0
votes
0answers
18 views
Proof by Farkas theorem
2) Show using duality that exactly one of the following systems has a solution:
I) Ax=b, 0 ≤ x ≥ e
II) A^T u + v ≥ 0, b^T u + e^T v=-1,v ≥ 0
Solution:
(P) ...
1
vote
1answer
24 views
Convergence of a series with factorials, parameters and ratio test
Question:
check when does $\displaystyle\sum_{n=1}^\infty {\sqrt[m] {n!} \over \sqrt[k] {(2n)!}}$ converge/diverge?
What I did:
Using the ratio test:
$$ {\sqrt[m] {(n+1)!} \over {\sqrt[k] ...
0
votes
1answer
29 views
What is derivation method of laplacian for spherical coordinate?
I know what laplacian of spherical coordinate is, but i what to know method of derivation laplacian for spherical coordinate:
$$dl^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$$
i know it is not ...
0
votes
1answer
30 views
Partial fraction
How to expand the following expression
$\frac{1}{(x^n-1)(x-1)}$
in partial fraction, I think it will be rewritten in terms of geometric series , but how to relate the undefined coefficients ...
2
votes
4answers
80 views
Improper Integral:$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$
I want show that this improper integral convergence: $$\int_{0}^{+\infty}\frac{\sin x}{x+\sin x}dx$$ please help me.
