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

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2
votes
2answers
145 views

Limit w/o L'hopital

$$\lim\limits_{x\to c} \dfrac{\sqrt{x}-\sqrt{c}}{\sqrt{x^2-c^2}} = ?$$ I tried factorizing and rationalizing numerator/denominator but no use. Any help/hint ? Thanks!
0
votes
1answer
26 views

Summation simplification explanation

I'm trying to understand summation for my algorithm course and it has been a while since I took discrete math. Could any body please explain how does summation simplification work from the problem ...
7
votes
2answers
241 views

What's the theoretical basis for integration using partial fractions?

Exercises involving integration using partial fractions depend on expressing a rational function $\frac{P(x)}{Q(x)}$ (where the degree of $P$ is less than the degree of $Q$) as a sum of ...
0
votes
1answer
91 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} ...
1
vote
1answer
51 views

Calculate the limit with l'hospital's rule

Calculate $$\lim_{\alpha\to 0}{-\exp(-\frac{\alpha t}{2m})\frac{F}{\alpha}\sqrt \frac mk\sin(\frac{t\sqrt{4km-\alpha^2}}{2m})+\frac F\alpha \sqrt{\frac mk}\sin{(t\sqrt{\frac km})}}$$ Here is what ...
3
votes
2answers
107 views

Integral of $x^{12}(1-x)^8$

Any one have a cute way of doing this integral: $$\int x^{12}(1-x)^8\,dx$$ I expanded using the binomial theorem, but wondering if their is a easier way?
2
votes
1answer
39 views

Finding Surface area of a curve rotated around the x axis

I need to calculate the surface area obtained by rotating $\sin\pi x$, $0\le x \le 1$ about the x-axis. So the surface area equation i think i have to use is: $$A = \int_0^1 2\pi y \sqrt{1+(dy/dx)^2} ...
3
votes
0answers
41 views

Working out the area of Australia through Calculus? [closed]

I was wondering if it would be possible, and if so how, to calculate the area of an abstract shape on a sphere using surface integrals and Parametric surfaces and such. I am looking in to this as ...
-1
votes
1answer
58 views

How can I disprove that? [duplicate]

Prove that 5=-5 $$ \sqrt{(-5)^2} = \sqrt{25} = 5 = \sqrt{(-5)^2} = \sqrt{(-5)\cdot(-5)} = \sqrt{(-5)} \cdot \sqrt{(-5)} = (i \sqrt{5})\cdot(i\sqrt{5}) = -5\,. $$
4
votes
1answer
58 views

Equal values given by Mean Value Theorems

So I was doing some calculus homework the other day, and the following question occurred to me: what functions have the property that the value of $c$ guaranteed by the Mean Value Theorem for ...
3
votes
4answers
41 views

Convergence of the series: $\sum_{n\geq1}\frac{(-1)^n\arctan (n)}{n+n^{1/2}}$

$$\sum_{n=1}^{\infty} (-1)^n \tan^{-1}(n)/(n+(n)^{1/2}).$$ I know that the series is not absolutely converges. I want to prove using Alternative test. I don't know how to prove that sequence $ ...
1
vote
1answer
40 views

How to maximize $a^2 + \delta^2(s-a)^2$ by inspection?

I need to maximize: $a^2 + \delta^2(s-a)^2$ where: $\delta\in(0,1)$ and $0\le a \le s$. The solution in my text simply states: Since $\delta^2 < 1$ , the maximum occurs when $a=s$. I ...
0
votes
1answer
44 views

Solving $x(x+1)y' + y = x(x+1)^{2}e^{-x^2}$ in $(-1,0)$

What I tried to do: I divided both sides by $x(x+1)$: $$y' + \frac{1}{x(x+1)}y = (x+1)e^{-x^2}$$ This has the form $$y'+P(x)y = Q(x)$$ The general solution would be (supposing $f(a)=b, a \in (-1,0)$ ...
0
votes
1answer
30 views

Jacobian conversion for double integrals

Evaluate $\displaystyle \iint_R (x^2+y^2) \, dx \, dy$, where $R$ is the region in the first quadrant bounded by the curves $$xy=2, \, xy=4, \, x^2-y^2=3, \, x^2-y^2=5.$$ I have to use the ...
0
votes
1answer
56 views

Integration of Fundamental Solution of Laplace's equation.

I am currently reading Evan's PDE and am getting hung up on many of the more "technical details". This question may be very basic (multivariable calculus). I am given that the fundamental solution of ...
1
vote
1answer
37 views

Strictly Decreasing Function

Suppose $f(x):R_+\rightarrow[0,1]$ such that $f'(x)>0$. There exists a function $H(x)=\frac{f(x)}{x}$. We know $H'(x)<0$ as long as $x\frac{f'(x)}{f(x)}<1$. Which general conditions function ...
0
votes
2answers
33 views

How to integrate $\frac{dx}{ \left(1+x^2 \right)^2}$ and$\frac{\sqrt[6]{x}}{1+\sqrt[3]{x}}$

How to integrate $\int \frac{dx}{ \left(1+x^2 \right)^2}$ and$\int \frac{\sqrt[6]{x}}{1+\sqrt[3]{x}} dx $ ? Which method (by parts or substitution )should be applied? How to use this method in those ...
0
votes
1answer
29 views

Algebra manipulation for a integral

When deriving the integral ${\int_0^3(x^3-6x) dx}$ in terms of reimann sums it has these two steps in my calculus book, Stewart 7th edition, and I don't understand how to derive the 2nd from the ...
1
vote
4answers
96 views

Suppose that $F'(x)\leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x)\leq G(x)$ for all $x \in \mathbb{R}$.

Suppose that $F'(x) \leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x) \leq G(x)$ for all $x \in \mathbb{R}$. Prove or disprove. I came up with counterexample that $\cos(x) \leq \cos(x)+1 $ for all ...
1
vote
3answers
53 views

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length.

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length. Prove or disprove. I got the idea that they are inverse functions and probably we can show ...
0
votes
1answer
26 views

Integral of trigonometric function using substitution

I'd like to get some feedback on the following calculation: $$\int{\frac{(\cos{\frac{1}{x}})^2}{x^2}}\,\,dx$$ Using substitution, let $$u = \frac{1}{x},\,\,\frac{du}{dx}=-\frac{1}{x^2},\,\,du = ...
1
vote
1answer
100 views

Riemann Integral of Unit Fraction Indicator Function

I am working on the following exercise for practice: Show that the Riemann integral $\int_0^1 f(x) dx = 0$, where $$f(x) = \begin{cases} 1 & \exists n \in \mathbb{N}, x=\frac{1}{n} ...
4
votes
1answer
39 views

Evaluate the definite integral using substitution

I've calculated the following definite integral, and would like to know if my calculation appears to be correct: $$\int_2^3{\frac{1}{x-\sqrt{x}}}\,\,dx = ...
-2
votes
1answer
70 views

Confusing arc length formula in calculus

So I am having problems with this section in my math book on the first example. In the first example it uses general constants and this makes it very confusing, at least for me. I understand up until ...
0
votes
0answers
26 views

Conflicting information about second derivative test and graphing

When performing a first derivative test, you have a point where the first derivative is $0$ or undefined. If the values to the left of that point are increasing and those to the right are decreasing, ...
0
votes
7answers
103 views

Loopy integral $\int xe^x\sin(x)$

How to find integral of: $$\int xe^x\sin(x)$$ I know it should be repeated integration by parts, but i do not know how many times i should do this and when to stop.
0
votes
0answers
40 views

Integral of trig fraction using substitution

I'm getting to grips with the process of integration by substitution, and would like to ask for feedback on my calculation: $$\int{\frac{\cos{x}}{\sqrt{1+2\sin{x}}}}\,\,dx$$ Let ...
0
votes
1answer
54 views

Horizontal tangent line

I need some verification on this problem, not sure if I did it right. Find all points on the level surface $$x^3 − 3x − y^2 + 4y + z^3 = 10$$ where the tangent plane is horizontal. So I take the ...
2
votes
2answers
58 views

Why is the integral of $\int {1 \over {1+ \sqrt{x}}} dx$ not the same as for $\int{1 \over u}$?

Why isn't the integral of this: $$ \int{ 1 \over {1 + \sqrt{2x}} } \cdot dx $$ Equal to $\ln|1 + \sqrt{2x}|$ when one lets u = $ 1 + \sqrt{2x}$ Isn't it true that $ \int{ 1 \over u} \cdot du = ...
0
votes
2answers
38 views

why the all the coefficient terms of this integral share the least common factor 1/594

why the all the coefficient terms of this integral share the least common factor 1/594? Refer to this: $\int 1/(x^{23}+x^{50}) dx$ There are a lot of weird terms in the answer but they all share the ...
0
votes
2answers
45 views

Optimization - Maximizing Profit

I have been struggling with the problem below for quite some time now and no one can seem to figure it out, so I am asking it here. The question is as follows: You own an apartment complex with 50 ...
0
votes
1answer
88 views

What is the derivative of $\frac{x^{n+1}}{n+1}$?

I am ask to find the most general antiderivative of $f(x)= x^n$ where $n \geq 0$. However, I wondering how the derivative of $\dfrac{x^{n+1}}{n+1}$ is equal to $x^n$ My answer is $x^n - x^{n+1} $ ...
0
votes
2answers
38 views

What is the limit of a rational function as it approaches its vertical asymptote?

For example, take the function $f(x)=\frac{1}{(x-3)^2}$. What is the the limit as x approaches 3? (sorry, I don't know how to format this question) My teacher says that there is no limit at x=3, ...
2
votes
3answers
98 views

How to integrate this function with ln and u substitution?

I can get started in the right direction, but cant seem to get all of the way there, and any examples I can find don't have the same complications. $$\int {2x\over (x-1)^2}\cdot dx$$ What I have ...
3
votes
4answers
175 views

Evaluate $\int \frac{dx}{1+\sin x+\cos x}$

Evaluate $$\int \frac{1}{1+\sin x+\cos x}\:dx$$ I tried several ways but all of them didn't work I tried to use Integration-By-Parts method but it's going to give me a more complicated integral I ...
0
votes
1answer
94 views

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? [duplicate]

What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? Using Langrange Multipliers, I've set up the standard equation with $$g(x,y) = (x/2)^2 + (y/3)^2 = 1$$ $$f(x,y) = ...
0
votes
3answers
38 views

Evaluating indefinite integral using substitution

I have the following integral to evaluate. I'm not sure whether to use the reverse chain rule or integration by parts, as my calculation hits a bit of a snag. Any suggestions would be appreciated! ...
0
votes
1answer
121 views

Can differentiation be used to find the average rate of change between two points rather than one?

I was wondering if I could use the derivative of a function to determine the average rate of change between two points, rather than one. I have a solution but I'm not sure whether it is valid or not: ...
0
votes
1answer
88 views

sphere-sphere intersection

Let $ S_1 : (x-1)^2 +y^2+z^2=1 $ $S_2 : x^2 +y^2 +z^2 =1$ $S_3 : (x+1)^2 +y^2 +z^2 =1 $ Find the volume of the solid inside $S_2$ and outside $S_1$ and $S_3$, using triple integrals. I have try ...
4
votes
1answer
56 views

Can some one help me parametrize $\frac{x^4}{a^4}+\frac{y^4}{b^4}+\frac{z^4}{c^4}=1$

Given a surface $$\frac{x^4}{a^4}+\frac{y^4}{b^4}+\frac{z^4}{c^4}=1$$how can I parametrize the surface using $X(u,v).$ I tried to use $$x=a\sqrt{\cos(\theta)\sin(\phi)}$$ ...
-1
votes
1answer
32 views

how to prove the following integral equation

Hi I was trying without success to prove the following, any ideas how? $$ \frac d{dx}\int_a^{e^x}f(t)dt=e^x\cdot f(e^x) $$
0
votes
2answers
42 views

Is Lagrange's mean value theorem is valid on $x^{1/3}$ in $[-1,1]$?

Suppose I have a function $f(x)=x^{\frac{1}{3}}$ in interval $[-1,1]$ , is lagrange's mean value theorem valid here ? $f(x)$ is continuous in this interval but there's a confusion in it's derivative. ...
1
vote
1answer
39 views

Evaluate $A=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \cos(n x) dx$.

Evaluate the limits $$A=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \cos(n x) dx$$ and $$B=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \sin(n x) dx$$ ...
2
votes
2answers
509 views

When is it allowed to take a constant out of a series?

When is it allowed to take a constant out of a series? Suppose we have a series $\sum ca_n$, when can we write it as $c\sum a_n$? It's pretty obvious when we know beforehand the series converges ...
2
votes
3answers
390 views

How to find a definite integral over a symmetric interval without finding the antiderivative?

How do I find the following without finding the anti derivative $$ \int_{-\pi}^\pi \ln(x^2+1)e^{\sin \lvert x\rvert}\sin x dx $$
5
votes
1answer
212 views

Sum the infinite series

How to solve this: \begin{equation*} \sum_{n=1}^{\infty }\left[ \frac{1\cdot 3\cdot 5\cdots \left( 2n-1\right) }{ 2\cdot 4\cdot 6\cdots 2n}\right] ^{3} \end{equation*} I can make the bracket thing, ...
1
vote
2answers
55 views

Proving the series doesn't converge: $\sum_{n=1}^{\infty}a_n$, $\lim_{n\to\infty}na_n=\infty$, $a_1=-1$

Let $\displaystyle\sum_{n=1}^{\infty}a_n$ and $\displaystyle\lim_{n\to\infty}na_n=\infty$ and $a_1=-1$. Prove the series does not converge. From the given that $a_1=-1$ we know that there has to ...
1
vote
4answers
202 views

If $A+B=\pi/3$ then what will maximum value of $\tan(A).\tan(B)$?

Suppose I am given that $$A+B=\frac{\pi}{3}$$ then what will be maximum value of $$\tan(A).\tan(B)=?$$ $$\tan(A+B)=\frac{\tan(A)+\tan(B)}{1-\tan(A).\tan(B)}=\sqrt{3}$$ then ...
8
votes
5answers
99 views

Evaluating $\lim\limits_{x \to 0}\left(\frac{\sin x}{x}\right)^{\frac{1}{1-\cos x}}$

How do I evaluate $$\lim_{x \to 0}\left(\frac{\sin x}{x}\right)^{\dfrac{1}{1-\cos x}}\ ?$$ I tried using the fact that $\left(\frac{\sin x}{x}\right)^{\frac{1}{1-\cos x}} = ...
0
votes
2answers
129 views

At 2:00pm a car's speedometer reads 30mph, and at 2:10pm it reads 35mph. Use the Mean Value Theorem to find an acceleration the car must achieve.

I'm only assuming that f(a) and f(b) are assigned to each respective velocity, but I'm not sure how the mean value theorem can be applied to distance rate and time.