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

learn more… | top users | synonyms

4
votes
0answers
22 views

How do I do this change of variables?

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$. I set $$u = y-x$$ $$v = z-y$$ $$w ...
0
votes
0answers
10 views

Stokes theorem problem

Let S be the torch-shaped surface defi ned by the cone z = 1 + √(x² + y²) for 2 ≤ z ≤ 4, the cylinder x² + y² = 1 for 0 ≤ z ≤ 2, and the disk x² + y² ≤ 1, z = 0. Orient the surface using the outward ...
2
votes
3answers
46 views

Convergence of $\frac{\ln(n)}{n^2}$

During a Calc exam, I needed to state whether $$\sum_{n=1}^{∞} \frac{\ln n}{n^2}$$ converged or diverged. I was going to try a strict comparison test (with $\sum_{n=3}^{∞} \frac{\ln n}{n^2}$ to ...
0
votes
1answer
13 views

Limit of sequence involving a product

This question is related to a post that was deleted. I want to calculate the following limit $$\lim_{n \to ...
1
vote
1answer
20 views

Can we do Taylor approximation in one direction

Let $f:\mathbb{R}^2\to\mathbb{R}$. Can we do Taylor approximation for only one variable $$f(x,y) \approx f(x_0,y) + \frac{\partial }{\partial x}f(x_0,y)(x-x_0) + \frac{1}{2}\frac{\partial^2}{\partial ...
1
vote
1answer
25 views

Calculus volume of revolution problem

Find the volume of the solid of revolution obtained by rotating the region bounded by $f(x) = x^3 + 1$, $g(x) = x^2$ and $0 ≤ x ≤ 1$ about the line $y = 3$. I know the gist of the problem, but ...
0
votes
0answers
8 views

Find the volume of solid D bounded by two surfaces.

Consider the solid $D$ bounded by the surfaces $$x^2+y^2+(z-1)^2=\frac{3}{2} \text{ and } x^2+y^2-z^2=-1$$ Find the volume of the solid $D$. I think it can be solved by using spherical coordinates ...
4
votes
3answers
69 views

Integration $I_n=\int_{0}^{1}\frac{dx}{(x^n+1)(\sqrt[n]{x^n+1})}$

$$I_n=\int_{0}^{1}\frac{dx}{(x^n+1)\large\sqrt[n]{\normalsize x^n+1}}$$ Could someone help me through this problem?
2
votes
4answers
50 views

Evaluate the limit $\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$

I have this assignment Evaluate the limit: 5.$\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$ I don't think we are allowed to use L'Hopital, but I can't imagine how else.
1
vote
3answers
40 views

How to solve for an unknown upper bound in a summation

I've always wondered: if you have something like $$ \sum\limits_{k=0}^x k $$ And you want to find x such that the value of the summation is the closest to a constant c, how would you proceed? And ...
0
votes
2answers
24 views

By applying the second version of the Fundamental Theorem of Calculus find the integral:

The second version of the Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ then $\int_{a}^{b} f(x) dx = F(b)-F(a)$. I need to use this to find a) $\int_{-2}^{-1} \frac{1}{x^3} dx $ and b) ...
3
votes
1answer
203 views

How to differentiate this negative power? [duplicate]

I'm reading the book "Calculus made easy" and I'm stuck with a step of a derivative with a negative power. Here is what is in the book: Case of a negative power. Let $y=x^{-2}$. Then proceed ...
0
votes
4answers
48 views

Proving an equation is a fuction

Prove that the equation $y^3 + 3xy -5x^3 + 1 = 0$ defines $y$ as a function of $x$ for all $x$ in the real numbers.
3
votes
1answer
47 views

Integrate $e^{-\frac{y^2}{2}}\left(\frac{1}{y^2}+1\right)$

I'm trying to find $$\displaystyle \int{e^{-\frac{y^2}{2}}} \left(\frac{1}{y^2}+1\right)dy$$ I tried using integration by parts and some substitutions, but nothing seem to work. The answer is ...
0
votes
3answers
42 views

Weird integration question

let = e(x) = $$\text{e(x) =}\int\frac{e^x}{e^x+e^{-x}}$$ and f(x) = $$\text{f(x) =}\int\frac{e^{-x}}{e^x+e^{-x}}$$ The question wants.... a) calculate e(x) + f(x) b) calculate e(x) - f(x) c) Use ...
0
votes
1answer
35 views

Integrating $x\cdot{\cosh(x^2)}$

How do integrate $x\cdot{\cosh(x^2)}$? Do i just use integration by parts? I know that integration by parts is $\int{u\cdot{\mathrm{d}v}} = uv - \int{v\cdot{\mathrm{d}u}}$ Making ...
0
votes
1answer
21 views

Calculus and Lagrange Multiplier

If I have the function $u(x_1,x_2) = (x_1-a_1)^{1/2}(x_2-a_2)^{1/2}$ If I take the derivative of the above with respect to $x_1$, would it equal $1/2 (x_1)^{-1/2} (x_2-a_2)^{1/2}$? And wrt to ...
0
votes
0answers
14 views

Setting up a volume-finding calculation

I'm asked to find the volume inside the sphere $x^2+y^2+z^2=25$ and outside the cylinder $x^2+y^2=1$. I approached the volume $V$ in the following way: ...
4
votes
1answer
35 views

Area and integration question, is this area under the curve?

Find the exact area between $x$ and the graph $f(x)=(x-1)(x-2)(x-3)$. $$f(x) = x^3-6x^2+11x-6$$ I found that this is an odd shaped positive polynomial with a maxima between 1 and 2 and minima ...
1
vote
2answers
111 views

How to evaluate $\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x$?

How to evaluate the following integral $$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x $$ The numerical result is $= -1.38104$ and when I look at it, I have no idea how to ...
0
votes
1answer
11 views

invertibility, derivative, and difference quotient

Suppose that $f$ is an invertible differentiable function, that the domain of $f^{-1}$ contains an interval around $a$, and that $f^{-1}$ is continuous at $a$ and that $f^{-1}$ is continuous at $a$. ...
0
votes
4answers
28 views

Derivative of $y=\tan(3)e^x$,

If, $y=\tan(3)e^x$, wouldn't the derivative be $y\;'=\sec^2(3)e^x \times e^x$? The outer function times the inner function, using the chain rule? The answer key gives the derivative as $y=e^x \tan ...
3
votes
1answer
18 views

If $f(1) = 3$ and $\int_{1}^{xy}f(t)dt = y\int_{1}^{x}f(t)dt+x\int_{1}^{y}f(t)dt\;\forall x,y \in \mathbb{R^{+}}\;,$ Then $f(e) =

Let $f:\mathbb{R^{+}}\rightarrow \mathbb{R}$ be a differentiable function with $f(1) = 3$ and satisfying:: $\displaystyle \int_{1}^{xy}f(t)dt = y\int_{1}^{x}f(t)dt+x\int_{1}^{y}f(t)dt\;\forall ...
4
votes
1answer
83 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
3
votes
0answers
30 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
1
vote
2answers
21 views

Even function divided by Odd function is an Odd function PROOF?

An Even function divided by Odd function is an Odd function,that is a fact. However is there a means to prove this?
0
votes
2answers
32 views

Struggling to find the second derivative of this function's first derivative

So I've found the first derivative of this function but now I have to find the second derivative. I've tried everything but I cannot seem to get it. Here's the original function: $x = a sec(θ)$, $y = ...
0
votes
2answers
21 views

Related Rates- Expanding Circle

I just wanted to see if I did this correctly. Only asking for B. So the question ask : The area of a circle increases at a rate of 1cm^2/s. a. How fast is the radius changing when the radius is ...
0
votes
0answers
9 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
0
votes
1answer
16 views

Calculate duration of task

Say I have some task to process 100 days of data, and it takes 5 hrs to process a day. But each day that it takes to process it a new day of data comes in. So for the initial set of data it takes: 5 ...
0
votes
2answers
27 views

How to prove functions are odd and even

Show that any function f on [-a,a] where a is a positive constant, can be written as the sum of an even and an odd function?
0
votes
1answer
22 views

how to show this manipulation in the integral

Let we have: $$G(t)=y_1(t)\int y_2(s)ds$$ when we take the limits as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds$$ then is it possible to write it as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds=\int^t_{t_0} ...
0
votes
0answers
20 views

Simplifying Gamma functions yet having a complication while graphing when the function was able to be graphed previous to simplification?

According to the Euler's duplication formula: $$ \Gamma(z) \Gamma(z+\frac{1}{2}) = 2^{1-2z} \sqrt{\pi} \Gamma(2z) \therefore $$ $$ \Gamma(2z) = \frac{\Gamma(z) \Gamma(z+\frac{1}{2}) ...
4
votes
0answers
38 views

Integral involving hyperfactorial

I'm trying to prove that: $$ \int_0^1 \ln\left(K(x)\right)\space dx =-\zeta'(-1)=\ln(A)-\frac{1}{12} $$ Where $A$ is Glaisher Kinkelin's constant and $K(x)$ is a generalization of the hyperfactorial ...
2
votes
3answers
46 views

Find derivative of $f(x)=\frac{1}{\sqrt{x+2}}+2x$ by definition

Use the definition of a derivative to find the derivative of: $$f(x)=\frac{1}{\sqrt{x+2}}+2x$$ my work: $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ ...
0
votes
1answer
17 views

Derivative of complex conjugate

In general, two different mathematical operations need not commute. Let f(x,y) be a complex valued function, taking in two real-valued inputs x and y. Then under what circumstances is the partial ...
0
votes
1answer
11 views

Volume of a region after some transformation.

Consider $$R=\{(x,y,z):x^2+y^2\leq 1,0\leq z\leq 2\}$$ and the transform $$T:(x,y,z)\to(x,y+\tan \alpha z,z)$$ where $0<\alpha<\pi$ Then what is the volume of $T(R)$? I tried myself but I ...
2
votes
0answers
23 views

Reference request regarding calculus exam

I'm currently a first year computer science student and I'm deeply interested in calculus . That being said, what we studied so far consists of: Cantor sets, sequences and a brief introduction to ...
0
votes
3answers
13 views

Integrate the differential equation of a simple rate equation

Could somebody please show me how to integrate the following: $dA/dt = -kA$ I'm told that the answer is: $A(t) = A(0)e^-kt$ but I do not know why. Could you be explicit in your answer and explain ...
0
votes
1answer
14 views

how to find the interval at which a derivative function is increasing

Alright, so here's the deal. I need to find the interval of this derivative function: f(x)= −5x2+12x−7 So far, I've gotten that the derivative is this: ...
1
vote
1answer
14 views

Summation of arithmetic-geometric series of higher order

There is a closed formula for the summation of arithmetic-geometric series: $\sum_{x=1}^{+\infty }(ax+b)r^x=\frac{(a+b)r-br^2}{(1-r)^2}$ when $-1<r<1$ But consider: $\sum_{x=1}^{+\infty ...
0
votes
1answer
19 views

How do I find the relative coordinates of a picture of a plane in 3d space.

I have a box, with corners $A$ through $H$, as depicted above. I'll consider $B$ the origin of a coordinate system, with the $x$ axis in the direction through $C$, the $y$ axis through $A$ and the ...
0
votes
1answer
23 views

What are the rules being used to compute $\lim\limits_{x\rightarrow \frac{\pi}{2}} (1-\cos x)^{\tan x}$?

I am given $\lim\limits_{x\rightarrow \frac{\pi}{2}} \frac{\ln(1-\cos x)}{\cos x} = -1$ So, $(1-\cos x)^{\tan x} = e^{(\tan x) \ln(1-\cos x)}$ and as $x\rightarrow \frac{\pi}{2}$, we have: $(\tan ...
7
votes
2answers
53 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
votes
0answers
35 views

Is there any way to solve this inequality by hand?

I have had this one equality problem I've been stuck on for a while. Could anyone share on what methods I would have to use to solve this by hand? No need for full answer, just a hint is okay. ...
0
votes
2answers
14 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
0
votes
1answer
20 views

Given that $x(\theta)=5\cos \theta,y(\theta)=5\sin \theta,z(\theta)=\theta$

Given that $x(\theta)=5\cos \theta,y(\theta)=5\sin \theta,z(\theta)=\theta$ $L(\theta)$ is the arclength at the point $P(x(\theta),y(\theta),z(\theta))$ and $D(\theta)$ is the distance from origin to ...
0
votes
1answer
17 views

Rate of Change with Derivatives

We just started with rate of change while using derivatives and I am stuck on a question, hope you can help. A particle moves on a vertical line so that its altitude at time $t$ is $y=t^3−12·t+3$, ...
0
votes
0answers
24 views

Maturity and Proficiency in calculus, linear algebra for successful research

Will the high level maturity and proficiency in basic calculus, linear algebra (both calculation and theorem aspects) be required or recommended as an important factor to be successful in mathematical ...
0
votes
0answers
27 views

Proving $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)} dx < \frac{1}{e}$

While i was playing around with very weird functions and came across this: $$ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)}dx \approx 0.3676932086...\approx \frac{1}{e} - ...