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

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-2
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0answers
34 views

Solve out $a,b$ from system of equations [on hold]

I encounter some question I don't know how to solve out $a,b$ from $$\begin{cases}(a^2-1)a=(b^2-1)b\\ (a^2-1)(3a^2+1)= (b^2-1)(3b^2+1)\end{cases}$$
2
votes
2answers
36 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
0
votes
0answers
9 views

Unit normal vector at inflection point for any curve: Defined or Undefined?

Consider an arbitrary parametric planar (for simplicity) curve: $ \vec{r}(t) = f(t) \,\hat{i} \, + \, g(t) \, \hat{j}$ Differentiable twice over its domain. $ \vec{r'}(t) = f'(t) \,\hat{i} \, + \, ...
2
votes
2answers
59 views

Taylor expanding $\frac{e^x}{x}$?

How can you taylor expand $$\frac{e^x}{x}$$ Can it be expanded at $x = 0$? Can it be expanded as $x \to 0$?
1
vote
3answers
57 views

Is $\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} = \infty$?

It was asked in our test, and below is what I did: $$\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} $$ $$=\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5}\times\frac{\sqrt{x^2+16}+5}{\sqrt{x^2+16}+5} $$ ...
0
votes
0answers
9 views

How can I solve the conservation of traffic PDE?

I'm trying to solve the conservation equation for traffic flow so that I can use it for an example. It is stated as follows: $$\frac{\partial \rho }{\partial t} + \frac{\partial \rho v(\rho ...
0
votes
1answer
40 views

Indefinite trignometric integral

I tried $u$-substitution and $uv$-substitution, can't seem to figure this out... any help would be appreciated! Question: $$\int\frac{x}{\cos(x)}\,dx$$ Thanks!!!
0
votes
0answers
35 views

Prove that $f'(c)= \frac{2}{2+3(f(c))^2}$ for some $c$

Problem: $f: [0, 1] \to \mathbb{R}$ is continuous on $[0, 1]$ and differentiable on $(0, 1)$. ALso, $f(0)=1$ and $(f(1))^3+2f(1)-5=0$. Prove that there exists a $c \in (0, 1)$ such that $f'(c)= ...
0
votes
3answers
65 views

x^x^x^…=2, what is the value of x?

I came up with this little simple exercise, stating: $x^{x^{x^{\dots}}}$ infinite times is equal to $2$, find $x$. As we're dealing with infinity, we can just separate the first $x$ and get $x^2=2 ...
0
votes
1answer
29 views

Finding Value of C to Maximize Area

f(x)=$xe^{-\sqrt x}$ Find the value of c, such that the area bounded between the graph, the x-axis, x=c, and x=c+1 is maximized. Find the maximum area. I don't know where to start with this one. I ...
2
votes
1answer
59 views

Doomsday Prediction

I have a calculus problem I can't seem to figure out. Any help would be appreciated! Doomsday prediction. In $1960$, three electrical engineers at the University of Illinois published a paper in ...
0
votes
3answers
21 views

How to account for solids of revolution around vertical lines to the right of the x axis?

I'm trying to find the volume of a solid created by rotating the region enclosed between $x=y^2$ and $x=1$ around the line $x=8$. Noting that the intersections of the functions occur at $(0,0)$ and ...
1
vote
2answers
31 views

Can someone explain why $(e,1)$ and $(t, \ln t)$ are the two points of intersection for this question?

I was just going through Khan academy and this question completely threw me. I've rewatched the prior videos a few times to try to understand what I'm suppose to do, but I still don't understand. The ...
0
votes
2answers
23 views

Solving a for a function within a known definite integral?

I have a problem in my physics class which seems to boil down to $$\int_0^1 f(x)x \,dx = C$$ where $C$ is a constant and I need to solve for $f(x)$. If possible, I need the solutions where $$f(0)=0.$$ ...
0
votes
1answer
22 views

Trouble getting between steps when solving integral

I've having a lot of trouble trying to figure out how they're getting from the step in blue to the one in red. Can some one please explain that?
3
votes
3answers
281 views

Finding the shortest distance between two Parabolas

Recently, a problem asked me to find the minimum distance between the parabolas $y=x^2$ and $y=-x^2-16x-65$. I proceeded with the problem as thus. Let $P(a,a^2), Q(b, -b^2-16b-65), a-b=x$. ...
0
votes
0answers
15 views

Orthogonal trajectory in $3$ dimension.

Find the orthogonal trajectories on the cylinder $y^2 =2z$ of the curves in which it is cut by the system of planes $x+z=c$, where $c$ is a parameter. I parametrized the equation. Orthogonal ...
0
votes
2answers
41 views

Show analytically that $te^{-t}$ is not decreasing monotonically.

How does one show analytically that $te^{-t}$ is not decreasing monotonically on $(0, \infty)$? One can consider numbers in the interval $(0, 1]$ and show a counterexample to monotonicity, but ...
2
votes
0answers
8 views

Can I still uses the method of logarithmic differentiation to simplify complicated functions if the the range of that function includes 0?

The method of log differentiation refers to taking the natural log of both sides of an equation to simply an complication functions evolving lots of multiplication and divisions and exponents. But ...
0
votes
1answer
21 views

Point to Plane Distance Questions

I'm reading from Marsden Vector Calculus 6th Edition and this picture is from page 43. I'm having difficulty understanding how they get to $$ \text{Distance} =|\vec v \cdot \vec n|$$ The way I ...
0
votes
0answers
13 views

Implicit - simplify last step

Please let me know if this link works. I'm pretty new at posting questions. Maybe there is a better way to post from the derivative-calculator.net site. I'm not sure how they simplify the last step ...
-3
votes
0answers
24 views

How do I get these values? [on hold]

I want to know how to get these answers: $A(12836.3)=42227.7$ $A=3.28971$ (phase) $\theta+46.7364=0$ $\theta=-46.7364$ from this equation. I am confuse is there a formula or a trig. identity? ...
12
votes
7answers
961 views

How does advancing through the math major work?

I am an undergrad math major that just completed Calculus 3 last semester. This semester I signed up for Discrete Mathematics, and will be taking Intro to Advanced/Abstract Math next. Of course-- I ...
1
vote
2answers
57 views

How to find the nature of this series?

What tests to use for this series: $$\sum_{m=1}^{\infty}(-1)^{m-1}\left(1+\frac{8}{m}\right)^m$$ I've tried alternating test and ratio test but were inconclusive. Can I apply nth term test to the ...
11
votes
3answers
644 views

Summation of a term to infinity

I read through many tutorials but no one mentioned this explicitly. Is the following conversion valid? $$\sum_{k=0}^\infty \frac{k-1}{2^k} = \lim_{n\to \infty} \sum_{k=0}^n \frac{k-1}{2^k}$$ ...
2
votes
1answer
19 views

Volume of region in the first octant bounded by coordinate planes and a parabolic cylinder?

Find the volume of the solid region in the first octant bounded by the coordinate planes, the plane $y + z = 2$ and the parabolic cylinder $x = 4 - y^2$. I have a final answer, I would just like to ...
0
votes
1answer
30 views

Water main construction. Find the angle using vectors.

A water main is to be constructed with a $12.5$​% grade in the north direction and a $25$​% grade in the east direction. Determine the angle $\theta$ required in the water main for the turn from north ...
0
votes
1answer
30 views

Intervals in which f(x) is Strictly Increasing/Decreasing

Find the intervals in which $f(x)=\sin x + \cos x, 0 \leq x \leq 2 \pi$ is strictly increasing/decreasing. First I find the derivative $f'(x) = \cos x - \sin x$, then put $f'(x)=0$, getting $\tan x = ...
1
vote
3answers
69 views

Distance between two circles on a cube

I found this problem in a book on undergraduate maths in the Soviet Union (http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf): A circle is inscribed in a face of a cube of side a. Another circle ...
0
votes
2answers
57 views

How do I evaluate a series? [on hold]

In this specific example, I don't understand the steps of evaluating this series: \begin{align} &\frac{12}{n}\left(\left[\sum_{i=1}^n-7\right]+\sum_{i=1}^n\left[\frac{-12}{n}\cdot ...
1
vote
2answers
45 views

Arc length of Archimedes Spiral $ r = \theta $ from $ 0 \le \theta \le 2\pi$

The equation of the Archimedes spiral is given by $$r = \theta$$ The formula for calculating the Arc Length is given by $$L = \int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$ The ...
-1
votes
1answer
27 views

Find the values of constants in piecewise [on hold]

Find the value of the constants a and b so that the function defined by $$ f(x) = \begin{cases} x+1 ,& 1<x<3 \\ x^{2}+bx+c, &|x-2| \geq 1 \end{cases} $$ is continuous in ...
2
votes
1answer
54 views

Why do you need absolute value when taking $\sqrt{\cos^2(x)}$

$$\sqrt{\cos^2(x)} = |\cos(x)|$$ Is this on the right track? If you have an underlying $\cos(x)$ that is negative, and then you square it, you will now have $\cos^2{x}$, which is positive. But, if ...
1
vote
3answers
74 views

Solving integral $\int _n^{2n}\frac 1x dx$ [on hold]

I want to solve the integral- $$\int _n^{2n}\frac 1x dx$$ From wolfram alpha I get it is $=log(2)\approx 0.69315$. But I am unable to solve it step by step.So I need help. Also,if the ...
12
votes
3answers
172 views

How to show the divergence of $\sum\limits_{n=1}^\infty\frac{\sin(\sqrt{n})}{\sqrt{n}}$

The 10 standard tests taught in class are: 1) $n^{th}$ term test for divergence.(Not applicable: $\lim =0$). 2) Geometric Series(Not applicable). 3) Telescoping Series(Not applicable) 4) Integral ...
1
vote
1answer
17 views

Identity regarding partial derivatives and polar representation

Let $f(x,y)$ be a differentiable function, and $g(r, \theta) = f(r \cos \theta , r \sin \theta)$. I need help showing that: $$ \left( \frac{ \partial f}{\partial x} \right)^2 + \left( \frac{ \partial ...
3
votes
0answers
40 views

How to calculate only the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...
0
votes
2answers
34 views

If $f(x)$ is a diff. function $ \int_{0}^{4}f(t)dt = 2\left[\alpha f(\alpha^2)+\beta f(\beta^2)\right]$

If the function $f:\left[0,4\right]\rightarrow \mathbb{R}$ is a differentiable then show that $\displaystyle \int_{0}^{4}f(t)dt = 2\left[\alpha f(\alpha^2)+\beta f(\beta^2)\right]\;\; \forall ...
0
votes
5answers
70 views

Evaluate $\lim\limits_{x\to 1}\frac{x^{10}-1}{\sqrt x-1}$ without L'Hopital's rule

I was faced with a problem about evaluating limits. I know for a fact that when a limit is indeterminate when substituted with the value $x$ is approaching, I differentiate by L'Hopital's Rule. But ...
2
votes
1answer
19 views

Splitting a sum to find a closed form of $\sum_{n=2}^\infty\frac{n}{n-1}x^{n-1}$

Find a closed form for $$S = \sum_{n=2}^\infty\frac{n}{n-1}x^{n-1}$$ My solution The radius of convergence is $R=1$ and the series does not converge in $\pm 1$. Rewrite the sum as ...
0
votes
0answers
12 views

Simplification of an expression with phasors

Given $\vec{A},\vec{P},\vec{Q},\vec{K}$ are phasors in complex plane. The goal is to minimize the below function $F(\vec{P})$ or find $\vec{P}$ such that $F$ is minimum. I am looking for an analytic ...
0
votes
1answer
37 views

$1+xy+yz+xz-x-y-z>0$ where $x,y,z \in (0,1)$

$f(x,y,z)=1+xy+yz+xz-x-y-z$, where $x,y,z \in (0,1)$. Show that: $f(x,y,z)>0$. $\begin{equation} \begin{cases} \frac{\partial f}{\partial x}=y+z-1=0 \\ \frac{\partial f}{\partial y}=x+z-1=0 ...
-1
votes
3answers
62 views

Find the general solution of the ODE $xy′′ − y′ + 4x^3y = 0$ [on hold]

Can someone help me figure out this ODE, its driving me crazy. I dont need a full solution beacuse that would take hours but maybe just the final answer? Find the general solution of the ODE $xy′′ − ...
5
votes
1answer
66 views

Find the limit of this sequence

Suppose $$ f_n(x)=\sum_{k=1}^n \frac{\cos(kx)}{k}, $$ and let $a_n=\min_{x \in [0,\pi/2]} f_n(x)$, find $\lim_{n \to\infty} a_n$. I wrote a program and found that the $\arg\min_{x \in [0,\pi/2]} ...
1
vote
1answer
32 views

$\exp\left({\frac{-1}{(x-a)(b-x)}}\right) $ is infinitely differentiable on $(a,b)$

Let $a<b$. I'm trying to prove that $$\exp\left({\frac{-1}{(x-a)(b-x)}}\right) $$ is infinitely differentiable in the open interval $(a,b)$. Induction seems like a good way to proceed, and I know ...
1
vote
2answers
55 views

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $\prod_{n=1}^\infty (1-a_n)=0$ converges if and only if $\sum_{n=1}^\infty a_n=\infty$.

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $$\prod_{n=1}^\infty (1-a_n)=0$$ converges if and only if $\sum_{n=1}^\infty a_n=\infty$. Proof of $\Rightarrow$: Assume that ...
1
vote
1answer
20 views

Motion in 3D Space: Finding Velocity from Distance, Launch Angle

The question asks: A bullet is fired from the ground at an angle of $45°$. What initial speed must the bullet have in order to hit the top of a $130 m$ tower located $190 m$ away? (Recall that ...
1
vote
1answer
25 views

Composition is infinitely differentiable

The funcitons below all map real numbers to real numbers. Suppose that $f(x) = h(g(x)) \ \forall x \in \mathbb{R}$. Suppose that $g(x) \neq 0 \ \forall x \in \mathbb{R}$ and that all derivatives of ...
1
vote
0answers
17 views

Complex integration by substitution

Integrate $ f(z) $ counterclockwise around the unit circle. $$ f(z) = 1/(4z-3) $$ My solution C(contour) : $ z(t) = \cos{t} + i\sin{t} = e^{it}, 0<t\leq 2\pi $ $$ \oint_C \frac{1}{4z-3} dz = ...
0
votes
1answer
17 views

Distance between two points with an stimated time.

I am working on a project and one problem is this: I have two points, $A$ and $B$, but the only thing I Know is, time. with a time I want a distance between $2$ points, I want to calculate meters... ...