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

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2answers
66 views

Let $\frac{dy}{dx}-\frac{y}{x}=xe^{-x}$. Find $\lim_{x \to \infty}\frac{y}{x}$

Let $$\frac{dy}{dx}-\frac{y}{x}=xe^{-x}$$ Find $$\lim_{x \to \infty}\frac{y}{x}$$ I have tried to separate variables and integral both side of the function. However, it seems impossible. Could ...
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2answers
154 views

Some Big-O complexity definition proofs

I'm trying to prove (by definition) the following but to no avail: $n^{n/2} \ne O(3^{n/2}) $ $n! \ne O(3^n)$ $(n-b)^a = \Theta(n^a)$ $a,b $ are both constants whereas $a > 0 $ and $b$ ...
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2answers
693 views

Little-o proof by definition

I'm trying to figure out how to prove the following but to no avail. Given the following functions : $f(n) = n^3 -4n$ $g(n) = 5n^2 + 3n$ I have to show that $g(n) = o(f(n))$ by definition, that ...
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1answer
237 views

Rate of change problem

Water is flowing into a large spherical tank at a constant rate. Let $V\left(t\right)$ be the volume of water in the tank at time $t$, and $h\left(t\right)$ of the water level at time $t$. Is ...
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1answer
25 views

How does $X_{n+1 }= (1-10^{4}h)X_n\quad X_n= (1-10^{4}h)^{n+1}$

How does $X_{n+1} = (1-10000h)X_n$ become $X_n= (1-10000h)^{n+1}$ I can't seem to understand the solution to one of my questions because of this transformation of $X_{n+1}$. I'm not sure how the ...
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1answer
92 views

find the critical value of r.

For the following equations sketch the bifurcation diagram, determine type of bifurcation, and find the critical value of $r$. $$\dot{x} = rx + \cosh x$$ I seem to understand how to do the first ...
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1answer
46 views

Autonomous linear ODE

Given $K(0) = 0,\!2P$. I'm supposed to solve the ODE $$ \frac{dK}{dt} = \lambda(P-K)$$ I have tried to seperate and integrate both sides $$ \int \frac{1}{P-K} dK = \int \lambda \space dt$$ to get ...
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1answer
35 views

Finding the function$g(x)$ which $f(x)=g(x), x\geq 0$

I found this question interesting: Which one of the following functions are equal to $f(x)=\lfloor \sqrt{x}-\lfloor \sqrt{x}\rfloor\rfloor$? ($\lfloor \cdot \rfloor$ is the floor function) ...
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2answers
68 views

a problem on differentiation

Consider the function $f(\theta)=|\cos\theta|+|\sin(2-\theta)|$ At which of the following points is $f$ not differentiable ? 1.$\{(2n+1)\frac{\pi}{2} : n \in Z\}$ 2.$\{n\pi :n \in Z\}$ 3.$\{n\pi+2 : ...
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2answers
139 views

Problem involving implicit differentiation with many exponents.

Find the derivative of the function $x^{x^{x^{x}}}$.
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3answers
48 views

Need help with the limit of sequence

I need help on a question from my homework, which asks me to find the limit of the sequence as n approaches infinity of $$a_n = \frac{\cos^2 n}{2^n}$$ Thanks
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2answers
110 views

Partial Fractions - Calculus

Evaluate the integral: $$\int \dfrac{9x^2+13x-6}{(x-1)(x+1)^2} dx$$ For some reason I cannot get the right answer. I split up the equation into three partial fractions but I cannot seem to find A, B, ...
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2answers
50 views

Number of valid elements Matrix*Matrix transpose

I am trying to find an analytical method to find the number of valid elements in a matrix*matrix^T from the matrix without the need of doing the multiplication. I ...
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1answer
84 views

Derivative of $\tan(xy^3)$

Can somebody tell me if I'm right on this? The math looks right, yet it just feels so wrong due to the obscene steps I had to take to get it. I hope I transcribed all that correctly from my paper. ...
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1answer
56 views

Continuos Function infinitely differentiable

What mean a $0$ in the set of functions infinitely differentiable $C_0^\infty$
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1answer
69 views

Does a contour of local extrema of a function $f : \mathbb{R}^2 \to \mathbb{R}$ need always be smooth?

Consider a smooth function $f : \mathbb{R}^2 \to \mathbb{R}$, I wonder that any contour (curve) in $\mathbb{R}^2$ where every point of it is a local maxima of $f$, need be a smooth curve? Edit : $f$ ...
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2answers
140 views

Derivatives of trig functions

How can I prove that $\frac{d}{dx} (\csc x)= -\csc x \cot x$? Specifically, how does one see the step $\cos x/\sin x = \cot x$?
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2answers
57 views

Is the given function exponential?

If a function of this form $f(x)=b\cdot a^x$ is called an exponential function, then is the function $g(d)=13.4\cdot \ln(d)-21.8$ also exponential? If yes why?
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1answer
49 views

How to find $p$ such that $A$ is independent of $c$.

Let A be the area of the triangle formed by x-axis, y-axis and the tangent line of $y=x^p$ ($p<0$) at $x=c$. Find $p$ such that $A$ is independent of $c$. P.S What does "$A$ is independent of $c$" ...
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2answers
58 views

radius of convergence for $a_n= (\log n)^2$

determine the radius of convergence of the power series for: $$a_n= (\log (n))^2$$ I know i am suppose to use the Ratio test and L'Hopitals Rule. I can't seem to figure out the steps
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2answers
238 views

Derivative of a function

$$G(y) = \ln \dfrac{(5y+1)^2}{\sqrt{y^2 + 1}}$$ Can we break this up like this: $$2 \ln(5y+1)\over \frac 12\ln(y^2 + 1)$$
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2answers
232 views

First derivative test for absolute extrema proof.

Prove that if $I \subset \mathbb{R}$ is an open interval and $f: I \to \mathbb{R}$ differentiable, and $f$ has only one critical point $x_0$ and this critical point is a local minimum, then $x_0$ is ...
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1answer
1k views

Find the derivative of $F =$ $(GmM)\over r^2$

Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass M on a body of mass m is $F =$ $(GmM)\over r^2$ Where G is the gravitational constant and r is the ...
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1answer
88 views

Convergence of $\sum (n - 1)a_n$ given convergence of $\sum a_n$

I haven't studied basic calculus/analysis for a while so I apologise if this is obvious. In the course of solving another problem I have found that a sum of the form $$\sum_{n=1}^{\infty}(n - 1)a_n$$ ...
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1answer
45 views

Basic Series Question

Let $(a_n)$ be a sequence such that $\liminf |a_n| = 0$. Then there exists a subsequence $(a_{n_k})$ such that $\sum_{k=1}^\infty a_{n_k}$ converges. any thoughts?
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3answers
133 views

Limit using L'Hopital's Rule

Find $$\lim_{x \to 1} \sqrt{x-1}^{\,\sin(πx)}$$ using L'Hopital's Rule. Initially I get $0^0$ so I know I need to use the rule, but I don't know where to begin. Could you help me out with some steps ...
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1answer
140 views

The derivative of $f(t, y(t))$ with respect to $t$?

Given a function $f(t, y(t))$, how can I express its derivative with respect to $t$ and $d f/{d t}$?
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2answers
72 views

How to solve these limits?

I can understand the concepts behind these limits, but I have no idea where to start to solve them. Here are my questions
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3answers
97 views

How do I find acceleration at $x=4$?

The question is a three part question. Part a) asked to find the average velocity with the equation $y= -\frac{1}{25}x^2 + \frac{4}{5}x$ on the interval $[6,10]$. Part b) asked to find the ...
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1answer
76 views

A particle of charge $-e$ orbits a particle of charge $Ze$, what is its orbital frequency?

A point particle $P$ of charge $Ze$ is fixed at the origin in 3-dimensions, while a point particle $E$ of mass $m$ and charge $-e$ moves in the electric field of $P$. I have the Newtonian equation of ...
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3answers
192 views

number of roots of an equation

Plotting the equation $x^3-x^2 \sin(x)+\cos(x)$ I see that $x^3-x^2 \sin(x)+\cos(x)=0$ has only one real solution, is there a simpler way to see that it cannot have 3 real solutions?
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1answer
66 views

Laplace-like operator

Help me please to apply a Laplace-like operator:$ \Delta f:= \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial z^2} + {1\over r}\,\frac{\partial f}{\partial r} - {f\over r^2} $ on the ...
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1answer
897 views

Using Comparison test to determine if $\int_0^{\infty} \frac{\arctan x} {2+e^{x}} \ dx$ converges

Only using the Comparison test, I am trying to see if the following integral converges: $$\int_0^{\infty} \frac{\arctan x} {2+e^{x}} \ dx$$ I first noted that $\arctan x \lt (2+e^{x}) \ \forall x \in ...
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1answer
411 views

Get Angle to Tangent that Intersects Point

I have a circle around a given point, call this point $(x_1, y_1)$. I know the radius of the circle around this point. I also have a second point $(x_2, y_2)$, that is a distance away, outside the ...
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2answers
293 views

How to do this Intermediate value theorem proof?

Use the Intermediate Value Theorem to show that the equation $x^3+x+1=0$ has a solution. How to do this? :S Thank you very much!
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1answer
2k views

A chain 64 meters long whose mass is 20 kilograms is hanging over the edge of a tall building…

A chain 64 meters long whose mass is 20 kilograms is hanging over the edge of a tall building and does not touch the ground. How much work is required to lift the top 3 meters of the chain to the top ...
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1answer
155 views

derivation using Leibniz rule

Differentiating $$\int_{(m-d-\mu)/\sigma}^{\infty}xf(x)dx$$ with respect to $\sigma$, where $\sigma$ is the standard deviation of the standardarized random variable $x$ and $\mu$ its mean. I guess ...
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2answers
101 views

Area between $y=\frac 13$ and$ y=\sqrt{x}$

For some reason I keep getting 15, and I've tried this question multiple times.
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1answer
153 views

Prove $\lim\limits_{n \to\infty} a_n =\lim\limits_{n \to\infty} \dfrac{2n-1}{3n+2} = \dfrac{2}{3}$

Prove: $\lim\limits_{n \to\infty} a_n =\lim\limits_{n \to\infty} \dfrac{2n-1}{3n+2} = \dfrac{2}{3}$ using the definition of the limit. This is what I have so far: Let $\epsilon > 0$ and take $N ...
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1answer
631 views

How to solve the recurrence $T(n)=3T(n/2)+n$

The exercise stated that i have to solve the recurrence using the Recursion-Tree Method. I have already finished the base part, which is $\Theta(n^{\lg3})$ But for the recursive part I'm having ...
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1answer
273 views

minimum distance between graphs of functions

Prove this : When the graphs of two differentiable functions have the minimum distance then the secants at those points are parallel .
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1answer
2k views

Find delta with a given epsilon.

Here is the problem. If $$\lim_{x\to-2}x^3 = - 8$$ then find $\delta$ to go with $\varepsilon = 1/5 = 0.2$. Is $\delta = -2$?
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2answers
97 views

Integral Problem with Partial Fractions

The problem is: $$\int\frac{7x}{(2x+5)^2}\;.$$ I got and am fairly confident in: $$\frac74 \ln|2x+5| + \frac{35}{8x + 20}$$ However this is apparently not the correct answer. Im not being graded ...
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2answers
295 views

Defining an upper/lower bound in lexicographically ordered C

If I have a lexicographic ordering on $\mathbb{C}$, and I define a subset, $A = \{z \in \mathbb{C} : z = a + bi, a, b \in \mathbb{R}, a < 0\}$. I have an upper bound, say $\alpha = 0 + di$. My ...
0
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1answer
72 views

Another Cross Product

So I understand most of the properties of cross products. However I ran into a small complication. I get that $i\times j = k$, $j\times k = i$. I also understand that $k \times j = -i$ and that ...
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3answers
89 views

is this the correct way to rewrite calculus limit?

Can $$\lim_{x \to 0} \frac{x^3-7x}{x^3}$$ be rewritten as $$\lim_{x \to 0} \frac{x^3(1-7x^{-2})}{x^3}?$$ These two seem to give different answers. Please help me I'm really confused :S
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4answers
89 views

Basic Calculus Function Limits question

We just had our first calculus lecture, and I'm kinda stuck at this proof right now: Prove that $\lim_{n\to\infty}\{(-1)^n\}$ diverges. Given: $\lim_{n\to\infty}(-1)^n = a$. Thus for ...
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3answers
120 views

What can this differential equation be used to model?

So, I can model growth and decay if I start with assuming that the growth rate is constant: $\frac{p'(t)}{p(t)}=\alpha$ and then I have $p'(t)-\alpha p(t)=0$ A general linear differential ...
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1answer
74 views

Where did I miss the minus?

If the $$e^{j\omega} = \cos(\omega)+j \sin(\omega)$$ and I have the following equation $$\phi_s = {2\over3}\left(\phi_u+\phi_ve^{{2\pi\over3}j}+\phi_we^{{4\pi\over3}j}\right)e^{-j\theta}$$ The ...
0
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1answer
139 views

Disk Method for Volume of Solid with negative exponent.

Use the disk method to find the volume of the solid generated when the region bounded by $y=(1-9x)^{-1/4}$, $y=0$, $x=0$, and $x=1/18$ is revolved about the x-axis. I know that to set this problem ...