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

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2
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1answer
1k views

Composition Taylor Series

Is there any theorem that specifies when we are allowed to compose the taylor series of two functions? Does it have a name? Thanks.
2
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3answers
85 views

Confused About Squareroots & Absolute Values

$${ x }^{ 2 }=4\\ \sqrt { { x }^{ 2 } } =\sqrt { 4 } \\ |x|=2$$ According to my professor, in the above case, the absolute value gives two solutions: $x=±2$ Consider the discriminant in the quadratic ...
2
votes
1answer
203 views

Finding the winding number of a curve

Let $\gamma(t)=(r \cos t,r \sin t)$, for some $r>0$, and let $\Gamma$ be a $C^2$-curve in $\mathbb R^2-\{\bf 0\} $, with parameter interval $[0,2 \pi]$, with $\Gamma(0)=\Gamma(2 \pi)$, such that ...
2
votes
2answers
95 views

Manipulating the equation!

The question asks to manipulate $f(x,y)=e^{-x^2-y^2}$ this equation to make its graph look like the three shapes in the images I attached. I got the first one: $e^{(-x^2+y^2)}\cos(x^2+y^2)$. But I ...
2
votes
2answers
9k views

How to find average rate of change

How would I find the average rate of change over $8$ minutes, of a person that runs at a rate of $v(t)=x\sin(x^2-7x)$ ft/min? I missed when this was taught and I have no clue on how to do it. Help is ...
2
votes
2answers
112 views

Finding angle which $f(x)=e^x$ intersects $y$ axis

How can I find the angle which the function $f(x)=e^x$ meets the $y$ axis? I think it is the slope of $y$ at the point $(0,1)$. Is that right?
2
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2answers
259 views

Finding an alternative to a no closed form integral

In my notes, I have learnt that some functions have no closed form integrals, for example $$f(x) = e^{-x^2}$$ has no closed form integral. I have 2 questions on this I understand that a closed form ...
2
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2answers
278 views

There exists an odd function such that $\int_{-1}^{1}f(x)g(x)dx \neq 0$

I'm stuck on this problem for a long time. I'm very grateful if some one can help me solve this.Thanks. Suppose $f$ is an odd non-zero continous function, prove that there exists another odd ...
2
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2answers
66 views

how to calculate a limit?

Here is a well-known function: $$f(x)=\begin{cases}\exp\left(-\frac{1}{x^2}\right) & x\not=0 \\ 0& x=0\end{cases}.$$ How to calculate : $$\lim_{t \rightarrow 0}\frac{f(t^2+t)-f(t)}{t^2},$$ ...
2
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2answers
2k views

Differential problem, find the maximum and minimum value

Find the maximum, minimum value and inflection/saddle point of the following function $f(x)=12x^5-45x^4+40x^3+6$ $f(x)=x+\frac{1}{x}$ $f(x)=(2x+4) (x^2-1)$ Give a little explanation or procedural ...
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2answers
5k views

How to graph gradient vector?

I'm working on a practice problem for my Calculus 3 course. It gives the function: $z=x^2+y^2$, and asks to graph the contours for $c=1,2,3$. Than asks to calculate the gradient at point $(2,1)$ and ...
2
votes
1answer
321 views

Taylor series problem

Find and state the convergence properties of the Taylor series for the following: a.) $f(z)=z^3\sin 3z$ around $z_0=0$ b.) $f(z)= \frac{z}{(1-z)^2}$ around $z_0=0$ I know that the ...
2
votes
3answers
821 views

find derivative of $e^{3\sqrt{x}}$ using chain rule.

im asked to find the derivative of this function using the chain rule. $$e^{3\sqrt{x}}$$ here are my steps. step 1 - identify the inner and outer functions. therefore I identified outer function ...
2
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2answers
60 views

Find the Taylor series of $\frac{1}{x+1} $ at $x=2$

This is what I did: $\begin{align*} f(x)&=&(x+1)^{-1}\\ f'(x)&=&-(x+1)^{-2}\\ f''(x)&=&2(x+1)^{-3}\\ f'''(x)&=&-6(x+1)^{-4}\\ f''''(x)&=&24(x+1)^{-5}\\ &\...
2
votes
3answers
110 views

Rewriting sequence from $X_{n+1}$ to $X_n$

I have the sequence: $$ \begin{align} X_{n+1} &= \frac{X_n^2 + 5}{2X_n} \\ X_1 &= 1 \end{align} $$ I have to prove that it converges and find its limit after I write it in terms of $X_n$, ...
2
votes
2answers
864 views

How do i find the equation of the tangent line?

Find the function of the tangent line to $$f(\theta)=5\left(\cos(\pi/3)\cdot\sin(\theta)\right)$$ at $\theta=2\pi/3$ Please show steps and explain. I need to understand this for an upcoming exam.
2
votes
1answer
146 views

Introduction to Calculus equality

I was reading through Apostol's Calculus where he has this equality, but I can't figure out how to prove this equality. I would appreciate a hint if you could help. Thank you. $$ [(\sum_{k=1}^{n-1}k)+...
2
votes
3answers
157 views

For what value is the local minimum the largest?

If $f(x)=e^x-kx$ for $k>0$, find the values of $k$ for which the local minimum at $x=\ln(k)$ is the largest. I found the derivative, which is $e^{x} - k$, and when I set that to $0$ I got $-e^{...
2
votes
3answers
102 views

Definite Integral with a discontinuty

I have the next integral: $$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$ I have no clue how to start. At $x=0$ there is a clear discontinuity and I don't know how to solve the integral. The main ...
2
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1answer
186 views

A question on Spivak's proof on manifolds

This is hard for me to ask since I have to write out the entire long proof done in Spivak's book on Manifolds. If anyone has the book, it would be great. This question concerns the Inverse Function ...
2
votes
1answer
75 views

How to prove this inequality : $(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}} \, ; x>0,\ y>0;\ 0<a<b$

Prove that when $\displaystyle x>0,\ y>0;\ 0<a<b$ $$\displaystyle(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}}$$
2
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1answer
187 views

Polynomial function, only 1 solution

I am given the following task: The graph of the polynomial function $f(x)$ is symmetric to the y-axis. It has exactly one local minimum on the x-axis and an inflection point at $x = 1$. Find the the ...
2
votes
1answer
115 views

Calculus trig derivative?

How would I evaluate the following two derivatives. $g(t)=\sin^2(x)+\cos^2(t)$ For this derivative I know $\sin^2+\cos^2(t)=1$ so there derivative of $1$ is $0$ For my second question I have to ...
2
votes
2answers
134 views

Constrained maximization problem

I need help with the following optimization problem $$ \max\;\alpha\ln(x(1-y^2))+(1-\alpha)\ln(z) $$ where the maximization is with respect to $x,y,z$, subject to \begin{align} \alpha x+(1-\alpha)z&...
2
votes
2answers
420 views

Create asymmetric 3D function by revolving a 2D function around an axis

Could someone please let me know how I can construct the equation for asymmetric torus similar to the figure below? The asymmetric torus seems to be a 2D function revolved around an axis while being ...
2
votes
1answer
124 views

Computing the double series

I need a starting point for $$\lim_{n\to\infty}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{\ln\left(1+\displaystyle \frac{i}{n}\right)\ln\left(1+\displaystyle\frac{j}{n}\right)}{\sqrt{n^4+i^2+j^2}}$$ What ...
2
votes
2answers
370 views

Minimizing arc length of $y=\ln(\cos x)$

Suppose you know that the difference between $a$ and $b$ is $2$. How can we find the values of $a$ and $b$ (with $-2\leq a,b\leq 2$) which minimizes the arc length of the curve $y=\ln(\cos x)$ from $x=...
2
votes
1answer
4k views

3D Fourier Transform

I'm trying to calculate the inverse of the following 3D Fourier transform. $$ \widetilde{f}= \frac{1}{(k^6-\alpha*k^2-\alpha*k_3^2)} $$ where $k = (k_1^2+k_2^2+k_3^2)^{1/2}$ the fourier transform is ...
2
votes
3answers
115 views

Two simple series

I dont know how to calculate these two series: $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( 1-\cos \frac{\pi }{n} \...
2
votes
1answer
88 views

An integral for the Earth's insolation

Consider the function $$ [-\pi/2,\pi/2] \ni \theta \mapsto s_\beta(\theta) = \int_0^{2\pi} \sqrt{ 1 - \left(\cos \theta \sin \beta \cos \gamma - \sin \theta \cos \beta \right)^2} \, d \gamma $$ for $...
2
votes
2answers
104 views

The solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=2xy^2$

Question from pg 32 of Barron's AP Calculus The solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=2xy^2$ for which $y = -1$ when $x = 1$ is (A) $y = -\frac{1}{x^2}$ for $...
2
votes
2answers
397 views

definite integral with periodic function of period = 1

If $f(x)+f\left(x+\frac{1}{2}\right) = 1$. Then $\displaystyle \int_{0}^{2}f(x)dx = $ If function $f(x)$ is Continuous and Differentiable in $x\in \left(0,2\right)$ My Try:: Using $f(x)+f\left(x+\...
2
votes
2answers
58 views

$\lim_{x\to \infty}A(x)$

If $A(x)=\int_{-1}^{x}e^{-|t|}dt$, then what is $\lim_{x\to \infty}A(x)$? I am not able to make any progress on this problem.I hope someone can help me with this.
2
votes
1answer
70 views

Convergence integral causal function

I have an exercise where there is the following given: $f$ is a causal function. $f$ is Laplace transformable:$\int_{0}^{\infty} f(t)e^{-zt} \, dt = L(z) $ with $Real(z)> -1$ I have to ...
2
votes
2answers
360 views

Population growth

Given that the initial population, $N_0$ of bacteria is $12$ and the population doubles every $20$ minutes. I wish to find a general formula for the population if one unit of time is $50$ minutes for $...
2
votes
1answer
109 views

A integral equation

Prove that : $$\displaystyle \int_0^{\frac{\pi}{2}}p(x)\cot x\text{d}x=2\sum_{k=1}^{\infty}\int_0^{\frac{\pi}{2}}p(x)\sin (2kx)\text{d}x$$ where $\displaystyle p(x)=x^n$
2
votes
1answer
370 views

Show the Hankel Transform of Rect is Jinc

I am trying to show that the Hankel transform of the rect function is the jinc function. The Hankel transform is defined as follows: $$ F(q) = \mathbb{H}\left\{f(r)\right\} = 2\pi\int_0^\infty ...
2
votes
2answers
296 views

Texts that define the derivative as the “Anti-integral”…?

Every text that I read starts by defining differentiation then integration... but does anyone know if there is one that goes the other way? Also is there any harm in taking this approach.... to me, ...
2
votes
3answers
141 views

How to prove that this point is undefined?

For the function $$f(x)=\lim_{n\to \infty}\;\large \frac {1} { \frac 1 {x^n} +1},$$ how do I show that the point $f(-1)$ does not exist?
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2answers
512 views

Calculating the mass of a Surface

Can someone help me with the following one? I need to calculate the mass of the surface $x+y+z=a $ , $a>0$ intersected with the cylinder $x^2 + y^2 = R^2 $ where the density is $ u(x,y,z)= (x^2+y^...
2
votes
2answers
683 views

Evaluate $ \int \frac{\sin^{-1}(x)}{\sqrt{1+x}} \; dx$

So I have to find the integral of $$ \int \frac{\sin^{-1}(x)}{\sqrt{1+x}} \; dx$$ I think I have to do this using the integration by parts..so I will take $f = \sin^{-1}(x)$ and $ \sqrt {1+x}=g' $......
2
votes
1answer
106 views

Prove that an odd degree polynomial must converge to $ \pm\infty $ when $ x\to\pm\infty$

While what I'm trying to prove is related to this topic but alas I cannot just assume that given: $ p(x) $ where $ \deg(p(x))=2n+1$ where $n\in\mathbb{N}$ that the only two cases are: $\lim \...
2
votes
1answer
269 views

What is the probability that the resulting four line segments are the sides of a quadrilateral?

Question: Divide a given line segment into two other line segments. Then, cut each of these new line segments into two more line segments. What is the probability that the resulting line segments are ...
2
votes
2answers
144 views

Prove that $f\circ g$ is differentiable in t=0, but the chain rule doesn't work.

Given $f(x,y)=\begin{cases}\frac{x^2y}{x^2+y^2} &\text{ if }(x,y) \neq (0,0)\\\\ 0&\text{ if }(x,y)=(0,0)\end{cases}$ and let $g(t)=(t,-2t).$ Prove that $f\circ g$ is differentiable in t=0, ...
2
votes
1answer
343 views

$\int_{0}^{\pi/2}\frac{\sin x \cos x}{x+1}~dx=\frac{1}{2}(\frac{1}{2}+\frac{1}{\pi +2}-M)$

If $\displaystyle{M=\int_{0}^{\pi} \frac{\cos x}{(x+2)^2}~dx}$, then show that $$\int_{0}^{\pi/2}\frac{\sin x \cos x}{x+1}~dx=\frac{1}{2}\left(\frac{1}{2}+\frac{1}{\pi +2}-M\right)$$
2
votes
3answers
86 views

How to find $\sum_{k=1}^{\infty}\frac{f(x+k\pi)}{2^k}$?

Let $\sum_{k=1}^{\infty}\frac{f(x+k\pi)}{2^k}=f(x)$, where $f(u)=c\sin u$.Find $c$. Trial:$\sum_{k=1}^{\infty}\frac{c \sin (x+k\pi)}{2^k}=c\sin x$.Then $c=0$ is a solution. Is there any other ...
2
votes
2answers
117 views

studying $\int^x_0\frac{1}{1+t^3}dt$

Let $F(x)=\displaystyle \int^x_0\frac{1}{1+t^3}dt$ 1)Prove that F is well defined and differentiable for all $x\in\mathbb{R}$ 2)If $n$ is a positive integer show that $F(x)=\displaystyle(\sum\...
2
votes
1answer
103 views

proving the sum in not integrable

This is a question from past comp exam, Let ${(g_n)_n }$ be a real continuous funtion with support in $(\frac{1}{n+1}, \frac{1}{n})$, such that $\int_0^1 g_n(t)dt =1$, for $n=1,2,3....$ Now define $$ ...
2
votes
1answer
265 views

How to prove this sum of integrals

How to prove this ? $$\displaystyle \sum\limits_{k=0}^{\infty }{\int_{2k\pi }^{\left( 2k+1 \right)\pi }{{{\text{e}}^{-\frac{x}{2}}}\frac{\left| \sin x-\cos x \right|}{\sqrt{\sin x}}}}\text{d}x=\frac{2\...
2
votes
2answers
43 views

May someone help me to find the vertical and horizonal asymptotes of this function?

I have found this question in the Leithold´s book, but there´s no any worked example of how i can compute it in order to find the horizontal and vertical asympototes.