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

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2k views

Find the horizontal tangent line

Fine the $x$-coordinate of all points on the curve $y=\sin(2x)+2\sin(x)$ at which the tangent line is horizontal. Consider the domain $x=[0,2\pi)$. There are three $x$ values. Can you explain how to ...
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166 views

Position of Particle; When is it parallel to x-axis.

I am going over a test question that was marked incorrect, but I am unsure as to why. The position of a particle (in mm) is given by $x=t^3-27t$ and $y=t^2-4t$ where time t is measured in seconds. At ...
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37 views

Show that this is one to one continuous and find its inverse which is continuous as well.

Let's define $\phi: \Bbb R^2 \to S$ for $S$ is subset of $\Bbb R^3$ For constant $a,b,c,d$ and $c\not =0$ $$\phi(x,y)=(x,y, \frac{d-ax-by}{c})$$ I want to show that the function $\phi$ is 1-1 ...
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99 views

prove that $\int(f(x)+g(x))dx= \int f(x)dx+\int g(x)dx$

Let $f,g$ be two functions defined on $A$. Supposed that $F$ and $G$ are anti-derivative of $f$ and $ g$. Prove that $\int(f(x)+g(x))dx= \int f(x)dx + \int g(x)dx$ Here is what I got. Let $H(x)$ ...
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1answer
128 views

Find the limit with L'Hopital's rule $\lim_{x\to0^+}\left(\frac{2}{\pi}\arcsin x\right)^{1/x} $

I was solving this problem and wolfram alpha said it was $0$.But i just can't get it to $0$ using L'Hospital. Can you please show me how to do it. $$\lim_{x\to0^+}\left(\frac{2}{\pi}\arcsin ...
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1answer
228 views

Strategy to recognize and solve sequence and series problems?

I've been reading my Stewart Calculus book and I honestly find most of the coverage of sequences and series easy to grasp (excluding power series, Taylor and Maclauren since we haven't covered those ...
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1answer
55 views

How to evaluate a limit with subtractions $\lim_{x \rightarrow -1}(\frac{3}{x^3+1}-\frac{1}{x+1})$?

I'm having trouble thinking of a way to solve this. $$\lim_{x \rightarrow -1}\left(\frac{3}{x^3+1}-\frac{1}{x+1}\right)$$
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1answer
39 views

How to calculate this mass.

Determinate the mass of portion of spherical surface given by the equation $x^2+y^2+z^2=2a^2$ who is inside of the cylindrical surface given by the equation $(x^2+y^2)=2a^2(x^2-y^2)$, knowing that the ...
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46 views

If $L = \lim_{t\to \infty}f(t)$ prove that $\lim_{t\to \infty}(1/t)\int_0^tf(x)dx = L$

Let $f:[0,\infty) \to \mathbb{R}$ be continuous and suppose $L = \lim_{t\to \infty}f(t)$ exists. Prove that $\lim_{t\to \infty}(1/t)\int_0^tf(x)dx = L$ My solution: Let $\epsilon > 0$ be given ...
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1answer
48 views

Finding slope of a curve at a point

Find the slope of the curve $$y=\frac{1}{x-4}$$ at $x=8.$ My try: $$m=\lim_{h \to 0}\frac{f(x_0+h)-f(x_0)}{h}$$ $$= \lim_{h \to 0}\frac{\frac{1}{(8+h)-4}-\frac{1}{(8)-4}}{h}$$ ...
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70 views

Help with limit of trigonometric function

$$\lim_{x \to 0}\frac{x \csc 10x}{\cos20x}$$ I'm unsure of how to solve this. I think if I were to simplify it, it would be: $$\lim_{x \to 0}\frac{x \csc 10x}{\cos20x} = \lim_{x \to ...
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61 views

how to tell if its a function in a function for calculus

in calculus you apply chain rule when you have function in a function. But i can't tell when it's true. (x+1) -> one function (x+1)^2 -> two functions what defines, how come +1 didn't make it two ...
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1answer
131 views

Related Rates explanation

A railroad track and a road cross at right angles. An observer stands on the road $70$ meters south of the crossing and watches an eastbound train traveling at $60$ meters per second. At how many ...
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2answers
399 views

Is there an unbounded function with a bounded derivative?

I know that there exists bounded functions with unbounded derivatives. For example, $\sin(e^x)$ is bounded and differentiable everywhere on $\mathbb{R}$, but its derivative is unbounded. Is it ...
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320 views

Computing surface area and volume of a unit sphere

When solving some exercises, I forgot the formula for the surface area of the unit sphere. However, I remember that the length of the perimeter of a circle of radius $r$ is $2 \pi r$. So I figured ...
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1answer
53 views

Confusion with this definition of the derivative

This function is from my text: $$p(\theta) = \sqrt{13\theta}$$ It states that the derivative of the function $p(\theta)$ with respect to the variable $\theta$ is the function $p'$ whose value at ...
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253 views

Find the radius of convergence and the interval of convergence of the series $\sum_{n=1}^\infty \frac{n(x-4)^n}{n^3+1}$

Series is: $$\sum_{n=1}^\infty \frac{n(x-4)^n}{n^3+1}$$ So, I understand that I use the ratio test to find r, but I can't simplify the equation to the point where I can do this. Here's where I am so ...
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1answer
169 views

Find the sum of $\sum_{n=1}^\infty \frac{x^n}{2^{n+1}}$

I need to find the sum of $\sum_{n=1}^\infty \frac{x^n}{2^{n+1}}$, But I don't really know how. I tried to manipulate integrals and derivetives but it's not helped me. Can you pleae help me to find ...
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512 views

Find the integral of $\tan^4(x) \sec(x)$

I want to know if there is a shorter way to find integral of $\tan^4(x) \sec(x)$ without using reduction formula of $\sec(x)$ because it's really takes a long time. Thanks all
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62 views

Integral of $y(x)$ when $y(t)$ is in the equation

I'm supposed to find the limit of $y(x)$ when $x \rightarrow \infty$ if $y$ is given by: $$y(x)=7+\int_0^x 4\frac{(y(t))^{2}}{1+t^2}dt$$ What I don't get is the $y(t)$ inside the integral. If I ...
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1answer
118 views

Help With Series (Apostol, Calculus, Volume I, Section 10.9 #9)

I am looking for help finding the sum of a particular series from Apostol's Calculus (Volume I, Section 10.9, Problem 9). The trouble is that I can find the correct answer, but only using methods that ...
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1answer
68 views

finding an 'e' based limit for this sequence

i need to find the limit for $$\lim_{n \rightarrow \infty} \left(1 + \frac{q}{n}\right)^n $$ where $q \in \mathbb{Q}$ how to i get this sequence to resemble $$\lim_{n \rightarrow ...
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2answers
302 views

Prove that if $f:A\to B$ is uniformly continuous on $A$ and $g$ is uniformly continuous on $B$, then $g(f(x))$ is uniformly continuous on $A$

Suppose that $f\colon A \to B$ is uniformly continuous on $A$ and $g$ is uniformly continuous on $B$. Show that $g \circ f$ is uniformly continuous on $A$. I tried to use the definition of uniformly ...
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27 views

Polar Equation Conversion

Change the polar equation $\theta=\frac{\pi}{3}$ to rectangular coordinates. How would I go about this question? I've tried $x=r\cos\theta$ and $y=r\sin\theta$, but I can't figure out $r$ since ...
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125 views

Double integral, solve by inspection

∫∫$_T (1-x-y) dA$ where $T$ is the triangle with vertices $(0,0), (0,1)$ and $(1,0)$. I got the answer $1/2$, since $dA=1$ and the area of the triangle is $(1 \cdot 1)/2 = 1/2$. But it's not correct. ...
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63 views

How do I know if my answer satisfies Rolle's theorem?

Given a function on $[0,2]$, $$f(x) = x^3 - x ^2 -2x +2$$ I know the answer has to be between $[0,2]$, but for some reason, my answer isn't being accepted. I derived the function and got the ...
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1answer
48 views

Determine the tangent plane on this paraboloide

given a paraboloide $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=\frac{z}{\gamma}$ The tangent on a point $(x_0,y_0)$ along the paraboloide is given by $(x,y,f(x_0,y_0) + \langle \nabla f (x_0,y_0), ...
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40 views

Maclaurin series for $ f(x)=a^x$

My friend is having trouble with these two questions on his homework. I want to help him out but I am not 100% sure how to do these. I took Calculus 3 a while back so its all old memory to me! ...
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32 views

Simple Derivative

I am wanting trying to remember how to solve a derivative of this nature: $$ \frac{dM}{dt} = rM(t)$$ $$ dM = rM(t)dt $$ Solving when t = 0 equals 1 we can get the solution as $$ M(t)= M(0)e^{rt} ...
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34 views

Critical points : $ z = \cos^2x + \cos^2y$ for $ y-x = \frac{\pi}{4}$

Find and classify the critic points (maximum, minimum or neither) of the function $$z=z(x,y) = \cos^2x + \cos^2y $$ if $y-x = \large\frac{\pi}{4}$ I've find $x = -\large\frac{\pi}{8} + \large\frac{k ...
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1answer
67 views

Prove that $\dfrac{|x+y|}{1+|x+y|}\leq\dfrac{|x|}{1+|x|}+\dfrac{|y|}{1+|y|}$ for any $x,y$

Use mean value theorem to prove the inequality: $$\dfrac{|x+y|}{1+|x+y|}\leq\dfrac{|x|}{1+|x|}+\dfrac{|y|}{1+|y|}\quad \forall x,y\in\mathbb{R}$$ I have no idea which function I should consider to ...
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79 views

Requesting feedback on proof of theorem

I'm trying to self study my way through Apostol's calculus and have just started. Having completed an undergraduate degree in physics some time ago, the math courses I took mostly focused on ...
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1answer
44 views

Simplifying logarithmic equation $0.1 = e ^{-(\ln 2)t/5700}$?

$$0.1 = e ^{-(\ln 2)t/5700}$$ How do I simplify this? I took the ln of both sides so does the $e^{\ln}$ cancel out? $$\ln 0.1 = \ln e ^{-(\ln 2)t/5700}$$ $$-2.3 = \ln e ^{-(\ln 2)t/5700}$$ but the ...
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69 views

prove that the sequence $x_{n+1} = \frac3{4-x_n}$ is converging and find its limit [duplicate]

prove that this sequence is converging and find its limit $x_1 = \frac32$ $x_{n+1} = \frac3{4-x_n}$ i believe that the solution entails proving the sequence is monotone descending and its ...
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151 views

limits of sequences such as $\lim_{n \rightarrow \infty}\sqrt[n]{n!}$

I have a problem figuring out the limit: $\lim_{n \rightarrow \infty}\sqrt[n]{n!}$. And other similar limits such as: $\lim_{n \rightarrow \infty} \frac{n}{\sqrt[n]{n!}}$ $\lim_{n \rightarrow ...
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1answer
76 views

Why does this series diverge? $\sum_{n=1}^\infty \frac{n-1}{4n-1}$

So taking my original problem: $\sum_{n=1}^\infty \frac{n-1}{4n-1}$ I treated it like a limit problem as took the sum to be $\frac{1}{4}$ and since that is $<1$ for this geometric series, I ...
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1answer
146 views

$\int\dfrac{\cos^5x\sin^3x}{1+\cos2x}dx = \dfrac{\sin^4x}{8}-\dfrac{\sin^6x}{12} +C$ or $\dfrac{\cos^6x}{12}-\dfrac{\cos^4x}{8} +C$?

$\int\dfrac{\cos ^5x\sin ^3x}{1+\cos 2x}dx = \dfrac{\sin ^4x}{8}-\dfrac{\sin ^6x}{12} +C$ or $\dfrac{\cos ^6x}{12}-\dfrac{\sin ^4x}{8} +C$? $\int\dfrac{\cos ^5x\sin ^3x}{1+\cos 2x}dx$ = ...
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1answer
67 views

Why is $\sum_{s=1}^{\infty }\left(-1\right)^{s+1} \left[\vphantom{\Large A}-2^{-s}\ \left(2^s-2\right)\ \zeta\left(s\right)\right]= 1-\log (2)$?

could you give and explanation why $$ \sum_{s=1}^{\infty }\left(-1\right)^{s+1} \left[\vphantom{\Large A}-2^{-s}\ \left(2^s-2\right)\ \zeta\left(s\right)\right] = 1-\log (2) $$ and $$\sum ...
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27 views

Where does the value 1200 come from in this equation for a uniformly continuous function

This example came from the $\epsilon-\delta$ criterion section of my book: $f(x)=x^{3}$ for $x$ in $[0,20].$ Then the function $f$ is uniformly continuous. To see this, observe that for all $u$ and ...
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1answer
89 views

2 ways of looking at $\nabla \cdot \vec r $, different answer?

I would like to calculate $\nabla \cdot \vec r $. I can think of 2 methods to do this, but they give different results strangely. Can someone help me out? $$\nabla \cdot \vec r =\nabla_r \cdot \vec ...
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1answer
41 views

Help with determining trigonometric limit

Use the relation $\lim_{\theta \to 0}\frac{\textrm{sin}\theta}{\theta}=1$ to determine the limit of $f(x)=\frac{\textrm{tan}(2x)}{x}$ I understand the identity ...
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1answer
41 views

Using a graphic calculator [closed]

I've been taking calculus and not doing so well, so I really need help. Here are my questions: Find the point on the line $y = x$ that is closest to the point $(0,2)$. You want to build a box ...
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1answer
309 views

Height formula of cylindrical shell method

With the equations y = sqrt(x) and y=2, rotated about the x axis, why would the height of the shell be just (y^2) and not (2 - Y^2)? Isn't the formula for the height the curve on the top - the curve ...
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1answer
133 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
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77 views

Why does determining the nature of local extrema for $\mathbb R \to\mathbb R$ functions require twice continuous-differentiability?

In the text Elementary Classical Analysis, why does Marsden specify the condition "twice continuously differentiable" here? Isn't mere twice-differentiability sufficient for the purpose indicated? ...
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79 views

Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$

How do I prove: Let $\lim\limits_{n\to \infty} a_n = \infty$ and $\lim\limits_{n\to \infty} b_n = \infty$ Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$ Thank you
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1answer
166 views

When to rationalize numerator and/or denominator?

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
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1answer
67 views

lagrange multiplier slope

I was reading the link given in the thread's last comment. I understood initial part. I understand that in case of the hill, if we take any point on that hill, the gradient of the original function ...
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1answer
21 views

Find $f(\dfrac{\pi}{2})$. Provided two functions

Suppose $f(x)=\int_0^{g(x)}\dfrac{1}{\sqrt{1+t^2}}~dt$ and $g(x)=\int_0^{\cos x}1+\sin t^2~dt$ Find $f(\dfrac{\pi}{2})$ I evaluated everything and ended up with $f(\dfrac{\pi}{2})=\dfrac ...
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28 views

Indefinite Integration (1) $ \int 2^{\log_{e}(x)}dx$ (2) $\int 2^{mx}\cdot 3^{nx}dx$

calculation of some Indefinite Integration (1) $\displaystyle \int 2^{\log_{e}(x)}dx$ (2) $\displaystyle \int 2^{mx}\cdot 3^{nx}dx$ $\bf{My\; Try}::$ for (1) $\log_{e}(x)=t\Rightarrow x=e^t$ and ...