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

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1answer
77 views

Derivatives of Series and the Fundamental Theorem of Calculus (Part 1)

The first part of the Fundamental Theorem of Calculus (FTC) states that: $$\frac{d}{dx}\int f(x)\,dx=f(x)$$ meaning that the indefinite integral of a function can be reversed by its equivalent ...
4
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1answer
54 views

Area in Polar Coordinates

I have tried to solve this problem by subtracting the area of the whole by the smaller area. I got up to $$\int_{1/2}^{11\pi/6} 12 (\cos (\theta)-6)^2 \mathrm{d}\theta- \frac12 ...
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1answer
66 views

how to determine delta epsilon figures for an equation involving the constant e

For the problem $\lim_{x\to 0} {e^x-1 \over x}=1$, I need to solve for x in order to solve using the delta epsilon definition. How do I go about solving for x in the equation $f(x)={e^x-1 \over x}$? ...
28
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20answers
3k views

Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?

As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself. thanks.
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1answer
36 views

taylor series with bounded derivatives.

given f(x) derivable infinite times, and exists L such that $|f^{(n)}(x)|\le L$ for $x\in\Bbb R$ and $n \in \Bbb N$, prove that if $f({1 \over n})=0 ~(n \in \Bbb N)$ than f(x)=0 $(x \in \Bbb R)$ ...
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3answers
147 views

Evaluate $\frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\cdots$

Evaluate $$ \frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\frac{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}{4}\cdots . $$ First, it is clear that terms tend to $1$. It seems that the infinity ...
4
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1answer
138 views

Integral $\int_0^\infty \ln x\,\exp\left(-\frac{1+x^4}{2\alpha x^2}\right) \frac{x^4+3\alpha x^2- 1}{x^6}dx$

$$I:=\int_0^\infty \ln x\,\exp\left(-\frac{1+x^4}{2\alpha x^2}\right) \frac{x^4+3\alpha x^2- 1}{x^6}dx=\frac{(1+\alpha)\sqrt{2\alpha^3 \pi}}{2\sqrt[\alpha]e},\qquad \alpha>0.$$ This one looks very ...
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4answers
698 views

Find a formula for $\sin(5x)$ in terms of $\sin(x)$ and $\cos(x)$.

I was asked to find a formula for $\sin(5x)$ in terms of $\sin(x)$ and $\cos(x)$. I thought about using euler formula which gives: $$\sin(5x) = e^{i\sin(5x)} = \cos(\sin(5x))+i\cdot sin(\sin(5x))$$ ...
2
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2answers
11k views

Using Calculus to find total and maximum revenue and profit

I'm grappling with understanding how to use calculus to find rates of profit, revenue, and cost. I have the following problem: $x = \text{ quantity }$, $12 < x < 48$ Total Cost: $C(x) = ...
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2answers
51 views

Show that $\lim_{x \to \infty} \frac{a^x}{x}=\infty$ as $x\rightarrow$ $\infty$ for $a>1$

Using Cauchy and Heine (Taylor expansion, L'Hospital rule, etc... is not allowed), prove that $\lim_{x \to \infty} \frac{a^x}{x}=\infty$ as $x$ approaches $\infty$ for $a>1$. Iv'e seen this - ...
11
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1answer
480 views

A variation of the isoperimetric problem in the plane

The isoperimetric problem in the plane: « The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed ...
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1answer
84 views

Proving that the function set $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set

I have the the following problem from my Fourier analysis book: Show that $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set in $PC(0,l)$, i.e. class of piecewise ...
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1answer
58 views

express $\frac{42}{ s^2 + 7s}$ as a partial fraction

express this rational function as a partial fraction $$ \frac{42}{s^{2} + 7s}$$ So i factories $$ \frac{42}{s(s+7)} = \frac{A}{s} + \frac{B}{s +7} $$ $$42 = A(s+7) + Bs$$ let s equal $-7$ ...
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1answer
43 views

Show that $\lim_{x \rightarrow +\infty}{(f(x+2)+f(x)-2f(x+1))}=0$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ twice differentiable and $\lim_{x \rightarrow +\infty}{f''(x)}=0$. Show that $$\lim_{x \rightarrow +\infty}{(f(x+2)+f(x)-2f(x+1))}=0$$ I have done the ...
2
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0answers
132 views

Derivative of Dirac delta behavior at 0

I calculated the curl of the vector field $\vec j = \delta(z) f(\rho) \vec e_\phi$ in cylindrical coordinates (ρ,φ,z). For the $\vec e_ρ$ unti vector I got: $$-\delta'(z)f(\rho)$$ The main question ...
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2answers
249 views

Need a theoretical textbook for calculus, proof based

The course descriptions is : A theoretical course in calculus; emphasizing proofs and techniques. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value ...
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2answers
62 views

given f(x) derivable twice in $[a,b]$ and $f'(a)=f'(b)=0$, prove that exists $c \in [a,b]$ such that $|f''(c)| \ge {4 \over {(b-a)^2}} |f(b)-f(a)|$

my partial solution: using taylor series: around a: $$f(x) = f(a) + f'(a)(x-a) + {f''(c) \over 2}(x-a)^2$$ when c is between a and x assign x=b: $$f(b) = f(a) + f'(a)(b-a) + {f''(c_1) \over ...
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1answer
63 views

Find the derivative of the integral

How can i find the derivative of the integral: $$ f(x)=\left(\int_0^x g(s,x)\ ds\right)$$ I tried use FTC but im confused with the function $g(s,x)$,please help.
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2answers
55 views

Given $H$ and $K$ are infinite. Show that $\frac{H-K}{H^2 + K^2}$ is infinitesimal.

I'm going through Elementary Calculus: An Infinitesimal Approach by Keisler. Problem set $1.5$ question $24$. I've tried every rearrangement I can think of but I always end up with an indeterminate ...
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2answers
76 views

Evaluating $\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$

$\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$ I suppose I don't fully understand the notation used within this problem. Using the second fundamental theorem of calculus: $\int_a^b f(x)dx = F(x)\bigr|_a^b = ...
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2answers
188 views

upper bound for $e^{ax^2}$

I want to find a upper bound for $$e^{ax^2}\leqslant \: ?$$ "a" is a constant and $a\geqslant 0$ . x is a variable. I prefer to have a polynomial function or power function (like $ x^{k}$) is there ...
1
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2answers
63 views

strict midpoint convex $\Rightarrow$ strict convex (help with a proof)

Hi everyone I have trouble with the following I think is something very simple, but I cannot figure out yet the correct approach for the strict inequality If $f$ is continuous and $f$ is strict ...
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0answers
80 views

Triple integral of a regular otctahedron

I have to express the volume of a regular octahedron as a single triple integral in rectangular coordinates, with vertices $(\pm 1, 0, 0),(0,\pm 1, 0 ), (0,0, \pm 1)$ centred at the origin. From ...
6
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1answer
145 views

Derivative of $\sec^{-1} e^{2x}$, my answer differs from wolfram alpha.

The problem: Find the derivative of $\sec^{-1} e^{2x}$ (\arcsec doesn't seem to work) My work: $u= e^{2x}$ $\mathrm{d}u = 2e^{2x}\,\mathrm{d}x$ The formula I know for the derivative of arcsec(u) ...
0
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1answer
35 views

Definition of Derivative And Exponential Functions

Given $f(x) = 5^{3x}$. Find $f'(x)$ using definition of a derivative. The definition of the derivative of $f(x)$ is $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ The derivative of $f(x) = ...
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2answers
82 views

reverse the order of integration

How do you reverse the order of this integral into $dy\,dx$? I feel like you need two separate ones but I don't know how to do it: $$\int_0^3\int_\sqrt{y}^3 f(x,y) \, dx \, dy$$ thanks
0
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2answers
44 views

Graphing a Parametric Equation

I need to graph and show the work for this problem. The graph needs to include arrows on the curve to show the direction of motion and I need to label the t-values graphed. $$c(t)=(2+4t, 3+2t)$$ So ...
2
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2answers
121 views

How to complete this epsilon delta proof

Prove $\lim_{x\to 1} {2+4x \over 3} = 2$ using the epsilon delta definition of a limit. if $0 < \left|x-1\right| < \delta$ then $\left|{2+4x \over 3}-2\right| < \epsilon$ scratch work for ...
2
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1answer
287 views

What does 'monotonically related' mean?

I am reading this paper. On page 2, it says that "the likelyhood is monotonically related to the average per-symbol log likelyhood." I know what a monotonic function is. But what does 'monotonically ...
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2answers
89 views

Are discontinuous functions strictly non-differentiable

I encountered a function, where for all values of $x<0$, $(x^2)+(y^2)=2$, while for $x\ge 0$, $(x^2)+(y^2)=4$. My question is although the function is discontinuous at $x=0$, the slope ...
0
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3answers
500 views

Finding the speed of a particle (parametric math)

I have to find the speed (as a function of $t$) of a particle whose position at time $t$ seconds is represtented by $$c(t)=(\sin t+t, \cos t+t)$$ How would I go about finding the maximum speed? ...
64
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14answers
10k views
0
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3answers
39 views

what are some of the methods to do divergent/convergent?

I have no problem with integration finding volumes, surface area and such. But the whole series / sequence chapters completely throw me off, I have no idea where to even begin with problems given. ...
8
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2answers
1k views

Integrate product of Dirac delta and discontinuous function?

Consider the piecewise constant function $\psi:I=[-1,1] \rightarrow \mathbb{R}$ given by $$\psi(x) = \begin{cases} \psi_1 & x \leq 0, \ \psi_2 & x > 0 \end{cases}$$ for some constants ...
2
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1answer
90 views

Definite integral involving arcTan and $e^x$

Evaluate $$\int_{-\pi/6}^{\pi/6}\frac{\tan^{-1}x}{\mathrm{e}^x+1} \,\mathrm{d} x $$ My try: I used integration by parts using $\tan^{-1}x$ as the first function and $\displaystyle ...
2
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3answers
293 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
0
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2answers
755 views

Second derivative change of variables

If I want to express $f''(x)+Af(x)=0$ in terms of a new variable $t$ where $t= cos(x)$ (So I want a new ODE with the same function $f(t)$ in terms of t now). How can I do this?
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0answers
33 views

Lower and Upper sums

Which functions have the property that every lower sum equals every upper sum? I think it is only the step functions, but I have no idea how to prove that it is the only type of functions that has ...
15
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2answers
737 views

A tricky integral

I'm trying to find the exact value of $$\int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{\arctan{(x^2)} }{1+x^2} \, dx$$ Ostensibly, I'd want to use this: $$\frac{d}{dx}\arctan{(x)}=\frac{1}{1+x^2}$$ But ...
0
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1answer
38 views

Find the length of the curve

I need to find the length of the curve $$c(t)=(3e^{t}-3, 4e^{t}+7)$$ for $$0\le t \le 1$$ If I understand correctly, I need to take the derivative of the y part of that coordinate over the ...
5
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2answers
68 views

Find the parametric equation to the curve

Find the parametric equation for the curve. $$x^{2}+y^{2}=10$$ I haven't learned parametric equations fully yet, so I wanted to check with you guys and see if you can confirm if I'm doing this ...
7
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2answers
731 views

Intuition behind arc length formula

I understand the arc length formula is derived from adding the distances between a series of points on the curve, and using the mean value theorem to get: $ L = \int_a^b \sqrt{ 1 + (f'(x))^2 } dx ...
3
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1answer
256 views

Integrate $\int_{-\infty}^{\infty}\exp\left(-\frac{\pi^2t(2x+1)^2}{2c^2}\right)\cos\left(\frac{(2x+1)\pi y}{c}\right)\exp(-2\pi i kx)dx$

By the poisson summation formula we have: $$\frac{1}{c}\sum\limits_{k=-\infty}^{\infty} \exp\left(-\frac{\pi^2t(2k+1)^2}{2c^2}\right)\cos\left(\frac{(2k+1)\pi ...
2
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2answers
91 views

Dirac's delta integration

What about the following integral? $$\int_0^a x^3 \delta(x-1) dx$$ If $a$ is more or less than 1 it's all clear, but what if $a=1$. Is the integral is equal to $1/2$ ? Edit: this is my motivation, ...
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3answers
64 views

Math question from calculus 2 course?

Prove the following inequality: $$2\sqrt{n+1} - 2 < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}} < 2\sqrt{n+1} - 1$$ This question was given in a calculus 2 course, ...
0
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1answer
44 views

Express the parametric equation in form of y=f(x)

I need to express the parametric equation in the form of $y=f(x)$ by eliminating the parameter. I haven't learned how to do this yet, I've attempted to read a few pages though but they didn't help me ...
0
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1answer
32 views

evaluating a flux integral

Question: "Region V, of unit volume, is bounded by the closed surface S. Given the vector field $\mathbf{F}=\langle 7x,2y,5z\rangle$, evaluate: $$\int_S \mathbf{F}\cdot\mathbf{dS}$$ I guessed that ...
0
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3answers
39 views

Dot Product and vector length

Hi! I am working on some online homework for my calc2 class that covers the dot product and I am really struggling with this one question. I understood how to solve part a, because we covered that ...
1
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1answer
53 views

Limits of triple integration (prism)

Integrate $f(x,y,z) = x^2 + y^2$ over the prism shown My problem isn't the integration process but just to determine what the limits are. $$\int_0^4 \int_{0}^{1-x} \int x^2 + y^2 dzdydx $$ I think ...
0
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1answer
24 views

Double integral in Cartesian coordinates

Using double integrals The volume of solid limited by the cylinder $x^2\ + y^2 = 9$ , and plans $z = 0$ and $z = 3 - x$.