# Questions tagged [calculus]

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

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### how to find minimum form piecewise function

Equation: $$(1+6.64x)^2+4(1+8.89x)^2-8(1+6.64x)-16(1+8.89x)-(\frac{1}{((1+6.64x)+(1+8.89x)-5)}+\frac{1}{((1+6.64x)-3)}-\frac{1}{(1+6.64x)}-\frac{1}{(1+8.89x)})$$ plot and solution form wolframalpha ...
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### Approximations using derivatives

I came across the following definitions in my textbook: The differential of $x$, denoted by $dx$, is defined by $dx = \Delta x$ The differential of $y$, denoted by $dy$, is defined by $dy=f'(x) dx$ ...
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### Please explain this integration relation [closed]

For $a>0,{\ }b>0$, $$\lim_{T\to\infty}\int_{a}^{b}\frac{-e^{-Ty}}{y}\,\mathrm{d}y = \lim_{T\to\infty}\int_{Ta}^{Tb}\frac{-e^{-y}}{y}\,\mathrm{d}y$$
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### Trying to find roots of $\arctan(2(x − 1)) − \ln |x| = 0$ analytically

I have found the roots graphically and also numerically. Apparently there is also a calculus root to finding them analytically. I was thinking to use the derivative but it doesn't seem to work. ...
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### Splitting $~x~$ and $~y~$ and solving for $~y~$

Solve for $~y~$: $$3xy+5y=2x+7$$ I have to do this for as an assignment going into Calculus, the problem is the teacher wants us to research how to do the problems on our own, and I don't know what ...
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### Swapping charge time for a capacitor doesn't behave as expected

This is a very basic question about equations but reading online courses about equations didn't clear up what I'm missing. TLDR; How to get from $T = RC$ to $R = \frac{T}{C}$ Working on a Raspberry ...
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I have been searching non-trigonometric approximations for some trigonometric functions and have found myself in need of showing that, $$x^2-2x+\sin\left(\frac{\pi}{2}x\right) \le 0,$$ in the range $... 1answer 43 views ### Why do all integrals equal to zero when substituting$ u = x(x-a-b)$. [duplicate] My friend showed me this substitution for an integral $$\int _a^b f(x)\, \mathrm dx:$$ Make substitution$ u = x(x-a-b)$. Now this changes the limit to $$\int _{-ab}^{-ab} \text {(something)}\,\... 0answers 31 views ### A generalized differential equation for a convolved differential operator. The solution to perhaps the world's very first differential equation$$f'(x) = f(x)$$is the well known exponential functions$$f(x) = k\exp\left[x\right]$$But what if we consider another ... 0answers 26 views ### What are some financial or economic applications of the improper integral? [closed] So basically I'm really interested in improper integration and also I am very passionate about finance. I just need some pointers, hints or references to where I can apply improper integration in the ... 1answer 65 views ### How to solve differential equation - x^2f''(x)+xf'(x)+f(x)=0\ , f(x_0)=x_0 Solve$$x^2f''(x)+xf'(x)+f(x)=0\ , ~~~~~~f(x_0)=x_0$$I didn't even know what method to be used to solve this problem. How to find solution for this problem. Please help me. 2answers 54 views ### Uniform convergence of f_n(x) = \cos^n(x/ \sqrt{n}) . I'm studying the uniform convergence of the following sequence : \begin{equation*} f_n(x) = \left\{ \begin{split} & \ \cos^n\left(\frac{x}{\sqrt{n}}\right) \ \ \textrm{if} \ x \in \left[0, \... 1answer 94 views ### I need help with this integration: \int_{0}^{\infty}x^{a}e^{x^{b}}e^{-(e^{x^{b}}-1)^c}dx I am trying to fined a closed form for this integration$$\int_{0}^{\infty}x^{a}e^{x^{b}}e^{-(e^{x^{b}}-1)^c}dx,$$where a,b,c>0 I think the generalized integro-exponential (E_{s}^{r}(z)=\... 2answers 58 views ### Question based on \epsilon - \delta definition of limits. Question: Let f :(a,b)\rightarrow \mathbb{R} be such that \lim_{x\to c} f(x)> \alpha, where c\in(a,b) and \alpha\in\mathbb{R}. Prove that there exists some \delta > 0 such that$$f(c+... 0answers 35 views ### Calculus system expansion to complex numbers Does calculus for complex numbers exists? did any one ever tried to make research on this? What would it mean to have complex numbers calculus? for instance: What is the meaning of $$\lim_{x\to z} f(... 0answers 68 views ### Volume of solid revolution of region bounded by y = x^{2} - 4, y = 8 - 2x^{2}, x=0, x=3 I have two curves: y_{1} = x^{2} - 4, y_{2} = 8 - 2x^{2}. I have a region bounded by y_{1}, y_{2}, y-axis, and x=3. What is the volume of solid revolve around the x-axis? Attempt: In the ... 1answer 18 views ### Complex Amplitudes and their Exp'l Increase/Decrease The following are quotes from a textbook, If S(t) is represented as a rotating phasor, the angular frequency of the phasor can be thought of as velocity at the end of the phasor. In particular ... 1answer 68 views ### Computer the flux of \nabla \ln \sqrt{x^2 + y^2 + z^2} across an icosahedron centered at the origin Let S be the surface of an icosahedron centered at origin) and let$$f(x,y,z)=\ln \sqrt{x^2+y^2+z^2} .$$Calculate the flux$$\iint_S (\nabla f \cdot n) d\sigma,$$where n is the outward unit ... 1answer 61 views ### Find \frac{d \rho}{d x} for \rho = \rho(t,x(t),p(t)) I got a question relating to this thread difference between implicit, explicit, and total time dependence Considering the reply in the top by Kostya, I konw what is the difference between \frac{\... 1answer 44 views ### What is the taylor series of this function at x =0? [duplicate] Let the f(x) =e^{-1 \over x^2} for x \neq 0 Plus Define f(0) = 0 By definition of the differentiation, f'(0) =0 But can't figure out the case of the f^n(0) 0answers 25 views ### Is it possible to show when the area of a polygon equals the area under its connected points? First, let me preface w/ what my own understanding consists of. I've only ever taken classes up to Discrete Math & Differential Equations, and it has been a while since these topics have been ... 2answers 27 views ### Solution verification on homework problem. Separable first order ODE IVP. The answer is supposedly y^2 = 1 + \sqrt{x^2 - 16} I don't know where I went wrong cause I know for a fact that my substitution of x = 4 \sec(\theta) is correct. I know for a fact that after ... 1answer 53 views ### Book of calculus [duplicate] Sometimes I want to test the concepts of calculus in practice, but the most popular books like Thomas/Maurice is a little approach to me.Does someone know a book that works with calculus in ... 3answers 51 views ### Limit of exponential function using natural log and change of variable without L'Hopital$$ \lim_{x\to \:0}\left(\frac{a^x-1}{x}\right) $$I know the answer to this is$$\ln \left(a\right)$$but I don't know how to reach that answer without L'Hopital. I just know the first step (given ... 3answers 70 views ### Calculating the area between two curves The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves.$$ y^2 = 9x, y = \frac{3x^2}{8} $$... 2answers 49 views ### What is the limit \lim_{x \to \infty} \frac{x^{\log x}}{c^x} where c > 1? My intuition tells me the answer is 0, but I can't figure out how to prove it. I've tried using L'Hopital's rule k times in a row, but since c^x doesn't change when being derived, this doesn't ... 0answers 42 views ### \sum_1^\infty \cos\left(\frac{\pi}{4k}\right)e^{-k} [closed] what is the closed form of \sum_1^\infty \cos(\frac{\pi}{4k})e^{-k} I don't have any idea how to start with it 2answers 30 views ### Order of factors in partial decomposition Is there a protocol for deciding which denominator fraction goes under A and which goes under B during partial decomposition? Doing this question: integral (5x-5)/(3x^2-8x-3) I factored the ... 2answers 38 views ### Implicit differentiation classic [closed] x^t y^m = (x+y)^{m+x} prove that \frac{dy}{dx} = \frac{y}{x} I tried and finally got this \frac{dy}{dx} = \frac{y-1}{1-x} Update: I found a solution without using logarithms, tx^{t-1}y^... 1answer 54 views ### Asymptote when x\to-\infty I have function f(x)=\sqrt{4x^2+5x} and need asymptote when x\to-\infty. I know that$$\;\sqrt{4x^2+5x}=\sqrt{x^2\left(4+\frac5x\right)}=|x|\sqrt{4+\frac5x}=-x\sqrt{4+\frac5x}$$since I assume ... 2answers 38 views ### Integral of positive Part Is$$\int_a^b \big(f(x)\big)^+\mathrm{d}x = \left( \int_a^b f(x) \mathrm{d}x \right)^+$$provided that f:[a,b]\to\mathbb{R} is integrable? This means, can taking positive part and integration be ... 1answer 58 views ### Find the convergent interval of \sum_{n = 0}^\infty {(3n)!\over (n!)(2n!)}x^n I am trying to find the convergent interval of this power series and I got the absolutely convergent interval to be (-{4\over 27},{4 \over 27}) by applying ratio test. But how can I verify the ... 1answer 41 views ### Differential equation related to energy conservation and Newton's law of gravitation I have been trying to determine, given the position of a point mass an initial distance x_0 from the surface of a spherically symmetric body with mass M and radius R, the position of the point ... 3answers 41 views ### How to check if a function is convex According to a calculus book I have been reading, we call a function g(x) a convex function if$$g(\lambda x +(1-\lambda)y) \leq \lambda g(x) +(1-\lambda)g(y)$$, for all x,y and 0<\lambda&... 4answers 87 views ### Verify the following limit using epsilon-delta definition: \lim_{(x,y)\to(0,0)}\frac{x^2y^2}{x^2+y^2}=0 Show that$$ \lim\limits_{(x,y)\to(0,0)}\dfrac{x^2y^2}{x^2+y^2}=0$$My try: We know that,$$ x^2\leq x^2+y^2 \implies x^2y^2\leq (x^2+y^2)y^2 \implies x^2y^2\leq (x^2+y^2)^2$$Then,$$\dfrac{x^2y^2}{x^... 0answers 68 views ### A generalization involving a logarithmic integral In the preprint, A note presenting the generalization of a special logarithmic integral by Cornel Ioan Valean, it is given the following generalization, Let$n\ge1$be a positive integer. Then, \... 4answers 63 views ### a hint for this Taylor series$ \frac{\cos\left(2x\right)-1}{x^2}$Compute the first three terms (nonzero)$\frac{\cos\left(2x\right)-1}{x^2}$the first term is$\cos \left(2\right)-1$but in the answer, the first term that I have to choose is...$-2$or$2$or$-1/...
I took $3$ random polynomials with non zero roots one having even degree and two having odd degrees $f(x)=\color{red}{4}x^2-(4\sqrt3+12)x+12\sqrt3$ having roots $\color{blue}{3,\sqrt3}$ and leading ...