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

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0
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1answer
16 views

How to prove that an integral converges

Let $(a_n)$, $(M_n)$ be sequences of positive real numbers such that ${a_n} \downarrow 0$, ${M_n} \uparrow \infty$ as $n\to\infty$. Let $\alpha>0$ and $\beta>1$. How to prove the following ...
0
votes
1answer
22 views

How would I use derivatives for suggesting an option to my user?

I was learning derivatives. I understood the theoretical concept behind it. When I was searching for the real-life example in machine learning I came across one of the answers in this question How do ...
2
votes
5answers
125 views

What is the value of the following sum

$$\lim_{n\to\infty}\sum_{k=1}^n \ln\Big(1+\frac{k}{n^2}\Big)$$ According to me, the answer is $0$. I'm curious as to what answers might others come up with, as well as the method of reasoning.
4
votes
1answer
28 views

Find $\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right)$

Find $$\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right).$$ I have tried rewriting the sum in a clever way, applying the Mean Value Theorem or Stolz-Cesaro ...
1
vote
2answers
31 views

Find equation of tangent line to a curve $g(x)$ at $x=4$

So I am trying to find the equation of a tangent line to the curve: $$y = g(x)\text{ at }\,x = 4$$ given $g(4) = -6,\;$ and $\;g'(4) = 2$.
2
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0answers
31 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) \end{equation} ...
2
votes
1answer
31 views

Confusing about the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$

What is the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$? A nice plot of $f(x)$ shows that the domain is $\mathbb{R}$ but we see that $x$ should be non-negative at the first sight. Of course, I ...
1
vote
4answers
113 views

Convergent or Divergent Integral

Convergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do ...
0
votes
2answers
21 views

How to show that the points $(0, 0)$ and $(\sqrt{2 \pi},−\sqrt{2 \pi})$ on the curve $e^{x + y} = \cos(xy)$ have a common tangent?

Show that the points $(0, 0)$ and $(\sqrt{2 \pi},−\sqrt{2 \pi})$ on the curve $e^{x + y} = \cos(xy)$ have a common tangent. How do I solve this question? First, I differenciated the curve and I got ...
0
votes
0answers
10 views

taylor series expansion, derivatives not continuous

As a part of an excercise I am supposed to find the Taylor series expansion for $(1-t)^{\frac{1}{2}}$ on $[0,1]$. According to the remainder theorem: ...
0
votes
3answers
69 views

Convergence or divergence of the integral $\int_0^1 dx/\sin x $

Is this Convergent or Divergent $$\int_0^1 \frac{1}{\sin(x)}\mathrm dx $$ So little background to see if I am solid on this topic otherwise correct me please :) To check for convergence I can look ...
0
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0answers
32 views

Need help with excel spreadsheet! [on hold]

So I am currently doing an assignment in which I have bought a house using a home-loan. For this part of the question I need to calculate how long it will take to repay the loan. So I have constructed ...
0
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4answers
41 views

Is intergration and an integral the same thing?

And if not whats the difference? I think the integral is the area under the curve and integration is an anti derivative (what ever that means)
1
vote
0answers
50 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
1
vote
1answer
21 views

How do i solve this to find PMT?

I know this may seem like a stupid question but i've been up late working on this math assignment and this question just isn't working when i transpose it. So this is the formula to find Present ...
2
votes
1answer
40 views

How to arrive at desired equality?

Why is the following second equality true? $$e^{1+1/2+...+1/(n+1)} - e^{1+1/2+...+1/n} \\= ...
1
vote
2answers
44 views

Evaluating the limits $\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$ and $\lim_{(x,y)\to(\infty,8)}(1+\frac{1}{3x})^\frac{x^2}{x+y}$

I got the following problem: Evaluate the following limits or show that it does not exist: $$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$ and ...
0
votes
1answer
29 views

Relation between limit of functions and sequences

I need to prove that the sequence $$ \frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
0
votes
4answers
43 views

General form for these types of integrals

I encountered this integral in physics-- $$2\int_{0}^{\infty} \dfrac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} dt$$ I know for certain that $a>0$, $b>0$. $a$ and $b$ are independent variables
1
vote
1answer
26 views

The general solution of first order ODE

How will i get the general solution for this $$y' = {-y^2 \over x} + {2 y \over x}$$ I have tried and come to this far by separating and equating $$\int {1\over-y^2+2y} dy = \int{1\over x} dx$$ which ...
1
vote
3answers
48 views

Solve $y' = x^4y+x^4y^4$

Solve the differential equation $$y' = x^4y+x^4y^4.$$ I'm not sure how to deal with the $x^4y^4$ term. So far I have only encountered differential equations where the exponent of $y$ was at most one. ...
2
votes
2answers
29 views

Interpolation between derivatives [duplicate]

Let $f$ be twice continuously differentiable on $[0,2]$, and $|f(x)|\leq 1$, $|f''(x)|\leq 1$. Prove that $|f'(x)|\leq 2$. If I use Lagrange intermediate value theorem, then ...
0
votes
1answer
33 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
0
votes
2answers
18 views

proves of parametric curves via parametric equations

Hi could anyone help me with this problem. An astroid is given by the equation $$x^{2/3} + y^{2/3} = 1.$$ Prove via parametric equations that the length of a piece of a tangent line between the ...
2
votes
1answer
34 views

Find the equation of the curve

I've been given the following math question; however, I don't understand the wording of what it's asking. if $ \dfrac{d^2y}{dx^2} = y''= \dfrac{dy'}{dx} = 3x^2 + x $ for any point on a curve and the ...
3
votes
2answers
46 views

Initial value problem for 2nd order ODE $y''+ 4y = 8x$

How can I go about solving this equation $y''+ 4y = 8x$? Progress I found the general solution for its homogeneous form. What I don't know is how to find its particular solution.
3
votes
1answer
28 views

Rate Of Change Of Shadow

A spotlight on the ground shines on a building $12m$ away. If a man $2m$ tall walks from the spotlight towards the building at a speed of $1.6m/s$, how fast is the length of his shadow on the building ...
4
votes
2answers
73 views

I need compute a rational limit that involves roots

I need compute the result of this limit without l'hopital's rule, I tried different techniques but I did not get the limit, which is 1/32, I would appreciate if somebody help me. Thanks. ...
0
votes
0answers
25 views

$n$th derivative of $f(x)$ using limit definition

After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative: Let $F^n=f(x+nh)$ ...
2
votes
3answers
57 views

Taking limits with a square root in a quotient

I'm trying to understand how to take the limit of $$\lim_{x\to 0}\frac{\sqrt{36+x}-6}{x}$$ Wolfram alpha said I should use the l'hospital rule and take the derivative of the numerator and ...
2
votes
1answer
17 views

Eliminating parameter to get Cartesian equation

$x = \sin(t/2)$ $y = \cos(t/2)$ $-\pi \le t \le \pi$ How would I go about getting the Cartesian equation of these?
1
vote
1answer
27 views

Volume of a ellipsoidal shape

I was given the following question: My approach so far was to create a parabolic function: y = 25/2 - (25^2)/392 Then I integrate from x = 0 to x = 14 Volume = 2 * pi * integral of y ^ 2 The ...
0
votes
2answers
26 views

How to find an 'optimum' minimum difference value to identify the closest similar points on a plot?

Data Set I have asked a similar question and the data set is same, here. The Goal In the given plot, you can see there is a loop (all the points in the blue tiles make 1 big loop) i.e. the plot ...
-3
votes
1answer
55 views

How can i resolve this limit without L'Hopital's Rule? [duplicate]

I have this limit: $$\lim_{x \to 0}\frac{\sin x-x}{x^3}$$ How can i resolve it without l'Hopital's rule?
3
votes
0answers
49 views

Evaluating the integral $\int \frac{dx}{\sqrt{a(1+x)^3+1}}$ [duplicate]

How can I solve this integral? $$\int \frac{dx}{\sqrt{a(1+x)^3+1}}$$ where $a>0$. I have tried by using Mathematica, but it fails. Someone has any sugestion?
0
votes
2answers
87 views

Evaluation of $\displaystyle \int \sec^3 (x)dx$

How Can I evaluate $\displaystyle \int \sec^3 (x)dx$ (Without Using Weierstrass Substution or Integration by parts.) $\bf{My\; Try::}$ Let $\displaystyle I = \int\sec^3(x)dx = \int ...
3
votes
2answers
60 views

Convexity of $e^{-2x}+e^{-bx^2}$, $\frac12<b<\frac32$

Prove that $u_b(x)=1-\frac12(e^{-2x}+e^{-bx^2})$ is concave for $\frac12<b<\frac32$. What about b=$\frac1{20}$? $b=2$? By disregarding the 1 and then the $\frac12$ we can turn it into the ...
4
votes
1answer
85 views

Calculate $\lim_{n\to\infty} n^\alpha \Big(\frac{\sqrt[n+1]{(n+1)!}}{n+1} - \frac{\sqrt[n]{n!}}{n}\Big)$

Let $\alpha$ be a positive number. Find $$\lim_{n\to\infty} n^\alpha \Big(\frac{\sqrt[n+1]{(n+1)!}}{n+1} - \frac{\sqrt[n]{n!}}{n}\Big).$$ I'd love to post a useful solution attempt, but all of my ...
1
vote
1answer
19 views

Need help solving an integral for Lagrange Remainder Proof

This image of part of a proof for the Lagrange Remainder for Taylor's Formula. I need help solving the last integral. Can anyone explain?
2
votes
1answer
44 views

Factoring a complex polynomial

Factorize the polynomial : $$ p(x) = x^{5} - x^{4}+ 4x - 4 $$ In real factors in the lowest degree possible. So in previous questions I have been given at least one rot so that I can factorize it ...
5
votes
1answer
75 views

Show $\sin(x+h) \cdot \cos x - \cos(x+h) \cdot \sin x = \sin h$ (without limits please - straight trigonometry only).

I've tried an algebraic approach using the identity $\sin(x) = \sin(x+h-h) = \sin(x+h)\cos(h) - \cos(x+h)\sin(h)$, leading to a complicated expression I'm having trouble simplifying: ...
1
vote
1answer
28 views

Proving IMVT using delta-epsilon

Let's assume $f(a)<0$ and $f(b)>0$. IMVT claims that there's $c\in(a,b)$ such that $f(c)=0$. The Proof: Consider $$A = \{ a\le x\le b : f(x) < 0 \}$$ That's a non-empty set and therefore, by ...
-5
votes
0answers
35 views

limits using (ε,δ )-definition of limit

Hi could anyone help me with this proof Prove that lim f(x)=0 whenever lim ||f(x)||=0 For x ε R^(n) x->a x->a f(x) is a ...
0
votes
2answers
30 views

Interval of convergence using ratio test on the series ln(1 - x)

I have to find the series expansion and interval of convergence for the function ln(1 - x). For the expansion, I have gone through the process and obtained the series: -x - (x^2/2) - (x^3/3) - . . . ...
2
votes
1answer
55 views

What is wrong with this separation of variables?

I know a number of ways of solving this basic DE: $\ddot{u} = -u$ Besides the fact that the solution is obvious, one can do: $\ddot{u} = \frac{d\dot{u}}{dt} = \frac{d\dot{u}}{du}\frac{du}{dt} = ...
1
vote
3answers
64 views

Complex Equations

The Equation: $$ z^{4} -2 z^{3} + 12z^{2} -14z + 35 = 0 $$ has a root with a real part 1, solve the equation. When it says a real part of 1, does this mean that we could use (z-1) and use ...
0
votes
5answers
78 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
5
votes
1answer
85 views

Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function

How to prove this identity for $n>1/2$? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$
2
votes
1answer
36 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...
6
votes
5answers
228 views

Alternate ways to find the limit of a given sequence

I need to find the $\lim_{n\to\infty}\{x_n\}$ where $\{x_n\}$ is defined as $$\{x_n\}_{n\ge1}=n^{\frac{1}{n}}\;,\;n\in \mathbb{N}$$ Now if I had a function $f:\mathbb{R}-\{0\}\to\mathbb{R},\quad ...