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

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1
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1answer
18 views

Object moving on x axis integration

So, I think I know how to set up this problem, but then I get stuck at the last part: An object moves along the x - axis such that its velocity at time $t$ is $v(t) = cos(2t) $. Suppose the object ...
0
votes
0answers
17 views

Volume of a region R when revolved about the x-axis (multiple problems)

based on this earlier question I had Volume of a region R when revolved about the x-axis region bounded by y=x^2+1,y=2,x=0 is revolved about y=-1 region bounded by y=x/2, y=sqrt(2x) is revolved ...
0
votes
3answers
23 views

Limit and L'Hopitals

I'm having trouble with this problem. $\lim{n \to \infty} (1+\frac 3n)^n$ My professor said to use a proof to figure out that the limit of the ln of the function is 3, but I can't figure out how to ...
4
votes
1answer
27 views

For what $p$ does the surface of revolution for $x^p$ have finite surface area?

I am trying to investigate the surface of revolution of the $x^p$ functions, in the domain $[1,\infty)$ Using the formula for surface of revolution, $$A=2\pi\int_1^\infty x^p ...
10
votes
4answers
123 views

Evaluating $\int_{0}^{\pi/2}\frac{x\sin x\cos x\;dx}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}$

How to evaluate the following integral $$\int_{0}^{\pi/2}\frac{x\sin x\cos x}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}dx$$ For integrating I took $\cos^{2}x$ outside and applied integration by parts. ...
0
votes
1answer
19 views

Differentiability of the local maximum of a function

On a test, I was asked whether the following statement is true or false: "If $x=c$ is a local maximum of $f$, then ${f'(c)}=0$" I thought it would be false- my reasoning was that the local max might ...
2
votes
0answers
79 views

how to use Matlab ifft to calculate the following integral? [duplicate]

$$R(t)=\int_{-\infty}^\infty\dfrac{\omega e^{i\omega t}}{(3-\omega^2)^{2}+4\omega^2}\,d\omega$$ where t is a integer and $t>0$ I used to calculate this integral by numerical integral,but it seems ...
0
votes
1answer
52 views

Who introduced the term indefinite integral and the notation $\int f(x)dx$?

I find the notation $\int f(x)dx$ for the indefinite integral of $f(x)$ on some interval $I$ is both suggestive and confusing. On the one hand, this notation is very suggestive when we calculate ...
1
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0answers
50 views

Is $F$ Riemann integrable, then $F'$ Riemann integrable?

I know Newton-Leibnetz theorem: if $f \in \mathcal{R}[a ,b]$($f$ is Riemann integrable on $[a ,b]$), and if exists differentiable function $F$ satisfy $F'=f$ on $[a, b]$, then ...
2
votes
2answers
58 views

Evaluating inverse of trigonometric function

I have this function, $$\sin\left[{\arctan\left({\frac{x}{\sqrt{1-x^2}}}\right)}\right]$$ I drew a right angled triangle putting $x$ on the opposite side and the square root on the adjacent which ...
3
votes
2answers
33 views

Using the fundamental theorem of calculus when the upper limit of integration is $t^2$

I have the find the derivative of the following function: $$F(t) = \int_1^{t^2} \frac{\sqrt{1+s^2}}{s} ds$$ If the upper limit of the integral was $t$ rather than $t^2$, this would be an easy ...
1
vote
1answer
28 views

Volume of a region R when revolved about the x-axis

Find the volume of the region when revolved about the $x$ axis $y= \sqrt{x-1}$, $y=2$, $y=0$, and $x=0$ Is this right? Also if you could help me with revolving this same region around $y=2$, ...
0
votes
2answers
35 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
3
votes
2answers
41 views

How to find $\int \sec^{3} \ dx$ [duplicate]

I am stuck trying to find $$\int \sec^3{x} \ dx.$$ Here is my attempt using integration by parts: $$\int \sec^3{x} \ dx = \sec{x}\tan{x} - \int \tan^2{x}\sec{x} \ dx.$$ At this point, I am stuck. ...
4
votes
1answer
63 views

Expressing “formally” $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series

I have to express $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series and give its interval of convergence. Knowing the binomial series is as follows this should be fairly easy: $$(1+x)^{\alpha}=\sum ...
1
vote
1answer
41 views

Geodesic equation

Assume that you have a parametrization of a surface $f:\Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3,(u,v) \mapsto f(u,v)$. Now if I have a curve defined by $g(t)=f(0,t)$. The geodesic ...
4
votes
2answers
70 views

How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?

I am looking to demonstrate the following result. Any ideas are much appreciated. $$ \begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx = &\;x \ln \left( b + \sqrt{b^2 +c^2 ...
1
vote
1answer
31 views

graphing $\frac{x^3-x+1}{x^2}$

I want to graph: $$f(x) = \frac{x^3-x+1}{x^2}$$ so I took the first derivative: $$f'(x) = \frac{x^3+x-2}{x^3}$$ but this function is hard to find the signals. In other words, it's hard to find ...
3
votes
2answers
50 views

Prove that $\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$ in the $\epsilon$-$\delta$ way [duplicate]

Given: $$\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$$ How to prove this limit using the $\epsilon$-$\delta$ way? (the biggest problem is to find $\delta$)
2
votes
1answer
39 views

using power expansion to find limit

I am preparing for my final exam, and stuck on this question. Using power series expansion, evaluate $$\lim_{x\to 0} \frac{x\cos(x) -\sin(x)}{x^2-x\ln(1+x)}$$ I have no idea how to proceed. ...
2
votes
1answer
30 views

Partial Derivatives versus Proper Derivatives

I'm having some difficulty understanding exactly what a partial derivative is. I had been content with the definition $$\frac{\partial F}{\partial x_i } = \lim_{\Delta x \rightarrow 0} \frac{F(x_0, ...
2
votes
1answer
18 views

recursive sub-sequences of sequence , one is increasing and one is decreasing to same limit -> the sequence converge?

Let $b_1=\:0$, $b_{n+1}\:=\:\frac{1}{1+b_n}$. I need to show that $\left(b_n\right)_{n\:=1}^{\infty }$ converge. I thought about dived $b_n$ to 2 sub_sequence : $b_{2n}$, $b_{2n+1}$. (i thought ...
0
votes
3answers
41 views

Differentiation and piecewise function

I have the following problem: $$f(x) = \cases {x\ln x , x \gt 0 \cr 0, x \le 0}$$ and I need to show the following: it is continuous at $x=0$ is $f(x)$ differentiable at $x = 0$? ...
2
votes
0answers
38 views

Calculate the distance between intersection points of tangents to a parabola

Question Tangent lines $T_1$ and $T_2$ are drawn at two points $P_1$ and $P_2$ on the parabola $y=x^2$ and they intersect at a point $P$. Another tangent line $T$ is drawn at a point between $P_1$ ...
1
vote
1answer
52 views

Is $ (x^3+1)^{30} $ the derivative of $(\int_0^{x} \ (t^3+1)^{10} \mathrm{d}t)^{3} $?

Is $ (x^3+1)^{30} $ the derivative of $(\int_0^{x} \ (t^3+1)^{10} \mathrm{d}t)^{3} $ I tried to enter this in WolframAlpha: But I have a solution here that says this is wrong. So this makes me ...
2
votes
1answer
24 views

Why can the integrate from 2 to x^2 sin(t^3) be written as y/x?

Why can the integrate from 2 to x^2 sin(t^3) be written as y/x? I expected the answer to be some cosines....
1
vote
0answers
22 views

How to find the $\delta$ when one tries to prove a limit of a trigonometric function?

For any given trigonometric function how do you prove the limit in the $\epsilon - \delta$ form? I've searched the web seems like no one is doing it in the $\epsilon - \delta$ form (I actually did ...
0
votes
1answer
14 views

2 Sub-limits of sequence converge epsilon proof

consider $\left(a_n\right)_{n=0}^{\infty }$, $L_1,L_2\:\in \mathbb{R}$. $\lim _{k\to \infty }\left(a_{2k}\right)\:=\:L_1$, $\lim _{k\to \infty }\left(a_{2k-1}\right)\:=\:L_2$ . How to prove using ...
9
votes
3answers
122 views

How to find $\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$

Evaluate $$\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$$ where $a$ is a real parameter $a\geq1$. I can easily find the definite integral for $a=1$. It is $\sin(1)$. In wolframalpha.com when I put ...
2
votes
2answers
33 views

Solids of revolution, how come we use the inverse function when we use method of cylindrical shells?

Doing my second course in college calculus, and we are doing integrals and volumes by slicing/solids of revolution. The question I had trouble with was: "Find the volume of a solid $S$, using the ...
2
votes
1answer
20 views

lagrange interpolation, polynomial of degree $2n-1$

Let $a_1, \dots, a_n$ and $b_1, \dots, b_n$ be real numbers. How would I go about showing the following? If $x_1, \dots, x_n$ are distinct numbers, there is a polynomial function $f$ of degree $2n - ...
1
vote
2answers
27 views

Polynomials of best approximation

The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm $$||f|| = \max_{x \in [a, b]}|f(x)|$$ ...
0
votes
0answers
31 views

Proving a limit of a trigonometric function

I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a trigonometric function I know only how to do it with polynomial functions
0
votes
1answer
16 views

Find $\frac{dz}{dt}$ if $z=z(x,y)$, $x=x(t)$ and $y=y(t)$

The total derivatives are: $$dz=(\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy$$ $$dx=\frac{dx}{dt}dt$$ $$dy=\frac{dy}{dt}dt$$ Substituting into the first equation gives: ...
2
votes
0answers
24 views

Real Analysis Problem Help! [duplicate]

Suppose that $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and that $f(a)= 0$ but $f'(x) > 0$ for all $x \in (a,b)$. Show that $x = a$ is the only zero of $f(x)$ on $[a,b]$. Im not ...
1
vote
2answers
66 views

Real Analysis Question? [on hold]

Suppose that $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and that $f(a)= 0$ but $f'(x) > 0$ for all $x \in (a,b)$. Show that $x = a$ is the only zero of $f(x)$ on $[a,b]$. Hint: Assume ...
8
votes
1answer
188 views

If $dx/dy =\sin(x)$ then is $dy/dx = 1/\sin(x)$?

If $\dfrac{dx}{dy} = \sin(x),$ then is $\dfrac{dy}{dx} = \dfrac{1}{\sin(x)}$? I'm trying to understand how to manipulate $dx$ and $dy$ quantities effectively.
0
votes
1answer
19 views

Can every power series be representated as a taylor series?

Can every power series be represented as a Taylor series? More concrete: Given an arbitrary power series $\sum_{n=0}^\infty a_n (x-x_0)^n$, is there always a $C^\infty$-function $f$ such that ...
1
vote
2answers
34 views

Find the area of the enclosed region

The region is enclosed by the curves $y=\sqrt{x+2}$, $y=\frac1{x+1}$, and lies between $x=0$ and $x=2$. Help please, any work would be helpful. Thank you!
0
votes
2answers
32 views

Solutions of differential euqations in term of definite integrals

In my textbook, the authors tried to solve the differential equation: $dy/dt+ay=g(t)$, where $a$ is a constant and $g(t)$ is a function. Why the authors of my textbook tend to leave the answer in ...
0
votes
0answers
20 views

Trig sub and Integration of Squareroot divided by polynomial squared

Question #2 What am I doing wrong? Do not give me the answer but rather a hint.
0
votes
1answer
14 views

Is there a difference in definition between an increasing sequence and a monotonically increasing sequence?

Is there a difference in definition between an increasing sequence and a monotonically increasing sequence? In some places I see they call it just inc/decreasing and some call it monotonically ...
0
votes
1answer
34 views

Evaluate integral by interpreting it in terms of areas

I tried (a) and I got 5, but I am suppose to get a 4. I really need a good explanation to understand how to approach these problems. I tried searching in youtube and stuff, but it was not helpful. ...
5
votes
1answer
61 views

An advanced integral $\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$

I'd like to ask you how you would like to approach the integral below $$\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$$ and then recommend me some tools you'd employ. It's ...
1
vote
0answers
31 views

How can I differentiate this equation? $\sin(x)^{\cos{y}}+\sin{y}^{\cos{x}}=3$

$$\sin(x)^{\cos{y}}+\sin{y}^{\cos{x}}=3$$ I solved a similar equation where those 2 functions were equal to each other by taking the natural log for both sides but now I don't know what to do, taking ...
0
votes
2answers
69 views

How to prove that $\lim_{x \to \infty} x = \infty$

Please refrain from using logic symbols, as I do not understand those. So, this is the question: $$\lim_{x \to \infty} x = \infty$$ Proving this using the actual formal definition of a limit. So ...
0
votes
0answers
31 views

Rewrite the integral $\int_{0}^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)\,dz\,dy\,dx$ in the orders $dx\,dy\,dz$ and $dy\,dz\,dx$.

Rewrite the integral $$\int_{0}^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)\,dz\,dy\,dx$$ in the orders $dx\,dy\,dz$ and $dy\,dz\,dx$. My try: We have $z=0,z=1-y,y=\sqrt{x},y=1,0\leq x\leq 1$ ...
1
vote
1answer
18 views

Derivative of vector valued function

The following is given for $ ∂x^TAx/∂x $ in a book on Matrix Algebra: What I cannot understand is: Where does $A^T$ come from in the second row (in the term $ ty^TA^Tx $)?
0
votes
0answers
42 views

Even and odd integrals

Find the definite integral $$\int_{-2}^{2} \Big(2f(x) + 3g(x)\Big)dx$$ where $f(x)$ is an even function such that $$\int_{0}^{2} f(x)dx = 3$$ and $g(x)$ is such that $$\int_{-2}^{4} g(x)dx = -3 ...
0
votes
2answers
49 views

Prove that $a_n = 2^n$

Let the recurrence relation $$ a_0 = 1 \\ a_{n+1} = \frac{2 \sum_{k=0}^n a_ka_{n-k}}{n+1} $$ I need to find a close formula for this recurrence. I've noticed that $a_n = 2^n$. I tried to prove it ...