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

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2
votes
3answers
86 views

asymptotic behavior of function

I'm preparing for my calculus exam. And I need prove this theorem. Theorem: Suppose that $f(x)$ is monotonic function at $x\in(a;\infty)$, integral $\int_{a}^{\infty}f(x)dx$ is converge. ...
3
votes
2answers
94 views

Proving $\lim_{x \rightarrow{1}^{-}} \int_{-x}^{x}\frac { f(t)}{\sqrt { 1-{ t }^{ 2 } } }dt$ exists

Let $\textit{f} :[-1,1] \rightarrow \mathbb{R}$ continuous on $[-1,1]$ I need to prove that $$\lim_{x \rightarrow{1}^{-}} \int_{-x}^{x}\frac { f(t)}{\sqrt { 1-{ t }^{ 2 } }}dt$$ exists But I have ...
2
votes
1answer
925 views

Integrating $\sqrt{\cos(x)}$

I need to calculate the area between $y=0$ and $y=\sqrt{\cos(x)}$ between $x=\frac{\pi}{4}$ $x=\frac{\pi}{2}$. I tried integrating $\displaystyle \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sqrt{\cos(x)}dx$ ...
4
votes
2answers
242 views

Limit of $f(x) = \lfloor x \rfloor + \sqrt{x - \lfloor x \rfloor}$

I'm doing Spivak's Calculus book and one of the exercises from 5th chapter says 4. For each of the functions [...], decide for which numbers a the limit $lim_{x \to a}\ f(x)$ exists. i) $f(x) ...
5
votes
2answers
78 views

Suppose that $f$ is continuous on $\mathbb{R}$ and $\int_{x-1}^{x}f(t)\,dt=x^{2}$. Find $f(x)$.

Suppose that $f$ is continuous on $\mathbb{R}$ and $\int_{x-1}^{x}f(t)\,dt=x^{2}$. Find $f(x)$. Using that Fundamental Theorem of Calculus, I get $f(x)-f(x-1)=2x$.
3
votes
2answers
132 views

Confusion over a limit. Different ways of solving give different answers?

Qn: If it is given that $$ \lim_{x\to\infty} \frac{x^2 - x - 2}{x + 1} - ax - b = 1 $$ then a and b must be? Now, I tried doing this by 2 methods. Method 1: $$ \frac{x^2 - x - 2}{x + 1} - ax - b $$ ...
1
vote
2answers
121 views

product rule for matrix functions?

Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when ...
3
votes
5answers
515 views

Is the limit not infinity?

Is the limit of this not infinity? No matter what the value of p is? Or is there a way to simplify that fraction? $$\lim_{k \to \infty} 2^{p}\left(\frac{k}{k+1}\right)^k$$
1
vote
1answer
107 views

Series of Functions - Pointwise and Uniform Convergence.

I was hoping for some help for the following questions. Prove that the series $\sum_{n=1}^\infty x^n(1-x)$ converges pointwise but not uniformly on $[0,1]$. Prove that the series $\sum_{n=1}^\infty ...
0
votes
1answer
233 views

Optimization. I need help finding the maximum profit

Josh wants to start a cell phone repair business. Josh determines that $x$ phones can be repaired daily at $p$ dollars per repair, where $x=175-p$. The cost of repairing $x$ phones per day is ...
1
vote
1answer
66 views

What is the exponential series representation of $x^x$?

I want to express $x^x$ in the form of an infinite series involving $qe^{sx}$ where $q$ is the $s$th coefficient of the series and $s$ is the power on $e^x$. Beyond just an answer I would like to know ...
1
vote
3answers
1k views

Compute cos(5°) to 5 decimal places with Maclaurin's Series

I'm working on a problem: Compute cos(5°) to 5 decimal places with Maclaurin's Series I know that that function cos(x) has a Mclaurin representation of: $ \sum_{n=0}^\infty \frac{(-1)^n ...
1
vote
1answer
181 views

Center of Mass double integral

A lamina occupies the region which is the intersection of $x^2+y^2-2y \leq 0$ and the first quadrant of the $xy$-plane. Find the center of mass if the density at a point of the lamina is twice the ...
2
votes
0answers
96 views

finding infimum of maximum of two functions

Let $r\geq 1$. Let $C>0$ be a constant. For $x\in R, x>0$ Find the following infimum: $$ \inf_{x}\max\left\{\frac{(1+x)^r}{x}; C\frac{1+x}{x}\right\}. $$
1
vote
1answer
158 views

Prove that $\int_0^x \int_0^y \int_0^z f(t) dt dz dy = \frac{1}{2} \int_0^x (x-t)^2 f(t) dt$

Prove that $$\int_0^x \int_0^y \int_0^z f(t) dt dz dy = \frac{1}{2} \int_0^x (x-t)^2 f(t) dt$$ Came across this problem and I'm not even sure how to start it. I figured that if the end goal is ...
0
votes
4answers
111 views

How to integrate $1/x$?

I am having trouble figuring out what to do in order to integrate $1/x$ over the interval from $1$ to $x$. I know the anti derivative of $1/x$ is $\ln x$, but how would I integrate this problem?
2
votes
1answer
188 views

Does $\int_0^1 \sum_{n=0}^{\infty}x e^{-nx}\;dx = \sum_{n=0}^{\infty}\int_0^1 x e^{-nx}\;dx$?

Does $$\int_0^1 \sum_{n=0}^\infty x e^{-nx}\;dx = \sum_{n=0}^\infty \int_0^1 x e^{-nx}dx$$ ? This exercise leaves me stumped. On the one hand, it seems the series $\sum_{n=0}^\infty xe^{-nx}$ is not ...
4
votes
2answers
69 views

Solving the limit of integrals $\lim\limits_{q \to 0}\int_0^1{1\over{qx^3+1}} \, \operatorname{d}\!x$

how do I solve this one? $$\lim_{q \to 0}\int_0^1{1\over{qx^3+1}} \, \operatorname{d}\!x$$ I tried substituting $t=qx^3+1$ which didn't work, and re-writing it as $1-{qx^3\over{qx^3+1}}$ and then ...
1
vote
3answers
64 views

Indeterminate powers and limits

The question is $\lim\limits_{x\rightarrow 0^+} x^{8 \sin(x)}$. It says, use L'Hospital's rule if necessary. Are there other methods to solve this? L'Hospital's rule would be complicated to evaluate, ...
0
votes
5answers
64 views

Derivative of $\sin(e^{-x})$

I'm am currently reviewing for a test and I'm not sure what the derivative of $\sin(e^{-x})$ is. The answer I got is $-e^{-x\cos(e^{-x})}$. Is this correct? If not what is?
1
vote
4answers
48 views

A small question: Defining integrals with two variables is 'allowed'?

For instance, is defining $$\int_{-a}^af(x)dx=\int_{-a}^0f(t)dt+\int_0^af(x)dx$$ an ok thing to do? Thanks.
1
vote
0answers
47 views

Norm of the derivatives of a polynomial

Let $M$ be an integer bigger than $2$, and $$f(x)=\sum\limits_{j=2}^{M}f_j+\frac{(x-\beta_2)(x-\beta_3)\cdots\cdots(x-\beta_ ...
1
vote
2answers
97 views

Find a real value $y$ for that the identity holds

Question: Find a real value $y$ such that the following identity holds $$\log(\log(\log(x)))=\int^x_y\frac{dx}{x\log(x)\log(\log(x))}\quad (x>y)$$ Then, find an algorithm to calculate ...
0
votes
2answers
186 views

Simplify the confluent hypergeometric function with parameters $(n,1,z)$

Could you please provide a simple closed form expression for the confluent hyper-geometric function including the following (simple) parameters? ${}_1F_1(n,1,z)$ where n is positive integer and z is ...
4
votes
1answer
164 views

Problem with a dipstick

The problem: Some houses are heated by burning oil. The oil is stored in a horizontal elliptical cylinder that is lying underground. To measure the residual volume, one has to use a dipstick ...
2
votes
1answer
85 views

Evaluating $\lim_{n\to \infty}\int^n_{-n}e^{-x^2}dx$ [duplicate]

How do I evaluate the following limit/integral: $$\lim_{n\to \infty}\int^n_{-n}e^{-x^2}dx$$
1
vote
3answers
61 views

Evaluating $\int^{\pi/2}_0\sin(x)\ \cos(x)\ \mathrm dx$

How do I calculate the following integral: $$\int^{{\pi/2}}_0\sin(x)\ \cos(x)\ \mathrm dx$$
1
vote
2answers
170 views

Evaluating Line Integrals!

$3xy^2dx+2x^3dy$ where is the boundary of the region between the circles $x^2+y^2=25$ and $x^2+y^2=64$ having positive orientation. Not quite sure how to evaluate this...
3
votes
2answers
49 views

Showing a function is not one-to-one near the origin

Let $$f(x)=\begin{cases} x+2x^2\sin\left(\frac{1}{x}\right) \text{ if } x \neq 0 \\ 0 \text{ if } x=0 \end{cases}$$ I'm trying to show this is not one-to-one near $0$. I was given a hint to consider ...
0
votes
2answers
33 views

Particular integrals question

Let $f(x)$ be a continuous function, and $a>0$. Prove $$\int_{-a}^af(x^2)dx=2\int_0^af(x^2)dx $$ This seems trivial.. I'm simply not sure how to solve it. Thanks in advance!
1
vote
3answers
62 views

Find $(a, b)$ such that $\lim_{x \to 0} \frac{ax -1 + e^{bx}}{x^2} = 1$

Find $(a, b)$ such that $\lim_{x \to 0} \frac{ax -1 + e^{bx}}{x^2} = 1$ I am able to find $b = \pm \sqrt{2}$ using L'Hopitals Rule, but unable to do anything for $a$.
1
vote
2answers
936 views

Linear approximation to ln(x) at x = 1, then estimate ln(1.08)

I know that the derivative of $\ln(x)$, or log of whatever base (x) = $(1/x)$ *the original function. If x is a more complicated expression, then the derivative would be $(x'/x)*f(x)$. If I knew the ...
3
votes
3answers
256 views

Understanding directional derivative and the gradient

I'm having trouble understanding the proof of directional derivative and the gradient. Could someone give me a easy-to-read proof of the directional derivative and explain why does the gradient point ...
1
vote
3answers
168 views

Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.

Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition. I'm not familiar with proving divergent sequence. Do anyone have any des? ...
0
votes
1answer
501 views

Integral of the modified Bessel function of the first kind zero order involving power and exponential functions

Is there a closed form expression for an integral of the modified Bessel function of the first kind zero order including the following? $\int_0^\infty x^a e^{-bx^2} I_0(cx)\ x \,dx$ where a is ...
15
votes
3answers
516 views

Prove $f(x)=ax+b$

Let $f(x)$ be a continuous function in $\mathbb R$ that for all $x\in(-\infty,+\infty)$, satisfies $$ \lim_{h\rightarrow+\infty}{[f(x+h)-2f(x)+f(x-h)]}=0. $$ Prove that $f(x)=ax+b$ for some ...
1
vote
1answer
54 views

How prove this equation(18)?

my book have this:let $f(x)=\sqrt{nx},n\in N,0<x<1$, then use Taylor we have $$f(x)=1+f'(x_{0})(x-x_{0})-\dfrac{g(x)}{2}$$ where $x_{0}=\dfrac{1}{n},g(x)=(\sqrt{nx}-1)^2$ my question: ...
1
vote
1answer
68 views

infinite sum limit how to find the following

Hi what is the limit of the following sum: $$\lim \limits_{n\rightarrow\infty}\frac{2}{n^2}\sum\limits_{j=0}^{n-1}\sum\limits_{k=j+1}^{n-1}\frac{k}{n}$$ Thanks a lot!
0
votes
4answers
88 views

Summation of n-squared, cubed, etc. [duplicate]

How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
0
votes
1answer
45 views

I need help Understanding Maximum profit

Josh has a cell phone repair business. The business has a $175$ monthly limit on the number of phones they can repair due to the lack of employees. HE currently charges $\$ 75$ Per Repair, But He ...
1
vote
4answers
97 views

L'Hospital's Rule and indeterminate powers

What is $\displaystyle \lim_{x\to\infty}\left(\frac{17x}{17x+9}\right)^{3x}$? I tried to solve this problem and could not understand this. I know that it is an exponential equation of the type ...
1
vote
1answer
2k views

How to maximize profit in this equation?

A 300 room hotel is filled to capacity at \$80 a night. If the charge is increased by \$3 it rents 9 less rooms. If it costs \$10 to clean a rented room the next day, how much should the inn keeper ...
4
votes
4answers
147 views

Evaluate $ \int^{\frac{\pi}{2}}_0 \frac{\sin x}{\sin x+\cos x}\,dx $ using substitution $t=\frac{\pi}{2}-x$

This problem is given on a sample test for my calculus two class. $$ \int^{\frac{\pi}{2}}_0 \frac{\sin x}{\sin x+\cos x}\,dx $$ I can find the value of this integral using other substitutions which ...
0
votes
1answer
308 views

How do I solve this Calculus Work problem?

A cylindrical water tank has height 8 m and radius 2 m. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank? I know that ...
0
votes
3answers
1k views

Linear approximation to $y = \sqrt{1-x}$ at $x=0$, then approximate $\sqrt{0.9}$ and $\sqrt{0.99}$

How do I find this? I know that the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$. Here, I would plug in $(1-x)$ instead of $x$. When $x = 0$, the slope would evaluate to $\dfrac{1}{2}$. I got ...
1
vote
0answers
287 views

Integral of the modified Bessel function of the first kind zero order

I would like to ask whether there exists a closed form formula for computing an integral of the modified Bessel function of the first kind zero order. In more detail, an integral including something ...
1
vote
0answers
43 views

How can I find $\int{\sqrt{\left(b^2-1\right)x^2+1\over-x^2+1}}dx$?

I got this from the perimeter of an ellipse. I came up with the formula: arclength of f(x) for x from a to b=$\int_a^b\sqrt{f'(x)^2+1}dx$. Since an ellipse has the equation: $$\left({x-h\over ...
1
vote
1answer
86 views

How does one find out the general solution of this second order differential equation?

I'm having trouble attacking this second order differential equation: $$y''-tP(t)y'+P(t)y=0$$ Any help would be appreciated!
0
votes
1answer
2k views

How to expand this taylor series and find radius of convergence

f(x)= √(1-x) at x=0 How do you find the taylor series and radius of convergence?
1
vote
4answers
1k views

Graph the polynomial $f(x)=x^4-3x^3-4x^2+3x-2$ using a graphing calculator [closed]

Graph the polynomial $f(x)=x^4-3x^3-4x^2+3x-2$ using a graphing calculator? a) The Range b) The Real Zeros c) The y-intercept d) relative minimum and Relative Maximum e) The intervel where f(x)< ...