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

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-2
votes
0answers
14 views

If given the limit that is a derivative, how do I find it's function and the point?

How would I solve for something like this?? $$\lim_{x\to 5} \frac{2^x - 32}{x-5}$$ using the definition of derivatives.
2
votes
1answer
26 views

To show the $\epsilon-\delta$ definition for limits holds.

Question: Check if the following limit exists, if so show that the $\epsilon$ $\delta$ definition for limits holds. $$\lim_{(x,y) \to (1,2)} \frac{(x-1)^2(y-2)^2}{x^2+y^2-2xy-4y+5}$$ My answer: So ...
1
vote
0answers
17 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
2
votes
5answers
46 views

Prove that $f$ has a minimum

Let $f$ be a positive and continuous function in $[0,\infty)$, such that $\lim\limits_{x\to \infty} f(x)=2$. Prove that if $f(0)<2$, $f$ has a minimum in $[0,\infty)$. I am stuck in the ...
0
votes
1answer
20 views

How to do this rather basic Surface area question

I am having a bit of difficulty evaluating the surface area of the region that consists of the part of the sphere $$x^2+y^2+z^2=3c^2$$, within the paraboloid $$2cz=x^2+y^2$$, where $c \gt 0$ I know ...
-9
votes
0answers
41 views

Real Analysis-Find the limit of this series [on hold]

Problem # 3 enter image description here Find the limit of this series.
3
votes
0answers
46 views

Prove $\lim_{n \to \infty} \frac{4n^3}{2n^2+1} \sin(\frac{\pi}{n}) = 2\pi$

For a beginning calculus student, prove $\lim_{n \to \infty} \frac{4n^3}{2n^2+1} \sin(\frac{\pi}{n}) = 2\pi$ I'm guessing this means something like Allowed: Pre-university maths, precalculus, ...
2
votes
0answers
22 views

The sum of two subspaces

Let $V_{1}$ and $V_{2}$ be two subspaces of V. Define the sum of $V_{1}$ and $V_{2}$ to be the subset of V $V_{1}+V_{2}=${$\overrightarrow v_{1} + \overrightarrow v_{2}:\overrightarrow v_{1} \in ...
0
votes
1answer
23 views

Surface are of a curve $y=\sin \left(\frac{\pi x}{6} \right)$ rotated about the $x$ axis.

I'm doing a problem involving finding the surface area of the curve for $y=\sin \left(\frac{\pi x}{6} \right)$, rotated about the $x$ axis, for $[0 < x < 6]$. I got as far as $\frac{72}{\pi} ...
0
votes
0answers
12 views

Integral of least squares and general rules of integration to solve the integral.

My calculus is very rusty and I am interested to know if the following is solvable: $$ \int_0^{\pi}( \log( \frac {(x_0 + e^{-i\omega})(x_0 + e^{i\omega})(x_1 + ...
0
votes
2answers
52 views

How would you calculate $(200\int_0^\infty e^{-0.8t}-e^{-1.8t}\,dt)/(250\int_0^\infty e^{-0.8t} \,dt)$?

$$\frac{200\int_0^\infty e^{-0.8t}-e^{-1.8t} \, dt}{250\int_0^\infty e^{-0.8t} \, dt}$$ I am confused as to how you would integrate the e's from zero to infinity. What steps would you take? By the ...
5
votes
2answers
36 views

Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$

Comparing the series expansion of $\arctan(x^2)$ and $\arctan(x)$ at $x=0$ it looks like one can take the result from $\arctan(x)$ and replace each $x$ with $x^2$ to deduce the series expansion of ...
3
votes
2answers
27 views

Show that $f(x):=\frac{2x^3+x^2+x\sin(x)}{(\exp(x)-1)^2}$ is continuously extendable to $x_0=0$.

What I know If $\lim\limits_{x \to x_0}f(x) := r$ exists, we can create a new function $\tilde f(x) = \begin{cases} f(x) &\text{if }x\in\mathbb{D}\setminus x_0 \\ r & \text{if }x = x_0 ...
0
votes
1answer
7 views

Scale series of number up by uniform amount

I have a series of numbers associated with a grid that determine the hue of each cell. Some of these cells are too dark and I'd like to scale them up slightly yet not to exceed the max value of $1$. ...
0
votes
0answers
19 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
1
vote
0answers
27 views

reduction formula for $\int \tan^n (2x)dx$

Establish a reduction formula for $$\int \tan^n (2x)dx$$ My attempt, Let $I_{n}=\int \tan^n (2x)dx$ $=\int \tan^2 (2x) \tan^{n-2} (2x)dx$ $=\int (\sec^2 (2x)-1)\tan^{n-2}(2x)dx$ $=\int ...
0
votes
4answers
52 views

How to find $\lim_{x\to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^2}$

By plotting $\dfrac{1-\cos x \sqrt{\cos 2x}}{x^2}$, we find that in sufficiently small domain near $x = 0$, $f(x)\to 0$ as $x\to 0$. So the limit seems to be $0$. Now I tried to evaluate it using ...
0
votes
1answer
58 views

What type of discontinuity is found in this graph?

$$ f(x) = \begin{cases} \dfrac{1}{x} && \text{when $x > 0$}\\ 4 && \text{when $x < 0$} \end{cases} $$ What type of discontinuity is present when $f(0)$ ? ...
1
vote
0answers
11 views

Induced Riemmanian metric and Differential of embedding

Suppose I have a manifold $M$ which is defined as the image of a 1-1 smooth map $G:\mathbb{R}^d\rightarrow H$ into a Hilbert space $H$. I want to understand the Riemmanian metric on $M$ concretely, ...
2
votes
2answers
50 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
0
votes
0answers
18 views

Unit normal vector at inflection point for any curve: Defined or Undefined?

Consider an arbitrary parametric planar (for simplicity) curve: $ \vec{r}(t) = f(t) \,\hat{i} \, + \, g(t) \, \hat{j}$ Differentiable twice over its domain. $ \vec{r'}(t) = f'(t) \,\hat{i} \, + \, ...
2
votes
2answers
68 views

Taylor expanding $\frac{e^x}{x}$?

How can you taylor expand $$\frac{e^x}{x}$$ Can it be expanded at $x = 0$? Can it be expanded as $x \to 0$?
1
vote
3answers
59 views

Is $\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} = \infty$?

It was asked in our test, and below is what I did: $$\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} $$ $$=\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5}\times\frac{\sqrt{x^2+16}+5}{\sqrt{x^2+16}+5} $$ ...
0
votes
0answers
9 views

How can I solve the conservation of traffic PDE?

I'm trying to solve the conservation equation for traffic flow so that I can use it for an example. It is stated as follows: $$\frac{\partial \rho }{\partial t} + \frac{\partial \rho v(\rho ...
0
votes
2answers
51 views

Indefinite trignometric integral

I tried $u$-substitution and $uv$-substitution, can't seem to figure this out... any help would be appreciated! Question: $$\int\frac{x}{\cos(x)}\,dx$$ Thanks!!!
0
votes
0answers
40 views

Prove that $f'(c)= \frac{2}{2+3(f(c))^2}$ for some $c$

Problem: $f: [0, 1] \to \mathbb{R}$ is continuous on $[0, 1]$ and differentiable on $(0, 1)$. ALso, $f(0)=1$ and $(f(1))^3+2f(1)-5=0$. Prove that there exists a $c \in (0, 1)$ such that $f'(c)= ...
0
votes
3answers
67 views

x^x^x^…=2, what is the value of x?

I came up with this little simple exercise, stating: $x^{x^{x^{\dots}}}$ infinite times is equal to $2$, find $x$. As we're dealing with infinity, we can just separate the first $x$ and get $x^2=2 ...
0
votes
1answer
38 views

Finding Value of C to Maximize Area

f(x)=$xe^{-\sqrt x}$ Find the value of c, such that the area bounded between the graph, the x-axis, x=c, and x=c+1 is maximized. Find the maximum area. I don't know where to start with this one. I ...
2
votes
1answer
60 views

Doomsday Prediction

I have a calculus problem I can't seem to figure out. Any help would be appreciated! Doomsday prediction. In $1960$, three electrical engineers at the University of Illinois published a paper in ...
0
votes
3answers
26 views

How to account for solids of revolution around vertical lines to the right of the x axis?

I'm trying to find the volume of a solid created by rotating the region enclosed between $x=y^2$ and $x=1$ around the line $x=8$. Noting that the intersections of the functions occur at $(0,0)$ and ...
1
vote
2answers
31 views

Can someone explain why $(e,1)$ and $(t, \ln t)$ are the two points of intersection for this question?

I was just going through Khan academy and this question completely threw me. I've rewatched the prior videos a few times to try to understand what I'm suppose to do, but I still don't understand. The ...
0
votes
2answers
24 views

Solving a for a function within a known definite integral?

I have a problem in my physics class which seems to boil down to $$\int_0^1 f(x)x \,dx = C$$ where $C$ is a constant and I need to solve for $f(x)$. If possible, I need the solutions where $$f(0)=0.$$ ...
0
votes
1answer
23 views

Trouble getting between steps when solving integral

I've having a lot of trouble trying to figure out how they're getting from the step in blue to the one in red. Can some one please explain that?
3
votes
3answers
288 views

Finding the shortest distance between two Parabolas

Recently, a problem asked me to find the minimum distance between the parabolas $y=x^2$ and $y=-x^2-16x-65$. I proceeded with the problem as thus. Let $P(a,a^2), Q(b, -b^2-16b-65), a-b=x$. ...
0
votes
0answers
15 views

Orthogonal trajectory in $3$ dimension.

Find the orthogonal trajectories on the cylinder $y^2 =2z$ of the curves in which it is cut by the system of planes $x+z=c$, where $c$ is a parameter. I parametrized the equation. Orthogonal ...
0
votes
2answers
41 views

Show analytically that $te^{-t}$ is not decreasing monotonically.

How does one show analytically that $te^{-t}$ is not decreasing monotonically on $(0, \infty)$? One can consider numbers in the interval $(0, 1]$ and show a counterexample to monotonicity, but ...
2
votes
0answers
9 views

Can I still uses the method of logarithmic differentiation to simplify complicated functions if the the range of that function includes 0?

The method of log differentiation refers to taking the natural log of both sides of an equation to simply an complication functions evolving lots of multiplication and divisions and exponents. But ...
0
votes
1answer
24 views

Point to Plane Distance Questions

I'm reading from Marsden Vector Calculus 6th Edition and this picture is from page 43. I'm having difficulty understanding how they get to $$ \text{Distance} =|\vec v \cdot \vec n|$$ The way I ...
0
votes
0answers
14 views

Implicit - simplify last step

Please let me know if this link works. I'm pretty new at posting questions. Maybe there is a better way to post from the derivative-calculator.net site. I'm not sure how they simplify the last step ...
-3
votes
0answers
24 views

How do I get these values? [on hold]

I want to know how to get these answers: $A(12836.3)=42227.7$ $A=3.28971$ (phase) $\theta+46.7364=0$ $\theta=-46.7364$ from this equation. I am confuse is there a formula or a trig. identity? ...
1
vote
2answers
57 views

How to find the nature of this series?

What tests to use for this series: $$\sum_{m=1}^{\infty}(-1)^{m-1}\left(1+\frac{8}{m}\right)^m$$ I've tried alternating test and ratio test but were inconclusive. Can I apply nth term test to the ...
13
votes
3answers
825 views

Summation of a term to infinity

I read through many tutorials but no one mentioned this explicitly. Is the following conversion valid? $$\sum_{k=0}^\infty \frac{k-1}{2^k} = \lim_{n\to \infty} \sum_{k=0}^n \frac{k-1}{2^k}$$ ...
2
votes
1answer
19 views

Volume of region in the first octant bounded by coordinate planes and a parabolic cylinder?

Find the volume of the solid region in the first octant bounded by the coordinate planes, the plane $y + z = 2$ and the parabolic cylinder $x = 4 - y^2$. I have a final answer, I would just like to ...
0
votes
1answer
34 views

Water main construction. Find the angle using vectors.

A water main is to be constructed with a $12.5$​% grade in the north direction and a $25$​% grade in the east direction. Determine the angle $\theta$ required in the water main for the turn from north ...
0
votes
1answer
33 views

Intervals in which f(x) is Strictly Increasing/Decreasing

Find the intervals in which $f(x)=\sin x + \cos x, 0 \leq x \leq 2 \pi$ is strictly increasing/decreasing. First I find the derivative $f'(x) = \cos x - \sin x$, then put $f'(x)=0$, getting $\tan x = ...
1
vote
3answers
74 views

Distance between two circles on a cube

I found this problem in a book on undergraduate maths in the Soviet Union (http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf): A circle is inscribed in a face of a cube of side a. Another circle ...
0
votes
2answers
64 views

How do I evaluate a series? [on hold]

In this specific example, I don't understand the steps of evaluating this series: \begin{align} &\frac{12}{n}\left(\left[\sum_{i=1}^n-7\right]+\sum_{i=1}^n\left[\frac{-12}{n}\cdot ...
1
vote
2answers
47 views

Arc length of Archimedes Spiral $ r = \theta $ from $ 0 \le \theta \le 2\pi$

The equation of the Archimedes spiral is given by $$r = \theta$$ The formula for calculating the Arc Length is given by $$L = \int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$ The ...
-1
votes
1answer
28 views

Find the values of constants in piecewise [on hold]

Find the value of the constants a and b so that the function defined by $$ f(x) = \begin{cases} x+1 ,& 1<x<3 \\ x^{2}+bx+c, &|x-2| \geq 1 \end{cases} $$ is continuous in ...
2
votes
1answer
57 views

Why do you need absolute value when taking $\sqrt{\cos^2(x)}$

$$\sqrt{\cos^2(x)} = |\cos(x)|$$ Is this on the right track? If you have an underlying $\cos(x)$ that is negative, and then you square it, you will now have $\cos^2{x}$, which is positive. But, if ...