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

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5
votes
1answer
21 views

How can I find the limit without using a closed form expression

I am trying to evaluate this limit without using the closed form expression for the sum of natural numbers raised to $k$th power. $$\lim_{n \to \infty} \dfrac{ 1^n +2^n+\cdots +n^n}{n^n}$$ So far I ...
0
votes
2answers
30 views

Unsure why ODE non-exact equation solution is wrong?

The question I'm trying to solve is $$\left(y-4y^6\right)=\left(y^4+5x\right)y'$$ where $y(0)=1$ I want to find the solution explicitly for $x$. I found the integration factor to be $u=y^-6$. ...
0
votes
2answers
29 views

Need help with a second-degree Taylor polynomial

It says to let T2(x) be the second degree polynomial for the functionf(x) = 6 + xe4x where a=0. I need to find T2(1). I thought it was just a taylor expansion and look at the second term, which I ...
2
votes
1answer
43 views

f: R → R and $|f'(x)| ≤ |f(x)|$

Let $f: R → R $ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
0
votes
1answer
17 views

Integral, partial fractions, need explanation for how to get from one step to another.

Can someone explain how they go from the red step to the blue one?
1
vote
0answers
55 views

Evaluating a Difficult limit!

I have to evaluate a very complicated limit, I've done this task already but I wanna make sure I did it right. The function I have in my hands is $$ F(\omega)= \tanh \Big[a\cdot ...
0
votes
5answers
69 views

How to find a simplified expression for $\binom{1/2}{n}$?

How to find a simplified expression for this specific binomial coefficient? $$\binom{\frac{1}{2}}{n}$$
0
votes
1answer
24 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
1
vote
1answer
24 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
1
vote
1answer
15 views

Proving that if the sequence $\{s_n-L\}$ converges to zero, then a sequence $\{s_n\}$ converges to a limit $L$

I am having trouble proving this statement without using the limit rules. I know I start by assuming that the sequence $\{s_n-L\}$ converges to zero, therefore, for every number $ ϵ > 0 $, there ...
-2
votes
1answer
38 views

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

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
46 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
2answers
30 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
48 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
29 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
44 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
53 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
24 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} ...
1
vote
0answers
14 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
55 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
20 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
30 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
73 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
60 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
55 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
19 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
69 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
60 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
52 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
41 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
68 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
44 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
32 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
26 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
289 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
16 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
25 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
15 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
25 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? ...