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

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10 views

Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$

Evaluation of Integral $\displaystyle \int\frac{1}{x^2\left(x^4+1\right)^{\frac{3}{4}}}dx$ $\bf{My\; Try::}$ Let $\displaystyle x = \frac{1}{t}\;,$ Then $\displaystyle dx = -\frac{1}{t^2}dt\;,$ ...
0
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1answer
14 views

Simultaneous equation

Solve that $x-y=a , 2x^2+y^2=9$ and let solution set $(x,y) = (\alpha,\beta)$ Find the maximum value of $|\alpha + \beta|$ where $a$ is a real number
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8 views

Are the signs of total derivatives the same as partial derivatives

I have a set of five equations which I have totally differentiated and solve for the relevant variables. I have another equation that I totally differentiated and have a set of partial derivatives ...
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0answers
4 views

Finding an Extermal of Hard Examples?

Who Can show me the calculation for solving extermal for $$\int_0^1 (x^2+ \dot {x}^2+2xe^t) dt \quad \text{ with }\quad x(0)=0,\;x(1)=free.$$ My TA say a short answer and I Couldn't reach to ...
0
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1answer
9 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
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0answers
11 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
2
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0answers
36 views

Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
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1answer
6 views

Using step 1 of the FTOC to solve the derivative of the integral.

I was going to say that the derivative would simply be the equation inside the integral, since if you're taking the derivative of the integral, but that's not right. Does anyone know how to go about ...
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1answer
17 views

Finding the variance of the time series defined as $x_t=\phi x_{t-1}+w_t$, for $t=2,3,4,…$.

Let $w_t$ be white noise with variance $\sigma_w^2$ and let $|\phi|<1$ be a constant. Consider the process $x_t=w_1$ and $x_t=\phi x_{t-1}+w_t$ for $t=2,3,...$. I need to find the variance. I ...
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17 views

Calculate the upper sums Un and lower sums Ln, on a regular partition of the intervals, for this integral:

sorry new to this site. Can someone please help me with this? I have tried for such a long time and have yielded no correct answers. $$\int_1^7 (3−5x)dx$$ We have $n$ rectangles, so what I did first ...
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0answers
18 views

What is $\frac{\partial}{\partial x}\int_0^t x(\tau)f(\tau)\, d\tau$?

If $F(x,t)=\int_0^t x(\tau)f(\tau)\, d\tau$, What is $\frac{\partial}{\partial x}F(x,t)$ ? And what is $\frac{d}{dt}(\frac{\partial}{\partial x}F(x,t))$?
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3answers
68 views

Finding the derivative of the integral using the Fundamental Theorem of Calculus.

I'm still not entirely solid on the concept of the Fundamental Theorem of Calculus, but I believe that the first step of the theorem will give us $$2x-1$$ which is the derivative of F(x). Usually, ...
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2answers
28 views

Find when the population is growing the fastest, under the logistic model

The population $P$ of an island $y$ years after colonization is given by the function: $\displaystyle P = \frac{250}{1 + 4e^{-0.01y}}$. After how many years was the population growing the fastest? ...
2
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3answers
66 views

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$?

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$? Source: calculus made easy by S. Thompson. I have looked up the formula for binomial theorem with negative exponents but it is confusing. ...
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2answers
28 views

Finding the derivative of the integral using FTOC.

Since they're simply asking for the derivative of $h$, would the answer simply be: $$\cos^2 (x) + x$$
-1
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0answers
29 views

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a$ is of the form $\xi(f) = \sum_{i} c_i \frac{\partial f}{\partial x_i}(a)$? [on hold]

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a = (a_1, \ldots, a_n) $ is of the form $\xi(f) = \sum_{i=1}^n c_i \frac{\partial f}{\partial x_i}(a)$? Thank you very much. Edit: ...
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3answers
26 views

Finding the local maximum from a definite integral

For this, is the First Derivative Test being used? If that's the case then wouldn't the equation be: $$(x^2 - 4) / (2 + \cos^2 (x))$$ I'm just not sure how to go about starting this problem.
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3answers
38 views

Evaluate the integral in terms of areas.

I understand that the first one is 4 from basically adding the squares inside the signed area, but I'm unsure on how to proceed in getting the other integrals. Any help would be appreciated, thank ...
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3answers
55 views

$y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?

My friend and I are currently debating the following question: Let $y_n$ be a sequence in a metric space and assume that the subsequences $y_{2n}$, $y_{2n + 1}$, and $y_{3n}$ all converge. ...
2
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1answer
28 views

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$ I've been messing with this problem for most of today and haven't managed to get ...
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1answer
32 views

Eigenvectors of derivative

I'm trying to consider how linear algebra relates to calculus. It seems to me that the only eigenvectors of the derivative operator on $\Bbb R$ are the functions $ce^{kx}$ for constants $c$ and $k$. ...
0
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1answer
15 views

Differentiability functions

If $f:A\subset \mathbb R^n\rightarrow \mathbb R^m$ and $g:B\subset \mathbb R^n\rightarrow \mathbb R^m$ are differentiable functions on the open sets A, B and $\alpha,\beta$ are constants. Prove that ...
1
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1answer
19 views

Solving a limit by recognizing the sum as a Riemann sum for a function defined on [0,1].

I understand that the change in x is represented by $$7/n$$ but how would you go about solving the actual limit here?
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1answer
46 views

Prove that if f is continuous in A then |f| is also continuous. [duplicate]

Prove that if $f$ is continuous in a then $|f|$ is also continuous. I have this exercise for homework of calculus I, and I was thinking that it could be treated by cases when $f>0$ and $f<0$, ...
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2answers
33 views

Raising/lowering with natural logs

I had a question on a test, and while I have already figured out that I should have done u substitution (I was running out of time and my brain froze), I was wondering if the following would be legal? ...
2
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0answers
70 views

Are “Transition Books” (Spivak/Apostol/Courant) really necessary?

Why do so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? ...
0
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1answer
24 views

Find an equation for a sinusoid with minimum and maximum

Here's my problem: Find an equation for a sinusoid that has a minimum at (30°,-1) and an adjacent maximum at (75°,7). Please help! I've tried everything I can think of, but I'm really drawing a ...
0
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1answer
36 views

How would I set up the Taylor's Inequality to prove that the function is equal to its Taylor Series expansion?

How would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but ...
1
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1answer
57 views

Can someone walk me through this PDE problem on my practice exam?

I have never seen a problem that resembles this and since it's on the practice test, I figure I need to know the steps to take to complete the problem in case there is a similar one on the test next ...
0
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1answer
20 views

Equation for the curve in terms of x,y

we got the equation $$r(t) = (t-2)i + (t^2+4)j$$ I got $$x = 1-2t$$ $$y = 1+4t$$ Would that be correct?
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7 views

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then$ A\in J \implies f(A)$

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then $ A\in J \implies f(A)\in J$; $J$- set are Jordan measurable sets in ...
3
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2answers
40 views

Upper and Lower Darboux integral of a piecewise function $f(x)=x$ and $f(x)=0$.

Let $0<a<b$. Find the upper and lower Darboux integrals for the function $$f(x)=x$$ if $x\in[a,b]\cap\mathbb{Q}$ and $$f(x)=0$$ if $x\in[a,b]-\mathbb{Q}$. I am so lost on this problem. Any ...
0
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1answer
35 views

Explicitly find all pairs $(a,b)$ such that $a^{1/a}=b^{1/b}$ and $a\ne b$.

Explicitly find all pairs $(a,b)$ s.t. $a^{1/a}=b^{1/b}$ and $a\ne b$. My multivariable calculus teacher posed this question as a fun brain teaser for the end of the semester. He said it was ...
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0answers
12 views

Are little-o and “error term” the same thing?

I'm reading The Elements of Infinitesimal Calculus by David A. Santos, and I stumbled upon this: 45 Definition Let $m$ and $n$ be non-zero natural numbers with $m < n$. We say that $x^n$ ...
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2answers
42 views

$\sum_{n=2}^\infty {(-2)^n \over n} $ How does this converge or diverge using the alternate series test?

$$\sum_{n=2}^\infty {(-2)^n \over n} $$ When I took the limit I got -2, I also tried using ratio and root test and got the same answer. The answer is supposed to be divergent I think but I thought if ...
2
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1answer
19 views

Sum of two divergent sequences with different number of partial limits

Suppose that $(a_n)$ is a sequence which has $1050$ partial limits, and $(b_n)$ is a sequence which has $2750$ partial limits. I'm asked to prove that $(a_n+b_n)$ diverges. So, in general the sum of ...
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1answer
17 views

Integral of 2-D Laplacian

I am so confused on these integrals. Here is the question. Problem $$G(x,y)=\ln(x^2+y^2)/2$$ Calculate the 2-D Laplacian $\Delta^2G$ For the interior $D$ of the circle $C$ of radius $a$ calculate ...
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0answers
37 views

proof that $\frac{e^{-t}}{2}(t^2+2t+2)\le1$ for $t\ge0$

Show that $\forall t\ge0,x\le1$ where $$x=\frac{e^{-t}}{2}(t^2+2t+2),t\in\mathbb{R}.$$ My proof we have $x=\frac{(t^2+2t+2)e^{-t}}{2}$ then ...
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2answers
60 views

Finding $\int \frac{1+\sin x \cos x}{1-5\sin^2 x}dx$

Find $\int \frac{1+\sin x \cos x}{1-5\sin^2 x}dx$ I used a bit of trig identities to get: $\int \frac {2+\sin (2x)}{-4+\cos(2x)}dx$ and using the substitution: $t= \tan (2x)$ I got to a long ...
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0answers
27 views

Asymptotic expression of $\int_{- D}^{D} \frac{\text{tanh}(\xi)}{\xi -\omega}\mathrm{d}\xi$

How to derive the following asymptotic expression ($|\omega| \ll D $)? $${\cal{P}}\int_{- D}^{D} \frac{\text{tanh}(\xi)}{\xi -\omega}\mathrm{d}\xi \approx ...
2
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0answers
27 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
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0answers
21 views

How to solve T(n) = 3T(n/4) + c

I found the pattern to this problem being the following... $$3^k T\left(\frac{n}{4^k}\right) + 3^{k-1}c + 3^{k-2}c + \cdots + c$$ I feel like this is wrong but if you can cancel common factors it ...
3
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1answer
280 views

Integration of trigonmetric function $(a\cos x + b\sin x + c)/(1 - d\cos x)^{2}$

Problem: Prove that the indefinite integral $$\int \frac{a\cos x + b\sin x + c}{(1 - d\cos x)^{2}}\,dx$$ is rational function of $\sin x, \cos x$ if and only if $ad + c = 0$. My Try: Looking at ...
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1answer
19 views

Solving for the derivative of this implicit function

I have $x$ implicitly given by $$ x + f(x) = y $$ and I want to solve for $\frac{d x}{d y}$. How should I approach this? I know that $$d(x+f(x))/dy = 1\\ d(x+f(x))/dx = 1 + f'(x) $$ but I don't ...
0
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1answer
423 views

Cauchy's definition of limit and Heine's definition of limit are equivalent..

Proof: Cauchy -> Heine $$ \forall x\epsilon D\ whiteout \ a\ |x-a|<\delta\Rightarrow |f(x)-L|<\varepsilon \\ \exists n_0\epsilon\mathbb{N} \ \forall n\ge n_0 \ |a_n-a|<\delta \Rightarrow ...
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3answers
46 views

I having trouble using partial fractions.

For the integral below I have to use partial fractions, however I am at a lost on how to do so. $$\int\frac{dt}{t^2-t-20}$$ The farthest I have gotten to is factoring the denominator to $(t+5)(t-4)$. ...
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0answers
16 views

What radius circle to remove from unit circle to make golden earring? [on hold]

A circular lamina of radius $x$ is removed from a circular lamina of radius $1$. If the center of gravity is at the edge of the smaller circle (along the line connecting the two centers) what is $x$? ...
4
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0answers
46 views

Why does $\frac{1}{r}\frac{dr}{d\theta} = \cot \psi$?

In the discussion of linear fractional equations in Birkhoff and Rota's Ordinary Differential Equations, the authors assert that if we convert a DE of the form $y' = F\left(\frac{y}{x}\right)$ to ...
15
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1answer
330 views

Evaluting $\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} \operatorname dx$

$$\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx$$ My try:: $\displaystyle \int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx = ...
1
vote
1answer
18 views

Convergence rate - Convex optimization

What is the best known algorithm in terms of convergence rate for unconstrained convex optimization and under what assumptions? $\min_{x} f(x)$ where $f(x)$ is a given twice differentiable convex ...