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

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1answer
80 views

What does $\ dx^2$ mean?

While writing the second derivative of y, $\frac{d^2y}{dx^2}$ what does the symbol $dx^2$ signify? I know that in case of the first derivative $dy$ means change in y and $dx$ means change in y and ...
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0answers
42 views

Prove by mean value theorem, $(1 - \frac{1}{x})^x$ is an increasing function for $x > 0$.

Prove by mean value theorem, $(1 - \frac{1}{x})^x$ is an increasing function for $x > 0$. I proved that it is increasing for $x > 1$. How to show that it is increasing when $0 < x < 1$?
1
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1answer
53 views

A possible new integral rule

Is there a non-constant continuous elementary function f defined everywhere on the reals, where f has elementary antiderivatives, such that for every elementary g with elementary antiderivatives, the ...
0
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1answer
46 views

Arc-Gamma Function.

Is there an arc-gamma function? Where gamma(x) = y... Arc-gamma(y) = x. I've searched and found something called DiGamma Function, but when I substituted it didn't seem to be "arc" but something ...
2
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3answers
397 views

Is the reverse of the second fundamental theorem of calculus always true?

The second part of the fundamental theorem of calculus states that if $$ f(x)=g'(x) $$ then $$ \int^b_a f(x)\,dx = g(b)-g(a)$$ And so I was wondering that if you solved $\int^b_a f(x)\,dx$ using some ...
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1answer
83 views

Volume of revolved solid using shell method: finding height

The problem that I am working with is: Find the volume of the solid of revolution formed by rotating the region $R$ bounded by $y = 4+ x^2,\;x=0,\;y=0,\;and\;x=1$ about the line $y=10$ I have ...
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2answers
127 views

Using logs to find numerical values

If $\frac{log(a)}{log(b)}=1000,$ then what is the numerical value of $\frac{log(a/b)}{log(b)}$? I was not sure on how to solve this. I was just looking at this problem to see how would you solve ...
2
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2answers
237 views

find y' if y=ln(7x^2+3y^2)

okay so i've asked this question before and i really appreciate the help you guys gave me. i want to see if what i've done so far is correct. 1st step: (7x^2+3y^2)'/ 7x^2+3y^2 2nd step: 14x+6yy'/ ...
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1answer
189 views

Volume of solid of revolution about y-axis

I need to find the volume of a solid of revolution formed by rotating the region bounded by these curves: $y=4+x^2,$ $x=0,$ $y=4+x^2,$ and $x=1$ about the y-axis Here is the graph: ...
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2answers
33 views

convolution problem given $h(x)=1/2$ for $0<x<2$ and $0$ otherwise

I have a convolution problem in the form $$g(x)= \int_{-\infty}^\infty h(y)h(x-y)\,dy$$ where they give me the function $h(x)=1/2$ for $0<x<2$ and $0$ otherwise. I have never done a ...
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2answers
48 views

Area functions, Find a formula for A(x)

Let $A(x)$ be the area between the function and the $x$-axis and between the $y$-axis and the vertical line at a given $x$. Consider the following function. $$f(t) = \begin{cases} - 2t + 8, & ...
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0answers
32 views

Finding the volume of a solid from revolution

Revolve $y=4+x^2$ bounded by $x=0,$ $x=1,$ and $y=0$ around $x=8$ I have started by splitting the area in two regions and using the shell method to get the part between y=4 and y=5 with: ...
2
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1answer
47 views

Given a set of sequences, compute a corresponding set of functions

Consider the following set of sequences: $ S_k(n)= \begin{cases} 1 & \text{$n \equiv0\pmod{k}$}\\ 0 & \text{$n\not\equiv0\pmod{k}$}\\ \end{cases} $ I want to compute a set of ...
1
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1answer
59 views

How to formally justify the existence of a limit with two variables?

Problem: Find the limit of the following functions a) $\displaystyle \lim_{x \to \infty, \ y\to \infty}$ $\frac{x+y}{x^2 + y^2} $ b) $\displaystyle \lim_{x \to 0,\ y\to 2} \frac{\sin(xy)}{x} $ I ...
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1answer
87 views

Does a word problem provide all information?

A while ago I asked a similar question about word problems and assumptions. Is it a definition or an accepted-fact that word problems provide all information about the relevant existence/situation in ...
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2answers
39 views

How to apply chain rule to:$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$?

How do I apply chain rule to the following: $$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$$ Where $$\Big( \frac{dy}{dx} \Big) = \Bigg(\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \Bigg)$$ I don't see the ...
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2answers
37 views

I need to find the formula for h(x) for all x

we're given a function $h(x)=\begin{cases}1/2&\text{for}&0\le x<2\\0 &\text{otherwise}\end{cases}$. Then we are told to define the function $\displaystyle g(x)= ...
-1
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2answers
39 views

Find $\int t\sin^{-1}t\hspace{1mm}dt$

Find $\int t\sin^{-1}t\hspace{1mm}dt$ How do we approach this question, is there a simple way to integrate
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1answer
44 views

Find the absolute maximum and absolute minimum values of f on the given interval, f(x) = x^2 e^{-x/2}, [-2,8]

Here's the function: f(x) = x^2 e^{-x/2}, [-2,8] Sorry for asking this question again, but i cant seem to move forward. Can i get some help again? so i graphed the ...
0
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1answer
32 views

Chain rule for partial derrivatives

Suppose $$t' = t$$ $$x' = x - vt$$ I need to prove that $$\frac{\partial{}}{\partial{x}} = \frac{\partial{}}{\partial{x'}}$$ $$\frac{\partial{}}{\partial{t}} = \frac{\partial{}}{\partial{t'}} - v ...
1
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1answer
41 views

Integral from 0 to 16 of $\sqrt{x}/(x-4)$?

$$\int_{0}^{16}\frac{\sqrt{x}}{x-4}dx$$ So I'm letting $u=\sqrt{x}$, $du=1/2\sqrt{x}$, $u^2=x$ and $dx=2\sqrt{x}du$. I just don't really know what to do from here. I am trying different things and ...
0
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2answers
58 views

How to evaluate a definite integral that contains a constant?

How to evaluate this definite integral with a constant? $$\int_0^{a^1/4} x^7\sqrt{a^2 - x^8} dx$$ I've never seen the constant 'a' before in this situation. But here's what a have so far at least ...
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2answers
40 views

Algebraic issues with the calculation of the second derivative of $(a+be^x)/(ae^x+b)$

I'm trying to work out the 2nd derivative of $\dfrac{a+be^x}{ae^x+b}$ I have $f''=\dfrac{(ae^x+b)^2(b^2-a^2)e^x-2ae^x(ae^x+b)(b^2-a^2)e^x}{(ae^x+b)^4}$ There are so many terms, and I'm seriously ...
0
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4answers
141 views

Finding the derivative of $v(r) = k(R^2 − r^2)$

The velocity (in centimeters per second) of blood r cm from the central axis of an artery is given by $$v(r) = k(R^2 − r^2)$$ where $k$ is a constant and $R$ is the radius of the artery. Suppose $k ...
0
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1answer
60 views

Division isn't associative

Consider the following fraction: $$\frac{\frac{\frac{a}{b}}{c}}{a}$$ How to explain that: $$\frac{\frac{\frac{a}{b}}{c}}{a} \ne \frac{\frac{a^2}{b}}{c}$$ Obviously, you get different results. But ...
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2answers
31 views

What is the domain of this function in calculus?

Given this function find the domain of it: $\sqrt{1-2^t}$ Now, just by looking at this anyone can tell that the domain is t>=0, but like all calculus students, we must show the work. I have this so ...
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2answers
129 views

Finding the equation of oblique asymptote of non-rational function

I have the function: $f(x)=2x-2^{x}+2$ I know that this function has an oblique asymptote, but all the tutorials I can find on google, are with rational functions with the form: ...
2
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2answers
61 views

Showing if $n \ge 2c\log(c)$ then $n\ge c\log(n)$

Is this true that if $n \ge 2c\log(c)$ then $n\ge c\log(n)$, for any constant $c>0$? Here $n$ is a positive integer.
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0answers
30 views

being $\mathbf{a}$ and $\mathbf{b}$ two vectors with same length, how do I expand $(\mathbf{a}^T\mathbf{b})^2$?

Let's say that I have two vectors $\mathbf{a}$ and $\mathbf{b}$. Assuming that they have same length, their product $\mathbf{a}^T\mathbf{b}$ and its square $(\mathbf{a}^T\mathbf{b})^2$ are scalars. ...
0
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1answer
30 views

If $|P_{n+1}-q|\le c|P_n - q|$ for all $n$, where $c<1$, then $P_n\to q$

Given $|P_{n+1}-q|\le c|P_n - q|$ for all $n$, where $c<1$ show that the $$\lim_{n\to\infty} P_n=q.$$ Was told to complete this problem by iteration. I'm terrible with proofs and they don't make ...
4
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3answers
102 views

Show that the solution of an initial value problem is always less than a given constant

My try is that $$\frac{dy}{dt} =(y-3)e^{\cos ty}$$ $$\frac{dy}{y-3}= e^{\cos ty}dt$$ $$\ln (y-3)=-\frac{e^{\cos ty}}{\sin ty} +c$$ my steps is correct or I made mistakes ? please help to solve ...
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2answers
543 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
1
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1answer
102 views

If$ x^2 + y^2 + Ax + By + C = 0 .$ Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

Also find the center and radius of the circle Here's my solution, I'm not sure if it's correct or not (specifically the conditions on $A$, $B$ and $C$. I feel that my conditioning is invalid and that ...
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4answers
116 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
1
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1answer
104 views

Rotate about the x-axis with respect to dy

How would I rotate the region bounded by $y = 4+x^2,\;x=0,\;y=0,\;and\;x=1\;$ along the x-axis in terms of dy. I have already solved this problem in terms of dx see here Here is the ...
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2answers
34 views

How do I find the points of discontinuity in a function with e?

Here is the function: $$\frac{1}{1+e^{1/x}}$$ I need to find the point(s) where the function is discontinuous. I already know how to do that with most functions, but this is the first time I've ...
2
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4answers
60 views

What is the rule behind this derivative?

$$\dfrac{\rm d}{{\rm d}t}\big(\sin^2(t)\big)=\sin(2t).$$ I don't understand what is the rule behind this derivation. I had tried to first rerivate sin() and then to derivate the square function, but ...
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4answers
59 views

Show that $\arctan a+\arctan(\frac{1}{a})=\frac{\pi}{2}$

$\arctan a+\arctan(\frac{1}{a})=\frac{\pi}{2}$ I have the mark scheme in front of me, and I understand where the numbers come from, but I don't understand why they do what they do. You need this ...
1
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2answers
95 views

Calculus limit epsilon delta

Prove using only the epsilon , delta - definition $\displaystyle\lim_{x\to 2}\dfrac{1}{x} = 0.5$ Given $\epsilon > 0 $, there exists a delta such that $ 0<|x-2|< \delta$ implies $|(1/x) ...
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1answer
109 views

calculus 2: volume of solid of revolution formed by rotation of region

Find the integral for the volume of the solid of revolution formed by rotating the region $R$ bounded by the curves $y = 4+x^2,\;x=0,\;y=0,\;and\;x=1\;$ about the $x$-axis in terms of $dx $ and $ dy$ ...
0
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1answer
23 views

Use of the second differential without an initial equation

I'm having some trouble finding minimums using the second differential without being provided an initial equation. Here is the question: A piece of wire of length 60 cm is cut into two pieces. ...
0
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1answer
275 views

How to derive demand function from a utility function without any knowledge of Lagrange Multipliers?

How do I derive the demand function for a utility function of, say, $U(x,y)=\sqrt{11x+11y}$ for goods X and Y in terms of $P_x$, $P_y$, and income $I$, with basic mathematics (basic calculus, but no ...
3
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1answer
270 views

Tricky proof problem based on definite integration

Let $a+b=4$, where $a<2$ and let $g(x)$ be a differentiable function. If the derivative, $\dfrac{dg}{dx} > 0$ for all $x$, then prove that $$\int_0^ag(x)\ dx + \int_0^bg(x)\ dx$$increases as ...
2
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2answers
63 views

Calculus Limits Problem

L'Hopital's Rule is not allowed. Question 1: $$\lim_{x\to -2} \frac{\sqrt{6+x}-2}{\sqrt{3+x}-1} = \ ?$$ I tried to cross multiply $\frac{\sqrt{6+x}-2}{\sqrt{3+x}-1}$with ...
0
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1answer
48 views

Radius of curvature polar

$$ρ = a sin^3({\theta/3}) , ρ=\sqrt{(x^2+y^2)}$$ Please help me find the radius of curvature of this problem I have solved parametric and polynomial forms but i am unable to get this
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2answers
35 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
4
votes
5answers
306 views

Simple differentiation from first principles problem

I know this is really basic, but how do I differentiate this equation from first principles to find $\frac{dy}{dx}$: $$ y = \frac{1}{x} $$ I tried this: $$\begin{align} f'(x) = \frac{dy}{dx} & ...
3
votes
2answers
66 views

$\tan \left(\sec ^{-1}(x)\right)$

$$\tan \left(\sec ^{-1}(x)\right)$$ I know that sec(?)=$\frac{x}{1}$ and that sec=hyp/adj, therefore I conclude that hyp=x and adj=1 and that op=$\sqrt{x^2-1}$ Since Tan = opp/adj I thought the ...
0
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3answers
788 views

Is a differentiable function always continuous?

Continuous Functions are not Always Differentiable. But can we safely say that if a function f(x) is differentiable within range $(a,b)$ then it is continuous in the interval $[a,b]$ . If so , what is ...