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

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9
votes
6answers
152 views

How to differentiate $y=\sqrt{\frac{1+x}{1-x}}$?

I'm trying to solve this problem but I think I'm missing something. Here's what I've done so far: $$g(x) = \frac{1+x}{1-x}$$ $$u = 1+x$$ $$u' = 1$$ $$v = 1-x$$ $$v' = -1$$ $$g'(x) = \frac{(1-x) ...
1
vote
4answers
132 views

Range of $f(x) =\frac {x -1}{x^2 -2x + 3} $?

Is my solution for finding the range of $$f(x) = \frac{x-1}{x^2 -2x + 3} $$ correct? Since its Domain is $ \mathbb{R} $, so transforming this equation into $x$ in terms of $y$ , we get $$ yx^2 - (2y ...
1
vote
1answer
154 views

Can I solve this with a Lambert Function?

New to W-Functions and do not understand it properly. How do I solve this equation? I know about numerical solutions (or graph solution), but I'm interested in pure algebraic solution if it exists. ...
2
votes
0answers
119 views

Sum of reversed numbers? [closed]

Here is the question that I'm confused with - Define $reverse(N)$ which reverses a given integer. For eg - $reverse(35)$ = $53$ Now, some natural numbers $N$ have a property that $N + ...
2
votes
0answers
42 views

How do I see that $U \cap f^{-1}(V)$ is open in $\mathbb R^n$, where $V \subset \mathbb R^m$ is an open neighbourhood of $f(x_0)$?

Suppose $U \subset \mathbb R^n$ is open, let $x_o \in U$, and let $f: U \rightarrow \mathbb R^m$ be differentiable in $x_0 \in U$. Let $V \subset \mathbb R^m$ be an open neighbourhood of $f(x_0)$. ...
1
vote
0answers
38 views

Verifying proof that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses.

Prove that if $f$ is a function $f(x): f'(x) > 0$ $\forall x: x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then ...
2
votes
0answers
33 views

Related Rates/Differentiation

A spherical snowball is melting in such a way that its volume is decreasing at a rate of $1~\frac{\text{cm}^3}{\text{min}}$. At what rate is the diameter decreasing when the diameter is ...
0
votes
1answer
73 views

The meaning of 'worst case'

When giving bound on convergence rate, complexity and so on, people sometimes will specify it by 'worst case'. What is the meaning of 'worst case'?
-1
votes
5answers
268 views

If $f(x) = x^4 − 32x^2 + 1,$ find the interval on which $f $ is concave up / down.

$f$ is increasing $(-8,0)\cup(8,\infty)$ and $f$ is decreasing $(-\infty,-8)\cup(0,8)$ Find the local minimum and maximum values of $f.$ local minimum $-8$ and local maximum $0.$ Am I right? Also, ...
1
vote
2answers
45 views

Maclaurin series not giving right answers when manually deriving?

Apologies about any formatting issues, I am new. I have to find the first four terms of the Maclaurin series for $$f(x) = \frac{1}{1-x}$$ So first I plug in: 1st term is 1 Then derive $$(1-x)^{-1} ...
0
votes
0answers
47 views

Find an optimal solution for $\min_{x} F(x)$ analytically

I want to find an analytical solution (exact/closed-form) for $x$ of the following minimization problem: $$\min_{x} b x \left[e^\left(\frac{a}{x}\right)-1\right]+d (1-c-x) \left[e^ ...
2
votes
3answers
219 views

AP Calc Integral Question

If $f$ is continuous on $[a,b]$, which of the following must be true? There is a $c \in [a,b]$ with $f(c) = 0$ $f'(c) = \frac{f(b) - f(a)}{b-a}$ $f(c) = \frac{1}{b-a} \int_a^b f(x)dx$. Choices: 2 ...
0
votes
1answer
93 views

What is the radius of convergence R of the Taylor series?

The following is a taylor series of a function centered at 8. $$\sum_{n=1}^{\infty}\frac{(-1)^n(x-8)^n}{7^n(n+8)}$$ I am trying to ind the radius of convergence of R of this series. My guess is ...
2
votes
1answer
70 views

Prove that given a uniformly continuous function $f\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}$ we have that $\lim_{x\rightarrow 0}f(x)$ exists.

So I have to prove what is in the title. I tried to go from the given definition of uniform continuity (for all $\epsilon>0$ there exists $\delta>0$ such that for all $x,y\in\mathbb{R}\setminus ...
0
votes
2answers
345 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - ...
0
votes
2answers
29 views

Projecting vector (3a) onto vector (-2b)

The problem: Given two vectors, $a$ and $b$, where $\text{proj}_{{b}} {a} = \begin{pmatrix} 4 \\ -7 \end{pmatrix}$, find the value of $\text{proj}_{-2 {b}} (3 {a})$. I'm not sure what to do to solve ...
2
votes
3answers
56 views

What's the supremum of the following set $\{ n + \frac{(-1)^n}{n} : n \in \mathbb{N}\}$

What's the supremum of the following set $\{ n + \frac{(-1)^n}{n} : n \in \mathbb{N}\}$? I know that the infimum is $0$, but what about the supremum? I have calculated with Maxima the first $1000$ ...
0
votes
0answers
47 views

Can anyony explain Why the $0^0\neq1$ [duplicate]

Why $$0^0\neq1$$ Before I thought that $0^0=1$ but when I went to WF to check it, I found it indeterminate. Can Anyone explain why.Thanks
2
votes
1answer
76 views

Non-trivial lower bound approximation of a convex function using the second derivative at the minimum

Say that I am given an infinitely differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. I am wondering if I can construct a meaningful lower bound approximation of $f$ using it's ...
1
vote
2answers
77 views

When do the curves $r=a(1+\sin\theta)$ $r=a(1-\sin\theta)$ intersect?

By converting the equations to $x$- and $y$-components, and setting them equal, I get they intersect at $\theta=0,\pi$, giving the points $(a,0)$ and $(a,\pi)$. But I don't get the point $(0,0)$--how ...
1
vote
1answer
126 views

Find where a curve crosses itself?

I have the curve $x=t^2,y=t^3-4t$. I made a $t_2$ such that $t_2>t_1$ and from $x$ found that $0=t_1^2 -t_2^2$, from here I solved for values by basic guess and test and then subbed them into the y ...
0
votes
2answers
332 views

Find points on curve $r=2\sin\theta$ where the tangent line is parallel to the ray $\theta = \pi/4$

I was thinking to convert to cartesian coordinates and then find when the slope of the tangent line is $1$, but I get a messy equation $2\cos^2\theta -2\sin^2\theta=4\sin^2\theta\cos\theta$ I was ...
1
vote
3answers
319 views

How to evaluate this exponential fraction limit?

I am trying to determine if 3$^n$ grows faster than 2$^{2n}$. One way I found online to do this was, from Growth was to evaluate $\lim_{n\to \infty} \frac{3^n}{2^{2n}}$ and if that limit evaluates ...
2
votes
5answers
380 views

Limit of a certain quotient

$$ \lim_{n\to \infty} {\sum_{k=1}^n {1\over\sqrt{k}}\over\sqrt{n}} $$ I understand that the summation is divergent and this is $\infty\over\infty$ form. But how to proceed further??
0
votes
1answer
112 views

the position vector $x(t_0)$ is orthogonal to the velocity vector $x'(t_0)$ if $x(t_0)$ is the point on the image of $x$ closest to the origin .

Let $x(t)$ be a path of class $C^1$ that does not pass through the origin in $R^3$. If $x(t_0)$ is the point on the image of $x$ closest to the origin and $x'(t_0)\neq 0$, show that the position ...
0
votes
2answers
167 views

Proving limit with epsilon delta

Prove that $$\lim_{x \rightarrow \infty} \frac{x^2+3}{4x^2-4x+8} = \frac{1}{4}$$ using the $\epsilon$-$\delta $ definition of a limit. So we want to find a $\delta$ such that for every ...
-1
votes
5answers
182 views

Why is $\lim \limits_{h \to 0} \frac{e^{h} - 1}{h} = 1$? [duplicate]

Can anybody show me how $\lim \limits_{h \to 0} \dfrac{e^{h} - 1}{h} = 1$?
3
votes
4answers
533 views

Calculus books recommendation (intermediate level)

I would like to ask for some intermediate level textbook for calculus (single variable), or, at least, some supplement to Spivak's Calculus for better understanding on how to approach and solve his ...
2
votes
4answers
168 views

the relationship between $f^{-1}(x)$ and $x$

A elementary question here. If I have $$f(x)<x\quad\forall\,\,x\in (0,1).$$ Can I deduce $f^{-1}(x)>x$ for $x\ne 0,x\ne 1$? If yes, could anyone show me how?
0
votes
1answer
66 views

Deduce alternate series test from Abel’s theorem

Show that the alternate series test can be deduced from Abel’s theorem. I know that Abel's theorem is Abel's Theorem Let $(a_n)$ and $(b_n)$ be two sequences of real numbers such that • $(a_n)$ ...
4
votes
2answers
180 views

Evaluating a definite integral by definition

I have an area function $A(x)$ defined as $$A(x) = \int_{-1}^{x} (t^2 + 1)\space dt$$ ... and I would like to use the definition of definite integral to evaluate it. I started this way $$A(x) = ...
1
vote
1answer
49 views

How to write the equation of this graph in the given figure?

By looking at the graph we come to know that whether the graph is linear or exponential or quadratic etc. If the graph is a straight line then we write the equation of the straight line for the ...
1
vote
0answers
68 views

Calculus book with interesting examples [duplicate]

I need to prepare presentation about calculus, paying special attention for examination of function of one variable. I am not very advances in the topic, but I'm looking for book which contains ...
1
vote
1answer
19 views

A question regarding the order of an asymptotic estimate

Suppose that $m, n \in \mathbb{N}$ such that \begin{equation} m \cdot \log m = n, \end{equation} where the logarithm is in the natural base. How can we estimate the solution $m = m(n)$ ...
0
votes
1answer
29 views

Change the order of integration:$\int_{0}^{1} dx \int_{0}^{1}dy \int_{0}^{x^2+y^2}f(x,y,z)dz$

$$\int_{0}^{1} dx \int_{0}^{1}dy \int_{0}^{x^2+y^2}f(x,y,z)dz$$ in the order : $$\int dz \int dx \int f(x,y,z)dy$$ I don't think it needs to be divided into two seperate integrals, but the professor ...
3
votes
0answers
135 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
1
vote
2answers
362 views

Is there another simple way to solve this integral?

$$\int \frac{x(2-x^3)}{(x^3+1)^2}dx$$ Is there some simple ways to solve this integral? As my solution for this integral is very long. It's not suitable for my student.
1
vote
2answers
52 views

Convergence of the sequence $\frac{x_{n+1}}{x_n}$

Let $x_n$ be a positive sequence such that the sequence $(\displaystyle\frac{x_{n+1}}{x_n})$ converges to $\lambda<1$. Show that $x_n$ converges to $0$. Hint: Show that there exists $c,r$ such ...
1
vote
3answers
96 views

Convergence of an infinite series involving conjugates

I have the infinite series $$\sum_{n=1}^\infty \left(1-\cos\frac{1}{n}\right) $$ I have to find if it converges or not, and I know I have to use the conjugate find it. So I get ...
1
vote
1answer
17 views

if $w$ is a normal distribution where $n(0,1)$, then find the mgf of $w^2$.

if $w$ is a normal distribution where $n(0,1)$, then find the mgf of $w^2$. I have looked it up and the answer is chi squared but i cannot seem to find a way to integrate this correctly. I start the ...
1
vote
1answer
83 views

integrate $dx/(a^2 \cos^2x+b^2 \sin^2x)^2$

Integrate $\dfrac{dx}{(a^2 \cos^2x+b^2 \sin^2x)^2}$. I can go up to the residue formula like in this example here but then I just can't end up with the result for when $n=2$. I keep messing up my ...
5
votes
1answer
268 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
0
votes
3answers
845 views

Norm of the sum of two vectors

This problem has two parts. Part a): $x$ and $y$ are vectors. If $||x|| = 7, ||y|| = 11$, what is the smallest value possible for $||x+y||$? (Note: the || || denotes the norm of a vector). This is ...
0
votes
4answers
49 views

When does $ \sum_{n=1}^{\infty} (-8)^nx^n $ converge?

I am trying to determine when the following series converges: $$ \sum_{n=1}^{\infty} (-8)^nx^n $$ When approaching these problems, do I simply just have to guess-and-check with cases or is there a ...
-3
votes
2answers
75 views

(SOLVED) Calculus Integration $\int_1^2 \frac{2}{((x^2)(x+1))}dx $, where's my mistake?

Integrate (upper limit =2, lower limit = 1) $\int_1^2 \frac{2}{((x^2)(x+1))}dx $ My ans: $1+2\ln(3/2)$ Lecturer ans: $1+2\ln(3/4)$ Checked multiple times, can't find out where's the careless ...
0
votes
1answer
30 views

How to stabilize cyclic tridiagonal matrix algorithm?

I've received a task which is: Solve equation by cyclic tridiagonal matrix algorithm: $$ \frac{\partial{f}}{\partial{t}} = \lambda*\frac{\partial{f}}{\partial{x}}, \\ x\in[0,1]\ t\in[0,1] \\ ...
8
votes
2answers
167 views

How find limit $\displaystyle \lim_{n\to\infty}n\left(1-\tfrac{\ln n}{n}\right)^n$

How find this limit $$\displaystyle \lim_{n\to\infty}n\left(1-\dfrac{\ln n}{n}\right)^n$$
0
votes
2answers
48 views

Evaluating a limit using Squeeze Theorem. [duplicate]

Consider a function $f (x)$ defined on $\mathbb{R}$ satisfying: $$ \left|f(x) - \frac{7^2 + 5x|x|+2}{x^2+16}\right| \leq \frac{1}{x^2} $$ for all $x\neq0$. Calculate: $\displaystyle \lim_{x \to ...
0
votes
1answer
35 views

Change the order of integrals..

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ to$$\int dz\int dx \int f(x,y,z) dy$$ Not sure if to split the shape into two shapes, or do it directly. Either way i would like to know how its ...
-1
votes
2answers
101 views

Velocity of a curve given by parametric equations

In standard Cartesian equations, $\frac{dy}{dx}$ is the velocity function because it's the derivative of position. $$\frac{dx}{dt} = \sin^{-1}\left(\frac{t}{1 + t}\right) ...