2
votes
1answer
24 views

Determine the values of real parameters …

If you have an idea, please, do not leave the page, just write it, I will be very thankful. We have the function $$f:R\setminus \{-1 \}\to{R}$$ ...
0
votes
0answers
12 views

Domain of function of form $f(x)=\frac{g(x)}{k(x)}$

I just want to know did we have a rule to find the domain of function in form of $f(x)=\frac{g(x)}{k(x)}$ .I know $k(x)\ne 0$ . but in general do we have any rule to compute domain of function like ...
0
votes
0answers
4 views

Non-monotonic function but Homothetic function

Is it possible for a function to be non-monotonic, but still homothetic? Thank you for your explanations.
2
votes
0answers
14 views

Quick way to determine the number of horizontal asymptotes

I understand how to calculate horizontal and vertical asymptotes, both by using the trick of comparing the degrees of the numerator/denominator and by using calculus. What I would like to know is ...
-4
votes
0answers
26 views

differentiate the given function. Simplify your answers [on hold]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
5
votes
2answers
121 views

Prove that no function exists such that…

The exercise goes like this: Find a continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall c \in \mathbb{R}$ the equation $f(x)=c$ has exactly 3 solutions; Prove that no ...
1
vote
2answers
38 views

Construct a specific function for a given sequence

Given the sequence $A_n=n$, I want to construct a function $f : R \to R$, such that for every $x \in N$: $f(x)=A_x$ $\int\limits_{0}^{x}f(t)dt=\sum\limits_{i=0}^{x}A_i$ How can do that? Thanks
1
vote
1answer
20 views

Single variable function derivative w.r.t. time?

I was studying calculus and I had doubts about this problem: (this is not homework) A circular wire expands due to heat so that its radius increases with a speed of $0.01 ms^{-1}$. How rapidly does ...
1
vote
3answers
98 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
2
votes
2answers
63 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
0
votes
1answer
21 views

Limits: $\lim_{x\to0^+}xe^\frac{-1}x$ and $\lim_{x\to0^-}xe^\frac{-1}x$

$$\lim_{x\to0^+}xe^\frac{-1}x \text{ and } \lim_{x\to0^-}xe^\frac{-1}x$$ Can anyone help me how to find these limits? Thanks
1
vote
2answers
19 views

A basic question on non-negativity of a function

how to prove that $f(x) = 2x\sin(\frac{1}{x}) - \cos (\frac{1}{x}) + 2$ is positive when $x\in (0,1]$. I can see that by plotting.
1
vote
1answer
31 views

$f$ is an even function defined on $(-5,5)$

If $f$ is an even function defined on $(-5,5)$ on the interval, then find four real values of x satisfying $f(x)=f(\frac{x+1}{x+2})$. My book gives the answer as $\frac{\pm3 ...
1
vote
2answers
29 views

Horizontal Asymptote & Range

Suppose y = c is a horizontal asymptote of a function y = f(x). Is it true that the number c does not belong to the range of f(x)?
1
vote
1answer
79 views

how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$

solve $ 3x^2+3xy-5y^2=55$ using number theory tools ,i have found the following $\Delta=3^2+4(5)(3)=9+60=69$ $d=69,u=1$ $w_{69}=\frac{1+\sqrt{69}}{2}$ ...
1
vote
1answer
55 views

Finding the integral of a radical function?

I had to create a problem in Calculus, and since I'm terrible at finding the limits of a definite integral, I decided it would be good to practice with that. So I created this: Two waffles are ...
0
votes
3answers
62 views

Can two function $f$ and $g$ have same values through out a given interval and different values outside that interval?

Is it possible that for two functions $f$ and $g$ and some interval $(a,b)$ we have $f(x)=g(x)$ for all $x\in(a,b)$ and $f(x)\neq g(x)$ for $x$ outside the interval $(a,b)$? $f$ and $g$ are ...
0
votes
1answer
29 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
0
votes
1answer
23 views

Taylor expansion with non integer exponents in the rest

Consider the function: $$f(x)=\sqrt[3]{8x^2+4x+1}$$ 1) Find $a,b,\alpha,\beta$ such that: $$f(x)=ax^\alpha+bx^\beta+o(x^{-1/3})$$ 2) Find $A=f([0,+∞[)$ and prove that $f:[0,+∞[\rightarrow A$ is ...
0
votes
1answer
39 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
15 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
1
vote
2answers
34 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
1
vote
3answers
67 views

How to prove that $f(x,y)=\ln{(y-x^2)}$ has no limit at $y=x^2$

I try to find a way to prove that $f(x,y)=\ln{(y-x^2)}$ has no limit at $y=x^2$ (it's border). Can you direct me and give my ways to show it?? Thank you!
1
vote
1answer
25 views

What is the Contour lines of $\ln{(y-x^2)}$

What is the Contour lines of $f(x,y)=\ln{(y-x^2)}$? I get $y=x^2+e^c$, but it doesn't seem to me the correct answer... Where I get wrong? Thank you!
1
vote
0answers
26 views

What is the contour lines of $\frac{x-y}{x^2+y^2}$?

What is the contour lines of $\frac{x-y}{x^2+y^2}$? I know hot they looks like but I have to describe them...(via $c$, i.e. $c=\frac{x-y}{x^2+y^2}$). I glad if you will help me with this... The ...
1
vote
2answers
72 views

Tangent line of the inverse function of $y = e^x + x$

I've been sitting on this problem for a while, hopefully you guys could give me a lead on what the hell is going on :) Let $f(x) = e^x + x$ Find the tangent line to $f^{-1}(y)$ (the inverse ...
2
votes
1answer
42 views

Differentiability of a convex function

Let $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$ be convex functions such that $f\ge g$ and $f(0)=g(0)$. Show that if $f$ is differentiable in 0, then $g$ is too and $$ f'(0)=g'(0)$$ I have no idea ...
0
votes
1answer
16 views

Progressive Linear functions

I have a problem and I'm not sure how to calculate it or write it. It uses Linear functon as: y = am + b So the problem is that evrey time the unknown-m increases by 1, The unknown-a will add to ...
3
votes
2answers
85 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
2
votes
1answer
38 views

I need help showing this inequality

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a twice differentiable function such that $f'>0$, $f''<0$, and $f(0)=0$. I need to show, that for every $x>0$: $\frac{f(x)}{f'(x)}>x$ Thanks ...
1
vote
2answers
26 views

Function composition and differentiability

This problem asks for an example of functions $f$ and $g$ such that $g$ takes on all values, $f \circ g$ and $g$ are differentiable, but $f$ is not differentiable. I'm having trouble jumping straight ...
0
votes
1answer
39 views

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ [closed]

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ Could anyone please help me? I dont have a clue how to start.
0
votes
1answer
49 views

Spivak problem on property of continuous functions.

Ok so problem goes like this: If f is continuous on [0,1] and f(x) is in [0,1] for each x.Prove that f(x)=x for some x. My proof goes like this but I am not quite sure of my result. Let ...
3
votes
2answers
44 views

Domain and range of a function.

Find the domain and range of the function $$f(x)=\frac{1}{\sqrt{[\cos x]-[\sin x]}}$$ Where [] denotes the greatest integer function. I started as $[\cos x]-[\sin x]\gt0$ $\implies \cos ...
1
vote
1answer
60 views

Very interesting multivariable calculus question.

If $\displaystyle z = \frac{f(x-y)}{y}$, show that $\displaystyle z + y \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$.
1
vote
1answer
43 views

Calculus book with good explanation on squeeze theorem?

I'm trying to learn squeeze theorem and it's quite difficult to get the concepts. Could anyone suggest some good books which explain this in depth, along with limits, continuity, etc? So far, I have ...
1
vote
1answer
64 views

Proof that $f(x)=x^{1/n}$ is continuous.

Here's what I've done: According to the definition, a function is continuous at $c$ if, for any $\epsilon>0$, there exists a $\delta>0$ so that, if $|x-c| < \delta$, then $|f(x)-f(c)| < ...
0
votes
1answer
35 views

Inferring a characteristic of a ratio of functions from the ratio of their derivatives

This is a strange one, but I need help trying to understand whether there is any logic behind this or not. Given $\frac {f(\sqrt{2})}{g(\sqrt{2})}=2$, and $\frac {f'(x)}{g'(x)}>2$ for all ...
-2
votes
0answers
41 views

The continuous function $f$ is defined on the interval $-4\le x\le 3$.

The continuous function $f$ is defined on the interval $-4\le x\le 3$. The graph consists of two quarter circles and one line segment. Let $g(x)=2x + \int_0^xf(t)dt$. Determine $g(-3)$. find $g'(-3)$ ...
0
votes
1answer
28 views

Finding the domain of a function of a single variable.

I was solving this problem: f is a function such that $f(x)=\sqrt{\frac{x-1}{x-2\{x\}}}$ where $\{x\}$ denotes the fractional part of x. Find the domain of f. So I began as: Case 1 ...
0
votes
3answers
46 views

Is function continuous $x\sin(y)/(x^2+y^2)$

I have the following function and I can't seem to prove that it is not continuous: $ f(x,y) = \begin{cases} 0, & {(x,y) = (0,0)} \\ x\sin(y)/(x^2+y^2), & \text{else} \\ \end{cases}$
0
votes
2answers
49 views

Property of a continuous function with $f(0)=f(2)$ [duplicate]

Suppose that $f$ is a continuous function on $[0,2]$ such that $f(0)=f(2)$.We have to show that there is a real number $c$ in the interval $[1,2]$, such that $f(c)=f(c-1)$. I am completely lost on ...
0
votes
2answers
32 views

Proving monotonicity of functions

For functions $f,g: \mathbb{R} \to \mathbb{R}$ prove the following: 1) If $f$ and $g$ are monotonic going up so is $f+g$ 2) if $f$ and $g$ are monotonic going up so is $f \cdot g$ 3) if $f$ and $g$ ...
0
votes
1answer
32 views

How to simplify $(3\sqrt{x})^3$

Simplify: $(3\sqrt{x})^3$. Where should I begin? I have tried to take to whole thing to the 2/3 power but that didn't seem to work.
1
vote
2answers
35 views

Convergent integral of divergent function

On one of my calculus lectures i was told that exist convergent inmproper integrals (in infinity) of divergent function. I was searching for an example in the internet, but I didn't find any. Has any ...
1
vote
2answers
53 views

Is this function inequality true?

Let $\lambda$ and $\lambda_L$ be the values of the function $f(x,y)$ at the optimum for problems \begin{align} \lambda=\max_{x}\min_{y}f(x,y) \end{align} \begin{align} ...
0
votes
1answer
18 views

Finding a function without knowing its structure but some conditions

I'm trying to find a function who meets this conditions but have no idea where to start. Just think it may be related to the function $Ca^{-\left(x-\mu\right)^2}$, If it really has this structure (or ...
0
votes
1answer
28 views

introductory calculus - Help me find a function with a few properties

I was asked to find a function $f: \mathbb R^2 \to \mathbb R$ such that: 1) $f$ is continuous at $(0,0)$. 2) $f$ has directional derivatives at $(0,0)$ (does this mean $f$ is differentiable at ...
2
votes
4answers
62 views

prove lim of a function

I need to prove that $$\lim_{n\to \infty}\frac{n^2-n+2}{3n^2+2n-4}=\frac{1}{3}$$ using the epsilon definition. I'm having specific trouble understanding how to make it less than epsilon once I've ...
1
vote
4answers
56 views

Question about a property that the derivative satisfies

I understand the derivative as a limiting process, and I understand that $$f^{\prime}(a)=\lim_{h \to0}\frac{f(a+h)-f(a)}{h} \tag{1}$$ But I'm slightly confused about this property that the ...