1
vote
1answer
29 views

Function Equivalent to a Constant Paradox

Say I define $z(x,y) = x^2+y = \text{constant}$ Then $\left(\dfrac{\partial z}{\partial x}\right)_{y} = 2x$ However, $\left(\dfrac{\partial \text{ constant}}{\partial x}\right)_y = 0$ Shouldn't ...
-3
votes
2answers
53 views

Clarification of Identification [on hold]

This is more of an observation question. When you see $x$, In $f(x) = x^2$ And when you see $g(x) = x^3$ You automatically identify $x = x$ Wouldn't the $x$'s be off by a little bit? But ...
0
votes
1answer
41 views

Limit of a function containing square root.

Q. $\lim _{x\to 0}\frac{\left(\sqrt{1-cos2x}\right)}{x}$ We can write this function as $\lim _{x\to 0}\frac{\left(\sqrt{2sin^2x}\right)}{x}$. Algebraically we have ...
0
votes
1answer
26 views

How to combine two functions into one continuous function so it can be integrated/differentiated?

I have a function like this : $$f(x)=\begin{cases}x \in)-\infty, 1)&,\;\;f(x)=x^2\\{}\\x\in(1,+\infty(&,\;\;f(x)=x^3\end{cases}\;\;\;\;\;\;$$ as you can see, the function as a whole is ...
0
votes
0answers
15 views

A proof problem about intergral equation's root

Several days ago,my junior asked me the following problem: Let $$F\left( w \right) = \frac{1}{T}\int_0^T {M{x_C}\left( t \right)\cos \left( {tw} \right)dt} - \frac{{\sin \left( {{T_s}w} ...
0
votes
1answer
35 views

Please explain the rules of differentiation?

I have an equation $f(x)=3\sqrt{\ x}$ and i have to find the derivative of the function f. What i have gotten so far is $3x^{-1/2}$, which then comes out to be ${3/2}x^{-3/2}$. I know the answer is ...
0
votes
1answer
15 views

Composite function domains

$f(x) = 1/x$ domain : all real numbers except $x=0$ $g(x) = \sqrt {x + 2}$ domain : $x$ is greater than or equal to $2$ I'm supposed to find the $f(g(x))$ and $g(f(x))$. This is simple ...
1
vote
1answer
31 views

Why are the following graphs discontinuous at $f(0)$ (epsilon-delta)

The caption for graph (f) is "Infinite jump". The caption for graph (h) is "Infinitely many infinite jumps". The graphs are meant to illustrate that we can pick arbitrarily small intervals around ...
1
vote
1answer
49 views

Prove that exponential functions grow faster than polynomial

I am asked to proof that being $r$ and $s$ two known fixed real numbers such that $r > 0$ and $s > 1$, there exists $n_0$ such that for every $n > n_0$ this happens: $n^r < ks^n$ where ...
1
vote
2answers
44 views

stereographic projections find the function

The problem is in the image. I need help. I have no idea how to do this.
-2
votes
2answers
38 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, & ...
1
vote
2answers
28 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)= ...
0
votes
2answers
24 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: ...
0
votes
1answer
27 views

Continuity and differentiability on piecewise function

Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
-2
votes
0answers
28 views

For a boat to float in a tidal wave the water must be 2.5 meters deep… (trig questions)

$$y=5+4.6\sin\left(\frac{t}{2}\right)$$ What is the period in hours? Simply $p=2\frac{\pi}{n}=4\pi$ which is $\pi$ per hour If the boat leaves the bay at midday what is the latest time it can return ...
-3
votes
0answers
29 views

Calculus net signed area [closed]

so this homework problem I have asks me to find the net signed area. So, what I did was 1+2+3+4+5+6-5-4-3-2-1 = 6, but that is wrong... Why is it wrong? Thanks!
1
vote
1answer
48 views

Function - Main Features?

I understand how to draw this function, but what does it mean by main features? any examples for the question below? Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = ...
0
votes
5answers
67 views

Showing that $f(x) = \ln x - e^x$ has no real roots

Show that $f(x) = \ln x - e^x$ has no real roots Since $\displaystyle\lim_{x \to 0^+} f(x) = -\infty$ and $\displaystyle\lim_{x \to \infty} f(x) = \lim_{x\to \infty} e^x \left ( \frac{\ln x}{e^x} ...
1
vote
1answer
21 views

How can I construct a specific sigmoid function?

The simple sigmoid function $$f(x)=1/(1+e^{−x})$$ approaches zero as x tends to negative infinity, and approaches $1$ as x tends to positive infinity. But I want to set $1$ and $20$ instead of $0$ and ...
0
votes
3answers
36 views

Making a Piecewise Function a Single Function

Is there a way to turn a piecewise function into one function. For example: $$\ f(x)=\begin{cases} g(x) & \text{if $a≤x<b $} \\ h(x) & \text{if $b≤x≤d$} \end{cases}$$ (Can you use the ...
0
votes
1answer
35 views

Finding the slope at two points.

I have been sitting at this for 2 days and I'm not getting anywhere, admittedly I might be just very dumb when it comes to mathematics, and as such I would really appreciate some help with this. I ...
0
votes
2answers
34 views

Finding asymptotes for $f(x)=\frac{x^2+3x-10}{3x^2+13x-10}$

$$f(x)=\frac{x^2+3x-10}{3x^2+13x-10}$$ I know that the horizontal asymptote is $1/3$. To find the vertical asymptotes, I set the denominator equal to zero and used the quadratic formula, and I got ...
1
vote
0answers
52 views

How to prove the uniqueness of a specific root?

Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$ where: 1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and 2). $0\leqslant f(x)\in C^{\infty}$ ...
4
votes
1answer
220 views

Proving a function is increasing

Is there a nice way to prove that $f(x)=x^3+x^2+x-3$ is strictly increasing without making use of derivatives or any other advanced concepts ? I'm trying to explain it to a 9th grader, but I can't ...
1
vote
3answers
145 views

Is there a function whose derivative is $|x|$?

Is there a function $y=f(x)$ such that $$\frac{df}{dx}|_{x=a} =|a|$$ for all $a\in \mathbb R$? I'm in a debate with my friend over it and we are stuck
0
votes
1answer
30 views

Notation Abuse for dependent variable differentiation

Let $z(x,y,t) = x^2+y+t^2$ where $x(t)=t$. $\left(\dfrac{\partial z}{\partial x}\right)_{y,t} = 2x$ If you substitute in x for a t, you get the following. $z(x,y,t) = x^2+y+xt$ ...
1
vote
3answers
37 views

Differentiating a function by simplification.

If we consider a function: $f\left(x\right)=\dfrac{x-1}{2x^2-7x+5}$ This function is not defined at x=1 and x=5/2. So if we differentiate this function by u/v method we have: ...
4
votes
2answers
69 views

How many numbers are less than million such that their digits sum is $\le 19$?

How many numbers are less than million such that their digits sum is $\le 19$? This question is a Generating-Functions exercise. The solution claims the answer is the coefficient of $x^{19}$ ...
1
vote
0answers
22 views

Representing a function by simplification.

The concept of 'functions and limits' always seemed confusing to me. I came across this question which occurred to me as a 'basic' of limits. Q. Check whether the following functions are the same: ...
2
votes
2answers
51 views

Finding the limit of $F(x)=\frac{x^2-4}{|x+2|}$

Let $F(x)=\dfrac{x^2-4}{|x+2|}$ and find the following limits $(a) \; \; \lim_{x \to -2^-}F(x)=$ $(b) \; \; \lim_{x \to -2^+}F(x)=-4$ $(c) \; \; \lim_{x \to -2}F(x)=DNE$ I substituted $-2$ to find ...
0
votes
1answer
16 views

Domain of a multiple logarithmic function.

Find the domain of the following function: $f\left(x\right)=log_4\left(log_5\left(log_3\left(18x-x^2-77\right)\right)\right)$ My text provides a solution which goes like: => ...
2
votes
1answer
51 views

Is root of a function differentiable?

Let's assume a function $f(\alpha,\theta)$ always has a single zero wrt $\alpha$: $\forall \theta, \exists \hat\alpha_\theta$ such that $f(\hat\alpha_\theta,\theta)=0$. Let's now consider this root ...
0
votes
1answer
12 views

Invertible or Non Invertible function?

If $f:$**R —> R** is defined by $f\left(x\right)=x^2+1$, then what are the values of $f^{-1}\left(17\right)\:$ and $f^{-1}\left(3\right)\:$ . My textbook arrives at the following answer: ...
1
vote
0answers
47 views

Solve the function from the composition [duplicate]

I have equations as follows $$f(f(x))=x^2+x$$ Then solve for $f(x)$. Can anyone give some hints about this question?
4
votes
1answer
65 views

Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$?

The exercise asks me this: Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$? ps: $f: \mathbb{R}\to \mathbb{R}$ I really don't know how to start :c, I appreciate hints.
0
votes
5answers
64 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
1
vote
1answer
47 views

$f\in C(\mathbb{R})$. What does it mean?

$f\in C(\mathbb{R})$. What does it mean? My guess is "Differentiable on $\mathbb{R}$" but I'm not sure.. Thanks.
2
votes
2answers
53 views

Properties inherited by $f\circ g$ from $f$

Suppose $f,g:\mathbb{R}\to \mathbb{R}$ Prop: Suppose $g$ and $f \circ g$ are ______, and $g$ achieves every value in $\mathbb{R}$. Then $f$ is ______. If in the blanks we put the word ...
0
votes
0answers
13 views

The inverse of a Moment generating function

The moment generating function of $X$ is $M_X(t) = \mathbb{E}[e^{tX}] = \int e^{tu}f_X(u)du$ where t is a complex variable and $f_X$ is the density of X. The cumulant generating funtion of $X$ is ...
1
vote
1answer
32 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
1
vote
2answers
31 views

Prove $f$ isn't uniformly continuous

I already proved (followed by an hint) that $f(y)-f(x) > x(y-x)$ for all $y>x>0$. I need to prove $f$ isn't uniformly continuous on $(0, \infty)$. What I did: Lets assume by contradiction ...
1
vote
1answer
48 views

Is there a uniformly continuous function such that $a_{n+1} = f(a_n)$?

Let $a_{n+1} = a_n - a_n^2$ and $a_1 = \frac{2}{3}$. I already proved that $a_n \to 0$ Now I was asked, is there a uniformly continuous function such that $a_{n+1} = f(a_n)$? All I can think of is ...
3
votes
2answers
89 views

Behaviour of the function $\ln(1+ x^2)$

Thus function has derivative equal to: $\frac{2x}{1+x^2}$. This indicates that it will flatten out while approaching infinity, ie, should have an asymptote. Yet, the function does not have any real ...
1
vote
1answer
40 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
2
votes
1answer
42 views

Confusing about the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$

What is the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$? A nice plot of $f(x)$ shows that the domain is $\mathbb{R}$ but we see that $x$ should be non-negative at the first sight. Of course, I ...
2
votes
1answer
44 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...
0
votes
2answers
38 views

How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$?

Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$ This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$ I say it is surjective because for all really $x$ there is all real $y$, $x \in ...
9
votes
2answers
489 views

Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...
4
votes
3answers
85 views

Intuitive and convincing argument that functions are vectors

Back to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge the ...
1
vote
2answers
85 views

Assumptions that can be made for $f(x) + xf '(x)\leq 0$

I am wondering if we can make any assumptions about a function $f$ i.f.f. it satisfies $$f(x) + xf '(x)\leq 0 \qquad\forall \;x>0\;?$$