1
vote
2answers
17 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
36 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
24 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
20 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
19 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
25 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
1answer
28 views

Calculus net signed area [on hold]

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
51 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
144 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
29 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
50 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
64 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
52 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
12 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
31 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
30 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
47 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
43 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 $$f(x) + xf '(x)\leq 0 \qquad\forall \;x>0\;?$$
0
votes
1answer
53 views

Limit of a function is unique [closed]

I have read the proof of this property. The uniqueness of a limit of a function: Spivak's proof I was thinking we can also prove this informally by using the definition of function. For a ...
2
votes
1answer
32 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
0
votes
2answers
47 views

Higher-order derivative test

Let $f:I\rightarrow \Bbb{R}$ $2007$ times differentiable at $x_0 \in I$. Also: $f'(x_0) = f''(x_0) = ... = f^{(2006)} = 0$ but $f^{(2007)} > 0$. Prove there's $\delta> 0$ such that $f$ is ...
1
vote
1answer
21 views

Intervals on which function is increasing and decreasing

Let $p(x)=x^5-q^2x-q$ , where $q$ is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below. We compute ...
1
vote
2answers
31 views

Sketching graphs abs value functions

how do I go forward with sketching the graphs of the following two functions? i) $y(t)=|2+t^3|$ ii) $f(x)=4x+|4x-1|$ thanks in advance!
3
votes
4answers
189 views

Prove/disprove: if $\lim\limits_{ n\to\infty} f(n)=\infty$ then $\lim\limits_{ n\to\infty}f(f(n))=\infty$

Let $f(x)$ a continuous function on $\Bbb{R}$. Prove/disprove: If $\lim\limits_{n\to\infty} f(n)=\infty$, then $\lim\limits_{n\to\infty}f(f(n))=\infty,$ where the limits are taken over $n \in ...
1
vote
3answers
89 views

What functions are most useful after the ones learned in high school?

I have learnt how to use trig functions, hyperbolic trig functions, exponentials and logs and simple things like polynomials, ellipses, hyperbolas and rational functions but lately when doing calculus ...
1
vote
1answer
28 views

Find the largest $n\in \Bbb{N}$ answering the following terms

Let $$f(x) = -\frac{1}{12}x^4 + o(x^5)$$ Also, Let $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C & \mbox{if } x=0 \end{cases}$$ I need to find the largest $n\in\Bbb{N}$ ...
0
votes
1answer
35 views

Unsure of definition of composite functions and integrals

Could someone explain to me what this function represents and how it is possible. Lets have $y :\mathbb {R} \rightarrow \mathbb {R}^2 $ and that $f: \mathbb{R}^2\rightarrow\mathbb{R}^2$, and lets ...