# Tagged Questions

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### Tangent to $e^x$

I have been asked to find the tangent to $y=e^x$ that passes through origin. This is what i came up with. Tangent $f(x)=e^x(x-a)+f(a)$, where a is zero, I therefore conclude with $e^xx$ to be the ...
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### 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.
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### Finding $\frac{\mathrm d}{\mathrm dx} x!$

I'm trying to differentiate $x!$ but I just can't seem to do it right. I define the function as follows: $$x! = \prod_{r = 0}^{x}(x-r) \quad,\quad x \in \mathbb N$$ I've tried attempted to try it by ...
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### How to find the slope of a secant line of the graph of a function?

The point $P(5,-2)$ lies on the curve $y=\large\frac{2}{4-x}$. (a). If $Q$ is the point $(X,\large\frac{2}{4-x})$, find the slope $M_{pq}$ of the secant line $PQ$ (correct to six decimal places). ...
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### sketch the region in the xy-plane

Sketch the region in the xy-plane defined by the equation or inequalities $|x|<7$ and $|y|<4$ Im not quite sure how I am supposed to do this.
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### perpendicular bisector of AB

Let $A(-6,9)$ and $B(10,-3)$ be points in the plane Find an equation of the perpendicular bisector of $AB$ the answer I get is $y=(4/3)x+(1/3)$ but it is not correct.
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### Find the equation for the circle

Find an equation for the circle that has center $(−3, 4)$ and passes through the point $(4, −2)$ I keep getting $(x+3)^2+(y-4)=113$
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### Rationalize the denominator and simplify

So the problem I have says rationalize the denominator and simplify. $$\frac{ \sqrt{15}}{\sqrt{10}-3}$$ My answer I got was $\frac{5 \sqrt6}{7}$. Am I doing this wrong or is this the wrong ...
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### process of summing series [on hold]

I'm trying to solve the following exercises, but I have a difficulty: a) Find the sum of the telescoping series : b) Find the sum of this series :
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### Show that the value of $\frac{\text{d}^{2r+1}y}{\text{d}x^{2r+1}}$ when $x=0$ is $\frac{1}{2^{2r}}\left(\frac{(2r)!}{r!}\right)^2$

The question originally asks you to prove that if $y=\sin^{-1}(x)+(\sin^{-1}(x))^2$ that: $(1-x^2)y''-x y'$ is independent of $x$. I get that $(1-x^2)y''-x y'=2$ hence proving the first part. The ...
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### proving that to every $\delta$ exist $n\in$ $\mathbb{N}$

I've got for homework this question: prove that to each $0<\delta<2$ there is $n\in$ $\mathbb{N}$ in which for him that happens $2\pi nsin0.5\delta>1$. I have no clues how to show that is ...
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### How should I self-study calculus?

So I already took Pre-Calc, and ended up with a B both semesters. I am an incoming senior in high school. My special-ed case manager won't let me take it because she doesn't want to see me panic ...
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### rationalize numerator and find conjugate

So in our math class today we were going over some problems on how to rationalize the numerator and finding the conjugates. I am completely lost on how you are supposed to go about it I could really ...
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### Polar form of complex numbers3

Let $z$ be the complex number $z=-2+i$ and let the angle $\phi$ be such that tan$\phi=1/2$ and $-\pi/2<\phi<-\pi/2$. Calculate the modulus $|z|$ and describe the principal argument arg$(z)$ ...
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### Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$\sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k}$$
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### Need help transform one Point to another.

This is probably ultra basic stuff, but I'm very rusted and don't know how to do this. Lets say I have 5 values: upperLimit $= 600$ value $= 589$ lowerLimit$= 500$ upperLimitY $= 250$ ...
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### Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
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### $\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor$ ? If a closed form doesn't exist then can we at least find an asymptotic ...
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### How do i solve this to find PMT?

I know this may seem like a stupid question but i've been up late working on this math assignment and this question just isn't working when i transpose it. So this is the formula to find Present ...
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### Volume of a ellipsoidal shape

I was given the following question: My approach so far was to create a parabolic function: y = 25/2 - (25^2)/392 Then I integrate from x = 0 to x = 14 Volume = 2 * pi * integral of y ^ 2 The ...
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### Why does $(a+b)^2= a^2+b^2 + 2ab$? Why is the $2ab$ there?

When I was doing research on finding the derivative I came across something strange. If $f(x) = x^2$ you find the derivative by going $$\frac{f(x+h)^2-f(x)^2}{h} =\frac{x^2+2xh+h^2-x^2}{h}.$$ Why ...
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### 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!
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### Is tutor essential for success in mathematics? [closed]

Everyone in my Pre-Calc - Calc I class is failing, except the kids who go tutor. They get top percentile ranks in the class. Should I drop maths all together so I don't have to invest in a tutor? I ...
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### What method is used to derive at the function to use in the squeeze theorem?

In every exam in the past $5$ years of Calculus A the question has popped up: Use the squeeze theorem to evaluate the $\lim_{x\to n} f(x)$ where $f(x)$ took many forms from a normal algebraic equation ...
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### Pre- calculus and calculus practice questions

I'll be taking pre-calculus this fall, and I am wondering if anyone on here can recommend a good problem solving workbook with lots of questions for practice.Also,any ideas for calculus I and calculus ...
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### How to find limits involving trigonometric functions as $x\to 0$?

Problem: find the limit as $x\rightarrow 0$ of $\dfrac{\tan(3x)}{\sin(2x)}$ $\dfrac{(\sin(2x) + 3)}{(\cos(7x)-8)}$ Note I am able to solve the first one using l'Hopitals, but I really want to be ...
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### Rearranging equation with algebra

I'm having a difficult time showing that the two are equivalent: $2(x_1-\theta)(1+(x_2-\theta)^2)+2(x_2-\theta)(1+(x_1-\theta)^2) = 2(\bar{x}-\theta)(1+(x_1-\theta)(x_2-\theta))$ I have multiplied ...
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### Trouble with integration using the definition of integral

I'm playing with integration for the first time and I can understand now why everyone tells me calculus II is the hardest calculus. I'm trying to solve this problem but I think I have the wrong ...
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### Is this function decreased with $x$?

Given three positive integers $a,b,c\ge 1$, I am wondering if the following $f(x)$ is decreased with $x$ ? $$f(x)=\frac{c+2x}{(a+x)(b+x)}, \quad x \in Z^+ \cup \{0\}$$ where $1\le c \le ab$.
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### Question on an algebraic inequality

Let $a,b,c,d,x,y,z$ be real numbers such that \begin{equation*} \begin{split} a+b+c+d=&0,\\ a^2+b^2+c^2+d^2=&1,\\ x+y+z=&0,\\ -1\leq x,y,z\leq& 2. \end{split} \end{equation*} Is it ...
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### Find the slope of a straight line that passes through the point (-2,0) and is tangent to the curve y=x^(1/2) [closed]

i) There are two distinct straight lines that pass through the point $(1,-3)$ and are tangent to the curve $y=x^2$. Find their equations. Hint: Draw a sketch. The points of tangency are not given; let ...
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### question on differentiable and continious function

How should the function $f(x)=x\operatorname{sgn} x$ be defined at $x=0$ so that it is continuous there? Is it then also differentiable? How should the function $g(x)=x^2 \operatorname{sgn} x$ be ...
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Calculate the derivate of the given function directly from the definition of derivative, and express the result using differentials $$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ when $f(x)= ... 4answers 56 views ### Finding the instantaneous rate of change of the function$f(x)=-x^2+4x$at$x=5$Finding the instantaneous rate of change of the function$f(x)=-x^2+4x$at$x=5$, I know the formula for instantaneous rate of change is$\frac{f(a+h)-f(a)}{h}$I think it's the negative in front of ... 1answer 17 views ### How Many Lines Passing Through$(0, c>\frac{1}{2})$Are Normal Lines To$y = x^2$How Many Lines Passing Through$(0, c>\frac{1}{2})$Are Normal Lines To$y = x^2$What I've got so far: Let$g$be the line that intersects the parabola perpendicularly. Let$P(p,p^2)$be the ... 4answers 116 views ### Solving the logarithimic inequality$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$I tried solving the logarithmic inequality: $$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$ several times but keeping getting wrong answers. 1answer 63 views ### inequality funny question I'm not sure what they want here: solve the inequality in realtion to$x$for various values of$a$:$\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$4answers 551 views ### How do I show that f is strictly decreasing on (0, infinity)? I have been asked to define$f: (0, \infty) \to (0, \infty)$by$f(x) = \frac 1 x$a) How do I show that f is strictly decreasing on$(0, \infty)$? I realize that I have to show that$f'(x)<0$, ... 0answers 24 views ### Alternative to Hungarian Algorithm to determine minimum cost? Is there a graphic calculator (CAS technology) method to solve minimum cost problems/allocations that are normally completed with the Hungarian Algorithm... Hungarian Algorithm is time consuming, ... 2answers 50 views ### How would I go about showing that f is one-to-one? How would I go about showing that f is one-to-one? (Excuse the image link, I am not savvy enough to use mathematical symbols on here!) Define$f: \mathbb R \setminus\{-\frac{3}{2}\}\to ...
I'm a little confused on this homework problem and I could use some explanation if anyone has seen something like it before. The question is: Use the Chain Rule to find $\frac{dy}{dt}$ at $t = 9$ ...