2
votes
6answers
104 views

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 ...
4
votes
1answer
61 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.
4
votes
4answers
189 views

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 ...
0
votes
2answers
24 views

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). ...
0
votes
0answers
27 views

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.
0
votes
4answers
66 views

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.
1
vote
2answers
49 views

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 $(x+3)^2+(y-4)=113$
3
votes
3answers
79 views

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 ...
0
votes
2answers
55 views

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 :
1
vote
1answer
28 views

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 ...
0
votes
0answers
36 views

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 ...
3
votes
5answers
589 views

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 ...
0
votes
1answer
35 views

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 ...
-1
votes
1answer
29 views

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$(z)$ ...
2
votes
1answer
43 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
1
vote
3answers
27 views

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 $ ...
1
vote
1answer
52 views

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 ...
2
votes
0answers
58 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

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 ...
1
vote
1answer
28 views

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 ...
1
vote
1answer
35 views

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 ...
2
votes
9answers
210 views

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 ...
0
votes
2answers
40 views

integrating $\ln(ax)$ in an equation.

The derivative $\frac{d}{dx}\ln{(ax)} = \frac{1}{x}$ What follows is that $\int{\frac{d}{dx}\ln{(ax)}} = \int{\frac{1}{x}}$ And so, $\ln{(ax)} + c_1 = \ln{|x|} + c_2$ where $a, c_1, c_2 ...
4
votes
2answers
24 views

what geometric object is represented (in the complex plane) by the solution of an equation?

The solution to the equation: _ z = 2/z can be described as a geometric object, which? anyone know how to go about this problem? thanks in advance ...
0
votes
3answers
52 views

Derivation of the “Combined Work Formula”

Before I get to my question, some background: Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$) Person $B$ can paint a fence at the ...
3
votes
3answers
56 views

Infinite Limit Question

I am just starting limits, really stumped on this one. How do I approach this? $$\lim_{x\to -\infty} (x-2)(x-3)$$
0
votes
1answer
32 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
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!
1
vote
3answers
149 views

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 ...
0
votes
1answer
37 views

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 ...
1
vote
1answer
45 views

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 ...
0
votes
1answer
35 views

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 ...
0
votes
1answer
27 views

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 ...
0
votes
1answer
34 views

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 ...
3
votes
4answers
81 views

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 $1\le c \le ab$.
1
vote
1answer
90 views

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 ...
-4
votes
2answers
44 views

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 ...
0
votes
0answers
37 views

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 ...
2
votes
1answer
184 views

Finding the derivative using the definition?

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)= ...
1
vote
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 ...
1
vote
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 ...
1
vote
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.
0
votes
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 $a$ : $\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$
3
votes
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 $f'(x)<0$, ...
0
votes
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, ...
0
votes
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 ...
3
votes
3answers
38 views

Using the chain rule with a composite function

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 $t = 9$ ...
2
votes
2answers
37 views

Maximum and minimum of $z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$

Find the maximum and minimum of the function $$z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$$ I have calculated $\frac{\partial z}{\partial x}=\frac{1+y^2+xy-x}{(1+x^2+y^2)^{\frac{3}{2}}}$ $\frac{\partial ...
0
votes
2answers
44 views

Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
0
votes
1answer
70 views

Solving the system with logarithms

I tried solving the system $ \begin{cases} (4x)^{\log_2 (2y)} = 64 \\ (8y)^{\log_2 (2y)} = 256 \end{cases} $ several times but still keep getting wrong solutions.
3
votes
1answer
32 views

Prove that $(x_n)_{n\geq1}$ is an arithmetic progression

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...