Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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0
votes
1answer
33 views

Where did I go wrong in this limit?

In answering this question regarding this limit: $$\lim \limits_{n\to \infty }\sin^2 \left(\pi \sqrt{(n!)^2-(n!)}\right)$$ I started by stating that $\sqrt{(n!)^2-(n!)} \to n!$, so that ...
2
votes
3answers
63 views

Evaluate $\lim \limits_{n\to \infty }\sin^2 (\pi \sqrt{(n!)^2-(n!)})$

Evaluate $$\lim \limits_{n\to \infty }\sin^2 \left(\pi \sqrt{(n!)^2-(n!)}\right)$$ I tried it by Stirling's Approximation $$n! \approx \sqrt{2\pi n}.n^n e^{-n}$$ but it leads us to nowhere. ...
0
votes
3answers
80 views

$x^y+y^x=19$ Find the value of $x-y$

If $x^y+y^x=19$. Is that possible that we can find the value of $x-y$? Can someone explain it? Thanks in advance.
2
votes
1answer
40 views

Evaluate $\lim \limits_{x\to 0}\sin^{-1} (\frac{\cos^{-1} x+\cos^{-1} x^{2}}{\pi })$

Evaluate $$\lim \limits_{x\to 0}\sin^{-1} (\frac{\cos^{-1} x+\cos^{-1} x^{2}}{\pi })$$ I tried it by direct substitution $$\lim \limits_{x\to 0}\sin^{-1} (\frac{\cos^{-1} x+\cos^{-1} ...
4
votes
2answers
121 views

Prove the following inequality without using induction: $\frac{1}{2^k-1}\leq \sin^{2k}\theta+\cos^{2k}\theta\leq 1$

How to prove the following inequality (without using induction)? $$\frac{1}{2^k-1}\leq \sin^{2k}\theta+\cos^{2k}\theta\leq 1,\quad k\in\Bbb N.$$
0
votes
4answers
77 views

A limit question [closed]

$$\lim _{x\to0}\dfrac{\displaystyle \int _0^{x^2}(\sin t)^{\frac{3}{2}} dt}{\displaystyle\int ^x_0t\left(t-\sin t\right)dt}$$ How to solve this limit problem? Notice that the numerator is $(\sin ...
3
votes
3answers
39 views

Graph and domain of $\frac{2}{7+\sqrt{x}}$

How to sketch the graph of $\frac{2}{7+\sqrt{x}}$? Can anyone give me some hints ?
2
votes
2answers
36 views

Zeros of the derivative of a polynomial.

If all the zeros of a polynomial $f: \mathbb{C} \rightarrow \mathbb{C}$ are real, does this tell us that the zeros of the derivative are also all real valued? i.e, if $f(z) = 0$ only has real roots, ...
0
votes
2answers
45 views

Polar form to cartesian

Let $\Gamma$ be a circle that passes through the origin. Show that we can find real numbers $s$ and $t$ such that $\Gamma$ is the graph of $r = 2s \cos (\theta + t).$ I know this has to be converted ...
29
votes
5answers
3k views

Trigonometry to the 24th power

How can I find the value of $$\sin^{24}\frac{\pi}{24} + \cos^{24}\frac{\pi}{24}$$ Specifically, is there some easy method that I am overlooking?
2
votes
2answers
127 views

Why is math so difficult for me? [closed]

I'm an aspiring software engineer and currenly in college for computer science. For some reason, no matter what I try, math is so unbearably difficult and indecipherable until I design a program for ...
1
vote
0answers
25 views

Functions satisfying $f(x) = y$ with $(x,y)\in\mathbb{Q}^2$

All continuous function $f$ from the open interval (0,1) to $\mathbb{R}$ takes a rational value: $f(x) = r\in\mathbb{Q}$; however, it is not sure that the pre-image $f^{-1}(r)$ of the rational r ...
2
votes
3answers
162 views

Coefficient Problem (polynomial expansion)

Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$ Just to begin, $(1-x)(1+2x) = -2x^2 + x + 1$ ...
2
votes
1answer
61 views

Maximum value of $x$?

Let $x,y,z,v,w$ be real numbers and $$x+y+z+v+w=8,\qquad x^2+y^2+z^2+v^2+w^2=16.$$ Find the maximun value of $x$? I've solved this question by using the average of the numbers and got $x\leq ...
0
votes
2answers
62 views

Identify the roots for the equation: $(x-2)(x-6i)(x+6i)= 0$

$$(x-2)(x-6i)(x+6i)= 0$$I'm not exactly sure if I fully understand this question. If I'm not mistaken, I managed to get the results: $r_1=2, r_2=6i$, and $r_3=-6i$. However I'm not entirely certain ...
2
votes
2answers
53 views

Write a polynomial equation of the smallest degree with roots $3$, $4i$, and $-4i$.

I already have the answer to this question (it's an example). However, I am still confused with steps 2-3. How does $x^2+4ix-4ix-16i^2$ simplify to $x^2-16$? $(x-3)(x-4i)(x+4i)=0$ ...
5
votes
4answers
48 views

How to determine the number removed from the list [duplicate]

One number is removed from a set of integers from 1 to n,the average of the remaining numbers is $\large{\frac{163}{4}}$. Which number was removed? I tried to find the mean of ...
1
vote
3answers
46 views

Continuous Withdrawal

I'm not familiar with business math,, but right now I'm thinking of a certain scenario..let say I invested 50 000 to a bank that offers an interest of 25% ,let say monthly,, and then I eventually ...
1
vote
2answers
28 views

Age Problem concerning Past and Future

Robert is $15$ years older than his brother Stan, $s$. However, $y$ years ago, Robert, $r$, was twice as old as Stan. If Stan is now $b$ years old and $b > y$, find the value of $b-y$. The answer ...
-1
votes
2answers
33 views

Exponentiation of real numbers [closed]

If $x>0$ is a real number such that $x^\alpha<1$ for some real number $\alpha$. Then $x<1$? Thank you.
1
vote
1answer
15 views

While finding points of discontinuity for a composite function do I need to consider the points of discontinuity of individual functions too.

I'm solving problems based on composition of functions and stuck in this problem. If $f(x)=\frac{1}{(x-1)(x-2)}$ and $g(x)=\frac{1}{x^2}$, then find the points of discontinuity of $f(g(x))$. We ...
5
votes
0answers
56 views

showing that an inequality holds

I am trying to figure out how to show that for $n\geq 3$, $$(2^n-1)^{\frac{n}{2(n-1)}}\geq (2^{n-1}-1)^{\frac{n-1}{2(n-2)}}+1.$$ I've tried basic algebra and induction, but the inductive hypothesis ...
-4
votes
6answers
96 views

Prove the following Identity: [closed]

Show that $$\frac{512}{(16-x^2)^\frac{3}{2}} ≡ \frac{x^2}{2(1-\frac{x^2}{16})^\frac{3}{2}}+\frac{8}{(1-\frac{x^2}{16})^\frac{1}{2}}$$ I having trouble proving this identity, I know that they are ...
0
votes
2answers
30 views

Minute Hand will be as much ahead of the hour Hand as it is Behind it

The time is past 2 o'clock in 10 minutes. The minute hand will be as much as ahead of the hour hand as it is behind it. What time is it? The Answer is 2:05.91 I am having trouble interpreting " ...
4
votes
2answers
83 views

Prove $\sqrt{2} + \sqrt[3]{2}$ is irrational

I solved the problem by way of contradiction. Suppose $x = \sqrt{2} + \sqrt[3]{2}$ is rational. Then we have $$2 = (x - \sqrt{2})^3 = x^3 - 3\sqrt{2}x^2 + 6x - 2\sqrt{2} = (x^3 + 6x) - \sqrt{2}(3x^2 ...
0
votes
2answers
54 views

Pie chart values don't equal 100%

On the website "ArtFCity" there is a pie chart with values taken from a data table. However, the values inside the pie chart do not equal $100\%$. What is wrong with this math? $55.59 + 31.1 + 2.19 + ...
-6
votes
2answers
57 views

In a group of sheep and ducks. How many sheep are there? [closed]

In a group of sheep and ducks, the number of legs is 20 more than twice the number of heads. How many sheep are there?
0
votes
0answers
27 views

Is it always possible to algebraically express a function defined by a set of rules?

Let's say you have an arbitrary function defined by a set of rules such that for example: Domain $\hspace{9mm}$ Range $\hspace{5mm}$ 1 $\hspace{23mm}$ 2 $\hspace{5mm}$ 2 $\hspace{23mm}$ 2 ...
1
vote
2answers
52 views

Why can't $x$ be negative in $x^{\ln{y}}$

According to wolfram alpha, the domain of $x^{\ln{y}}$ is $x>0$ and $y>0$ but putting $(-1,1)$ for $(x,y)$ I get a perfectly fine answer of 1? $y>0$ makes sense, since $\ln{y}$ is only ...
2
votes
3answers
63 views

Find the shortest distance between the point and a parabola

Find the shortest distance between the point $(p,0)$, where $p> 0$, and the parabola $y^2=4ax$, where $a>0$, in the different cases that arise according to the value of $p/a$. [You may wish ...
1
vote
1answer
24 views

How do you solve this system of equation?

if $J_x= \oint y^2 ds $ and $J_y= \oint x^2 ds $ and $J_{xy} = \oint xy ds $ how I can find $a$ and $b$? $$\left\{\begin{matrix} a.J_{xy}+b.J_x=-M_x\\ a.J_y+bJ_{xy}=M_y \end{matrix}\right.$$ ...
0
votes
1answer
36 views

The height of right isosceles triangle decreases with the speed proportional to the area of this triangle

The height of right isosceles triangle decreases with the speed proportional to the area of this triangle. At time $t=0$ the area of triangle is $2$, and at time $t=1$ the area of triangle is ...
1
vote
1answer
26 views

Struggling with a problem in functions.

Suppose '$f$' is a continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $f(f(a))=a$ for some $a \in \mathbb{R}$ then find the number of solutions of the equation $f(x)=x$. Options given: ...
0
votes
1answer
52 views

Find $a,b$ such that $x_1,x_2,x_3,x_4$ to be in an arithmetic progression

Consider $a,b\in\mathbb{R}$ and $x^4-8x^3+ax^2+8x+b=0$ with $x_1,x_2,x_3,x_4\in\mathbb{C}$. We need to find $a,b$ such that $x_1,x_2,x_3,x_4$ to be in an arithmetic progression. Here is all my ...
0
votes
2answers
59 views

Is $\sqrt{\log (n)}=\frac{1}{\sqrt{2}}*(\log n)$? [closed]

Is $$\sqrt{\log (n)}=\frac{1}{\sqrt{2}}*(\log n)$$
0
votes
3answers
55 views

For which values of $a$ does $||x^2-16|-7|-2a=0$ have 3 roots?

Here is the equation: $$||x^2-16|-7|-2a=0$$ Please help solve this. For which $a$ does the equation have 3 roots?
0
votes
3answers
59 views

How to find $\log{x}$ close to exact value in two digits with these methods?

I'm trying to find the result of $\log{x}$ (base 10) close to exact value in two digits with these methods: The methods below are doing by hand. I appreciate you all who already give answers for ...
1
vote
0answers
26 views

Intercepted at the Coordinate Axes

A line passes through point $(2,2)$. Find the equation of the line if the length of the line segment intercepted by the coordinate axes of the square root of $5$. The correct answer among the choices ...
0
votes
0answers
62 views

Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
2
votes
1answer
53 views

value of $\arctan (\cosh u)$ as $u \to -\infty $

I am interested in the value of $\arctan (\cosh u)$ as $u\to -\infty $ $$\arctan (\cosh u)= \dfrac i 2 \log \left| \dfrac {1-i\cosh u}{1+i\cosh u} \right|$$ and since $$\cosh u= \dfrac ...
3
votes
2answers
54 views

First contest problem

I downloaded a contest and worked the first problem which is: There exists a digit Y such that, for any digit X, the seven-digit number 1 2 3 X 5 Y 7 is not a multiple of 11. Compute Y. My ...
0
votes
5answers
63 views

Find the Inverse of this function: $y= \frac{1-\sqrt{x}}{1+\sqrt{x}}$

$y= \frac{1-\sqrt{x}}{1+\sqrt{x}}$ I multiplied both sides by the denominator. However, I am stuck at that point. Help?
17
votes
1answer
3k views

Strange old multiplication table

Today I read an article about chalk boards from 1917 discovered in an Oklahoma school. One of the chalkboards included the following curious image: (Oklahoma City Public Schools) The article ...
2
votes
1answer
65 views

Show $\int \dfrac{\sinh x}{\cosh 2x}=\dfrac 1 {2\sqrt 2} \ln\left|\dfrac {\sqrt 2 \cosh x-1}{\sqrt 2 \cosh x +1}\right| + C$

Show by means of the substitution $u = \cosh x$, that $$\int \dfrac{\sinh x}{\cosh 2x}=\dfrac 1 {2\sqrt 2} \ln\left|\dfrac {\sqrt 2 \cosh x-1}{\sqrt 2 \cosh x +1}\right| + C$$ $$\int ...
2
votes
2answers
52 views

Polar Equation area

The graph of $r = \frac{4}{2 - \cos \theta}$ forms a closed curve. The area of the region inside the curve can be expressed in the form $k \pi$. What is $k^2$? How would I do this? I have tried to ...
0
votes
2answers
22 views

Finding Coefficient given 2 Equations of Lines and an Angle

A line has equation $$3x - ky = 0$$ Find the value of k if this line makes an angle of 45 degrees with the line $$2x + 5y - 17 = 0$$ The answer among the choices is supposed to be $7$. But I keep ...
1
vote
1answer
29 views

How do I convert these conics to standard form?

There are two conics I need to convert from general form to standard form but I am not sure if I am going about it right. They are $9x^2 + 5y^2 + 18x - 36 = 0$ and $2x^2 - 8x + y + 6 = 0$ The ...
1
vote
3answers
88 views

equations of sides of triangles; find largest angle

If the sides of a triangle are $2x+3$, $x^2 + 3x + 3$, and $x^2 + 2x$, find the greatest interior angle of a triangle. The answer is $120$ degrees. I was hoping to find a formula to relate all the ...
0
votes
1answer
20 views

A shooting game; target hit both when shot

In a shooting game the probabiltiies that Roger and Joel will hit a target are 2/3 and 3/4 respectively. What is the probability that the target is hit when both shoot at it? The answer among the ...
0
votes
4answers
38 views

Simplify and Evaluate Trigonometric Functions

Here is the question: Simplify and evaluate without using a calculator: $$\sin\pi/12 + \cos\pi/12$$ So, I think this is a trig identity question, but I am unsure of how to get started. If ...