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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9
votes
1answer
129 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
3
votes
1answer
34 views

Inequality $a^2b^2+2(a+b)\geq 4ab+1$

Let $a,b\geq 1/2$. Prove that $$a^2b^2+2(a+b)\geq 4ab+1.$$ We know that $(ab-1)^2\geq 0$ implies $a^2b^2+1\geq 2ab$, so the inequality reduces to $2(a+b)\geq 2ab+2$, or $a+b\geq ab+1$. But this is ...
4
votes
2answers
32 views

Evaluating $\sum_{i=a+1}^{N}\frac{i(i-1)}{i-a}$

I am trying to solve the German Tank Problem. There might be numerous ways of finding the expected value of N. However, the way in which I am proceeding, I need to find this sum. However I am stuck at ...
6
votes
1answer
83 views

Given $f(x)$ and $g(x)$, find $(fg)(x)$

I've attempted to solve the problem below, and here is what I got for a solution: Given $f(x)=x^2-9$ and $g(x)=x^2+3x-1$, find $(fg)(x).$ $$ \begin{align} (fg)(x)&=(x^2-9)(x^2+3x-1)\\ ...
5
votes
6answers
117 views

Solve $\sin2x +\sin x = 0$ algebraically

I am studying for a final and came across a review question that I have no idea how to do. The question is "Solve the equation $\sin(2x) + \sin(x) = 0$ on the interval $[0, 2\pi)$. I can graph it ...
2
votes
2answers
48 views

Is parametric form of a given function unique? [closed]

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form ...
2
votes
0answers
39 views

Macaulay duration for a coupon bond. Proof

I am working on showing the following. There is a coupon bond redeemable at par with annual coupon rate $r$ per year. The yield to maturity is $i$. The total number of coupons is $n$. Show ...
1
vote
2answers
128 views

How much thrust is required to move a boat of 120 kg / 265 pounds to a speed of … [closed]

How much thrust is required to move a boat of 120 kg / 265 pounds to a speed of 10 km / 6 miles per hour in 7 seconds. I found the following: http://en.wikipedia.org/wiki/Thrust-to-weight_ratio ...
5
votes
9answers
271 views

If $e=\lim\limits_{x\to \infty} (1+x^{-1})^x$, why doesn't $e=1$?

I'm sure that this is a very basic question, but it has been bothering me for a while: If $e=\lim\limits_{x\to \infty} (1+x^{-1})^x$, shouldn't $e=1$? If $x$ is tending towards infinity, why ...
1
vote
1answer
58 views

Airplane Wind problem

Airplane flying at 400 mph at an angle of 30 deg encounters a wind. The resultant velocity of the airplane is 475.3 mph at an angle of 27.18 deg. What was direction of the wind. I set this up as ...
2
votes
1answer
56 views

The function f is defined as follows: $f:A \to A$

The function f is defined as follows:$f:A$ to $A$ where$$ f(x)=\frac{3(x +1)}{x^2-1}$$ Along my proof in showing that show that there exists an x ∈ A with $f(x) = y$ (showing f is onto) ,I ran into ...
1
vote
1answer
51 views

Show that $f(x,y,z)=0$ if and only if $(\sqrt {x^2+y^2}-1)^2+z^2=r^2$.

Define $f(x,y,z)=(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2)$, where $0<r<1$ Show that $f(x,y,z)=0$ if and only if $(\sqrt {x^2+y^2}-1)^2+z^2=r^2$. Here is what I have tried: Let ...
1
vote
2answers
99 views

Reverse an equation with ln and power

I'm trying to solve for $x$ in the following equation: $\ln(y) = a \cdot (\ln(x)) ^ b + c$ $a = 0.0838 b = 2.6275 c = 0.2506$ but my results look bad. Can anybody show me his demonstration ?
1
vote
1answer
83 views

No real $x,y$ such that $(x+y)^2+(x-2)^2+(y-2)^2=4$

Here's the context of this problem. Solve: $x^2=y^3-3y^2+2y$ $y^2=x^3-3x^2+2x$ We subtract the second equation from the first and obtain $$(x-y)(x^2+y^2+xy-2x-2y+2)=0$$ The first ...
0
votes
1answer
60 views

Reasoning behind multiplying by conjugates

What is the reason behind multiplying by conjugates? I am currently studying single variable calculus and throughout the lessons from the text I'm using, the reasoning as to why one would multiply by ...
0
votes
0answers
35 views

Factoring a Polynomial to Find Tangent Line

I have a polynomial equation $ x^n + a x^{n-1} + bx^{n-2} ... + z =0$ for which the coefficients depend on a parameter $ t $. The equation has one real root that I am interested in. For this real ...
0
votes
1answer
33 views

How would I graph this polar equation?

$$r=-2cos\theta $$ Steps I took: $$r^{ 2 }=-2x$$ $$x^{ 2 }+y^2=-2x$$ $$x^{ 2 }+y^{ 2 }+2x=0$$ Usually I can complete these problems by completing the square in order to find the equation of the ...
0
votes
2answers
21 views

Expressing a polar equation in rectangular form and then graphing it

$$\theta =-\frac {\pi}{ 2} $$ This question confuses me because the only way to find the Cartesian coordinates for this must be by using tangent. And this is where I get confused: $$ \tan\theta ...
2
votes
3answers
55 views

Arithmetic progression with deceleration

A train is travelling at $180 \text { km/h }$, $500\text { m }$ away from a train station, what is the constant deceleration needed to get to a complete stop at the station. A continued question ...
0
votes
2answers
86 views

How did they solve for a here?

Consider the following algebraic steps: $$ F - (M_1 a + \mu_k M_1 g) - \mu_k M_2 g = M_2 a $$ $$ F - \mu_k M_1 g - \mu_k M_2 g = (M_1 + M_2) a $$ $$ a = \frac{F - \mu_k M_1 g - \mu_k M_2 g}{(M_1 + ...
-1
votes
2answers
37 views

When to apply rules of logarithms, order of operation

Sometimes I get a little confused with what order to do things in when it comes to $ln$ being raised to the natural base. For example $e^{\int -A\ln{x} dx}$ where $A$ is an arbitrary constant. Should ...
3
votes
4answers
89 views

How to solve the system $x y^5=8000$ and $x y^4>4100$?

I need help getting this equation solved for a website I am building. I am pretty bad at math and am only in pre-algebra. I don't know how I would go about canceling out the ^5 and ^4 because I can't ...
2
votes
1answer
46 views

Periodicity of an infinitely differentiable function

Consider $f:[-\pi,\pi] \to \mathbb{C}$ be an infinitely differentiable function with $f^{(n)}(-\pi) = f^{(n)}(\pi)$ for all $n \in \mathbb{Z}^+$. Is this a periodic function ? I think it is a ...
3
votes
2answers
74 views

Intersection of two circles.

Let $C_1$ and $C_2$ be the circles: $\rho=a\sin\theta, \rho=a(\cos\theta + \sin\theta)$ respectively. The graphs of these two circles are From the graphs, we see that the intersection points are ...
2
votes
2answers
98 views

How can I understand solving the equation?

$$\begin{align} &\left[(\sqrt[4]{p}-\sqrt[4]{q})^{-2} + (\sqrt[4]{p}+\sqrt[4]{q})^{-2}\right] : \frac{\sqrt{p} + \sqrt{q}}{p-q} \\ &= ...
0
votes
1answer
52 views

Complex roots of Complex polynomal

Apologies if this is a repeated thread I just couldn't quite find anything that helped. how do I go about finding the complex roots of a complex polynomial? such as $$x^3 + (1-i)x^2 + (1-i)x - i$$ ...
2
votes
1answer
37 views

Simple computation question about the limit of a function including little oh

Consider a sequence $$c_n:= t + o(t/n)n$$ where $o(\cdot)$ denotes little-oh I want to compute $\lim_{n\to \infty} c_n =?$ I guessing the result should be $\lim_{n\to \infty} c_n = t$ but not sure. ...
1
vote
4answers
334 views

Rewrite $\sin(\cos^{-1}(x)-\tan^{-1}(y))$ as an algebraic function of $x$ and $y$.

Rewrite the expression as an algebraic function of $x$ and $y$: $$\sin(\cos^{-1}(x)-\tan^{-1}(y)).$$ I am unsure of how to change this into an algebraic function, yet I am able to simplify inso sin ...
0
votes
0answers
72 views

Find all solutions in the interval $[0, 2\pi)$: $5\cos(2\theta)=2$

Find all solutions in the interval $[0, 2\pi)$ rounded to five decimal places: $5\cos(2\theta)=2$. I began by using the double angle formula for $\cos(2\theta)$ and substituting with $1-\sin^2 ...
0
votes
2answers
29 views

How many of each ticket were sold in one day?

Child tickets - $\$7$ Adult Tickets - $\$10$ Senior Tickets - $\$5$ Day one sold $678$ tickets for $\$5,812$ Day two sold $535$ tickets for $\$4,541$ How many of each ticket were sold on day one ...
1
vote
2answers
24 views

Solved ODE, how did answer key rewrite solution to be in this form?

I was solving the ODE $\frac{dx}{dt} = 4(x^2+1)$ with initial condition $x(\frac{\pi}{4})=1$ I got $\tan^{-1}{x} = 4t+c$ Then I plugged in the initial value and rewrote to get ...
1
vote
1answer
77 views

Series and Sequences Train Question

There's a question here that put me off, it differs from the normal AP/GP questions asked. A train is travelling at $180 \text { km/h }$, $500\text { m }$ away from a train station, what is the ...
0
votes
2answers
33 views

Rules regarding exponents

Given the following algebra problem: $$2^{n+1}-1+2^{n+1}=2^{n+1+1}-1$$ I know $2^{n+1}=2^n2^1$ but just to confirm the truth of the problem above, I just assumed the left hand side is $2^{n+2}-1$ ...
1
vote
2answers
27 views

Question about converting a polar equation to a rectangular equation

$$\sec\theta =2$$ So I went through all the steps and got: $$\cos\theta =\frac { 1 }{ 2 } $$ $$\sin\theta =\pm \sqrt { 1-\frac { 1 }{ 4 } } $$ $$\sin\theta =\pm \frac { \sqrt { 3 } }{ 2 } $$ ...
1
vote
5answers
83 views

Why sometimes we get only one root of quadratic equations?

What is logic behind getting (sometimes) only one root of a quadratic equation which satisfies the equation?
3
votes
2answers
34 views

Converting a polar equation to a rectangular one

$$r=\frac { 4 }{ 1+2\sin\theta } $$ Steps I took: $$(1+2\sin\theta )r=\frac { 4 }{ 1+2\sin\theta } (1+2\sin\theta )$$ $$r+2r\sin\theta =4$$ $$r+2y=4$$ $$(r+2y)^2=16$$ ...
3
votes
7answers
182 views

Why Is $y^{-1}$ = $\frac{1}{y^1}$?

Basically, I'm asking 'Is there any place where I can access a compendium of formal mathematical proofs'? I need to know what processes mathematicians went through to declare $(-1)(-1)=1$ and so on. I ...
0
votes
2answers
55 views

derivatives $\frac{dy}{dx}$

Sand is falling from a rectangular box whose base measures $40$ inches by $20$ inches at a constant rate of $300$ cubic inches per minute. (Include units in your answers.) a) How is the ...
3
votes
1answer
41 views

Question about recursive algorithm

I have following problem: $$f(n)=\frac{1}{1^2+1}+\frac{2}{2^2+1}+\frac{3}{3^2+1}+\cdots+\frac{n}{n^2+1}$$ Write recursive algorithm for $f(n)$ Prove that recursive algorithm is correct ...
0
votes
2answers
45 views

How to prove by induction that $2^{2^n} + 1$ has $7$ in unit's place?

It must be true,first of all, for $n \geq 2$ . So, firstly I proved that for $n = 2$, it's true. Taking for $n = m$, is true, how can I prove that for $n = m + 1$ ie. $2^{2^{m + 1}} + 1$ has $7$ in ...
2
votes
3answers
80 views

If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$

Question : If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$ What I did : $$|z-3i| \leq |z|+3 \tag i$$ Also $$|z-4| \leq |z| +4 \tag{ii}$$ Now adding (i) and (ii) we get $$ ...
2
votes
4answers
196 views

How to prove a right angle if i have two tangents?

I would appreciate your help, it is long time since I solve trigonometric, like if I have the tangent of angle B equal to $\sqrt{2}-1$ and the tangent of angle C equal to $\sqrt{2}+1$, how can I prove ...
1
vote
1answer
35 views

Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ …

Problem : Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ My approach : Let $z = x +iy$ $ \Rightarrow (x^2+y^2)=1$ $\Rightarrow |z| =1 $ represent a circle with centre at (0,0) and radius 1 ...
0
votes
5answers
63 views

Equation of the straight line equidistant from $(2,-2)$ & $3x - 4y + 1 = 0$?

I have tried this as: $$\sqrt{(2 - h)^2 + (k + 2)^2} = \dfrac{3h - 4k + 1}{5}$$ where $(h,k)$ is the point on the required straight line. But on expanding, the equation contains terms of $h^2$ & ...
5
votes
3answers
112 views

Rules for whether an $n$ degree polynomial is an $n$ degree power

Given an $n$ degree equation in 2 variables ($n$ is a natural number) $$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$ If all values of $a$ are given rational numbers, are there any known ...
1
vote
0answers
50 views

Where is the fault in this approach for transforming this Dirichlet series?

Mathematica knows that: $$\lim_{s\to 1} \, \zeta (s)\left(-2^{1-s}-3^{1-s}+6^{1-s}+1\right)=\sum _{n=0}^{\infty } \left(\frac{1}{6 n+1}+\frac{-1}{6 n+2}+\frac{-2}{6 n+3}+\frac{-1}{6 n+4}+\frac{1}{6 ...
0
votes
2answers
72 views

Explain sandwich theorem

I was reading my math book trying to understand "limits and derivatives". I understood almost everything till this. Below is the statement from my book. Can anyone please explain this to me. If ...
0
votes
2answers
38 views

Interesting problem about abc being a perfect power

For natural a,b,c this equality holds: $a^3c^2+b^3a^2+c^3b^2=3 \sqrt[3]{(a^5b^5c^5)}.$ Show that abc is a perfect fifteenth power. I got to this point so far in my problem. I tried to make a ...
0
votes
1answer
45 views

Train overtake time

I am having trouble solving this problem. In particular I am having difficulty translating this problem into an equation. Question: A train leaves a station and travels east at 75 km/h. Two and ...
3
votes
1answer
82 views

If $f(x)+2f(1/x)=3x$, find all $y$ such that $f(y)=f(-y)$.

The function $f(x)$ is not defined when $x=0$. This function has the property that $f(x) + 2f\left(\frac 1x\right) = 3x$. Find all such values of $y$ such that $f(y) = f(-y)$. (This means it is an ...