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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1
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1answer
48 views

Determine the derivative of the reciprocal function, only with the product rule.

Following is given: $( \alpha )$ the derivative of a inverse function: $$\left(\frac{1}{f}\right)' = -\frac{f'}{f^2}$$ $( \beta )$ The the product rule: $$({f}*g) = ({f'}*{g})+({f}*{g'})$$ ...
1
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2answers
41 views

Does the arithmetic mean minimize the sum of absolute values of deviations?

We have $x_1,x_2,\ldots,x_n \in \mathbb{R}$. I conjecture there to be a number $M \in \mathbb{R}$ such that for any $i=1,2,\ldots,n$ the quantity $$|x_i - M|$$ is as small as possible. How do you go ...
1
vote
1answer
38 views

Solve $\sqrt{5-12i}$ by square root definition

I KNOW it can be solved by the trig formula, but I want to solve it by the square root definition, so please don't just post an alternative way to do it. By the square root definition: $$z = 5-12i$$ ...
1
vote
4answers
28 views

length of path travelled on $(t, \cos t, \sin t)$ from times $t = 0$ and $t = 2\pi$

Let the position of a particle in three dimensional space at time t be $(t, \cos t, \sin t)$. Then the length of the path traversed by the particle between the times $t = 0$ and $t = 2\pi$ is (A) ...
1
vote
1answer
27 views

Q: Spivak, basic question, existence of multiplicative identity

The properties explain up to this point are: I don't understand why is it important to make the assertion 1 ≠ 0 . Could someone elaborate on that and maybe give some examples ?
-9
votes
0answers
37 views

Proving I/D with a constant backwards [closed]

178-0/4005=0.56 2422-178/4005 =0.41 183-769/4005=0.15 In order to get the increase/decrease values i divide the difference between the last two numbers by a constant 4005 My question is ...
3
votes
2answers
102 views

Ball and urn method (counting problems)

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Since, $500 = 2^2 5^3$ I believe this can be solved using Ball and Urn let $a = ...
3
votes
5answers
564 views

Binomial theorem application

I have a question about the bonomial theorem, and in specifically, a question that I want help on. I have worked out the answer, but by manually expanding each and every alternative. However, I ...
0
votes
2answers
49 views

Trouble understanding algebra in proof

Can someone explain to me the step where the integral lower bound turns from n to 1? I was trying to read this proof and I am having trouble understanding that step.
0
votes
1answer
38 views

Why does the vector field $(\sin (\theta), - \cos(\theta), 0)$ indicate sideways motion?

If I study a physical system, such as a car, and let it drive forward a little bit, say a distance $m$, then I can draw out the right triangle and find the car's position at $(m\cos \theta, ...
0
votes
3answers
40 views

How to get values of $a$ from $3 > \log_{5/4}a$?

I cannot come up with the solution to how to get $a$ from $$3 > \log_{5/4}a.$$ If it would be equality, we would get $(5/4)^3 = a$ But what should I do for inequality? Thanks
0
votes
2answers
34 views

How to get $a$ from $2 = \log_{5/4}a$

I do not know what algebra to apply to get $a$ values from $2 = \log_{5/4}a$ I will appreciate you help!
0
votes
1answer
41 views

For $x$, $y$, and $z$ positive real numbers, what is the maximum possible value for [closed]

For $x$, $y$, and $z$ positive real numbers, what is the maximum possible value for $ \sqrt{\frac{3x+4y}{6x+5y+4z}} + \sqrt{\frac{y+2z}{6x+5y+4z}} + \sqrt{\frac{2z+3x}{6x+5y+4z}}? $
0
votes
1answer
42 views

Word - Problem… [closed]

Liveleen and a friend drive from Vancouver to Toronto in 56 h of steady driving. They leave Vancouver on May 30 at 8 a.m. Give the month, day, and time that they arrive in Toronto.
0
votes
2answers
72 views

Lagrange theorem- demonstration

Would you help me demonstrate with Lagrange's theorem that $$\sin x < x < \tan x \quad\text{for }x\in \left(0,\frac{\pi}2\right)$$ I am not so advanced in mathematics, we haven't seen this ...
-1
votes
0answers
8 views

Ranked weighted averages of different sample sizes [closed]

Given the following data sets: x1 = [1,1,1,2,2,2] x2 = [1,2] x3 = [1,1,1,1,2,2,2,2] I would like to rank the data sets based on their weighted averages. In addition, it is required that the sample ...
0
votes
1answer
42 views

Trigonomnetric equality involving tg

I need help proving the following identity. $\tan^210^\circ+\tan^250^\circ+\tan^270^\circ=9$. I am not sure if it is even true.
0
votes
2answers
45 views

Logarithm problem [closed]

How do I solve the following equation: $$(1.5)^x = x^4$$ ?
1
vote
2answers
45 views

How to turn sin(arcsinh(x)) into algebraic form?

How can I turn $\sin({\sinh^{-1}{x}})$ into explicit algebraic form ? I've tried to plug in $\sinh^{-1}{x}$ into sine's exponential form $\frac{e^{ix} - e^{-ix}}{2i}$, but then I cannot think of any ...
0
votes
2answers
190 views

In the problem find the minimum value of $(a + b)$.

In the problem a and b are positive real numbers , and two equations of $x^2 + ax + 2b = 0 $ and $x^2 + 2bx + a = 0$.Where $a,b \in \mathbb{R^{+}}$ has real roots. find the minimum value of $(a + ...
1
vote
2answers
118 views

Solve a system of two nonlinear equations

$$ \begin{cases} x^2 - y^2 + 12y - 21 = 0\\ 2x^2 + y^2 + 2xy + x = 0 \end{cases} $$ I've tried the change of variables: $u = x + y$, $v = x - y$ After it I've got: $$ \begin{cases} uv + 12\frac{u - ...
2
votes
2answers
65 views

Show that $\frac{(n-a)^2}{n}$ can be written as $1-\left(\frac{n}{a}\right)^2\cdot\frac{n}{(n/a)^2}$

\left(\frac{n}{a}\right)^2\cdot\frac{n}{(n/a)^2}$. I have got so far to $(a^2/n)-2a+n$ But I can not see how to proceed. Can anyone help please?
0
votes
1answer
26 views

Some tricks to simplify the following expression that has max between variables.

I am trying to simplify the following where $\alpha,\beta>0$ $$f(\alpha,\beta)=\max\{\alpha,\beta,1\}+\max\{\alpha,\beta\}- [1-\max\{\beta,\alpha\}]^+ =0 $$ where $[x]^+=\max(0,x)$. I know also ...
0
votes
3answers
108 views

Problem on clocks.

A man enters his home from some time between $6$ to $7$ pm . when he leaves his home sometime between $7$ to $8$ pm, he observes that the minute hand the hour hand have interchanged ...
2
votes
0answers
61 views

Question on Time Speed and Distance .

Two friends started from A to B in the same direction along a straight line simultaneously. The faster friend john was on bike , while ron was on bicycle. The ratio of their speeds was $1:5$. ...
0
votes
2answers
52 views

What is X and Y in this Dr. Math solution? [closed]

I am trying to use this solution for my problem and Dr. Math does not explain what is big X and big Y in this step: y = m3*(x - X) + Y, y = m4*(x - X) + Y. ...
0
votes
1answer
17 views

Rearrange forumla to isolate 1/A

I would like to rearrange this formula : $$A.(B1 + B2) = xYZ $$ to this one : $$ 1 / A = [expression 1]B1 +[expression 2]$$ Where [expression 1] and [expression 2] could contains one or plus from x, ...
0
votes
2answers
51 views

Equation with $x$ and $\exp(x)$

Let consider the equation: $$e^{0.075x} - 1 = 0.09x$$ I set: $f(x) = e^{0.075x}-1$ and $g(x) = 0.09x$, the I find the cut points of two curves $f$ and $g$. But Is there any way to solve this ...
-3
votes
2answers
53 views

How to prove: When the product of two complex numbers is a real number, the complex numbers are proportional to each other's conjugates. [closed]

Looking for a solution to the problem: Given $ z_1, z_2 \in \mathbb{C}$ and $\mathbf(z_1)\cdot(z_2) \in \mathbb{R}$, prove $ z_1 = p\cdot\overline z_2$ for any $ p\in\mathbb{R} $.
0
votes
2answers
60 views

Why is this simplification wrong?

I was simplifying an equation on khan academy and simplified: (z-6)/(z+3) (1-6)/(1+3) (because z/z=1/1) -5/4 But got it wrong, the top equation is as simple as it gets. Can someone point out ...
2
votes
3answers
102 views

How many times between $2$ pm and $4$ pm does the minute hand coincides with second hand.?

How many times between $2$ pm and $4$ pm does the minute hand coincides with second hand.? options $a.)\quad 118 \\ b.)\quad 119\\ c.)\quad 120\\ d.)\quad 121\\$ Number of rounds of full circle ...
-3
votes
3answers
123 views

Calculating $1 + \sqrt{2} + \sqrt{3} + 2 + \sqrt{5} + \sqrt{6}$ … + $\sqrt{98} + \sqrt{99} + 10$ [closed]

Could I please have some help with calculating this problem? $1 + \sqrt{2} + \sqrt{3} + 2 + \sqrt{5} + \sqrt{6}$ ... + $\sqrt{98} + \sqrt{99} + 10$ To make it simpler: $$\sum_{k=1}^{100} \sqrt{k}$$ ...
0
votes
1answer
33 views

Simplifying a function that has max and min expressions

I am trying to FIND under what conditions are the following functions equal $$F(\alpha,\beta)= G(\alpha,\beta)\leftrightarrow F(\alpha,\beta)- G(\alpha,\beta)=0$$ where $$F(\alpha,\beta)= ...
1
vote
2answers
75 views

Calculate infinite sum $\frac{12}{5} + (\frac{4}{5})(\frac{12}{5}) + (\frac{4}{5})^2(\frac{12}{5})+\dots$

Calculate the infinite sum $\frac{12}{5} + (\frac{4}{5})(\frac{12}{5}) + (\frac{4}{5})^2(\frac{12}{5}) + (\frac{4}{5})^3(\frac{12}{5}) + \cdots$ I actually managed to work it out since posting. ...
0
votes
2answers
49 views

can we simplify $a^{\log_e^{(1/a)}}$? [closed]

I try to simplify $a^{\log_e^{(1/a)}}$, where $\log_e^{(1/a)}$ is a natural logarithm with irrational (transcendental) number $e\approx 2.718)$. Can anyone give some help?
2
votes
0answers
43 views

Simplifying an expression with absolute values

I am trying to simplify the function $D(\alpha,\beta)$ shown below (with $\alpha,\beta>0):$ $$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{|\alpha-1|}{2} - 2 ...
-2
votes
0answers
21 views

Determine the domain range of the function: $y=-4\cos\frac 32(x-1)+3$ [closed]

Determine the domain and range of the function: $y=-4\cos\frac 32(x-1)+3$ My guess for the domain: D$= \{x\mid x\in \Bbb R\}$.
1
vote
1answer
47 views

Lost on “Simple Computations”

I have come across the follow assertion: $$\text{for } x,y,z >0, xyz = x+y+z+2$$ may be rewritten as $$\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}=1$$ and that proving this is a matter of 'simple ...
0
votes
1answer
43 views

Find the common ratio of a geometric series knowing value of first term,number of terms, and partial sum [closed]

does anyone how to transform the following equation... $X = a\dfrac{1-r^t}{1-r}$ into something like $r = \ldots$
-1
votes
1answer
25 views

percent problem [closed]

A recipe of trail mix has 3 ingredients wich includes 20% raisins,20% m&M's and 60% nuts.if I had a total of 8 cups of trail mix. How much (in cups) do I have of each ingredients?
0
votes
1answer
49 views

Inequalities with floor function

How large should $n$ be in order for the following inequality to hold? $$\left\lfloor \frac{n}{m} \right\rfloor \leq 2 \left\lfloor \frac{n}{2m} \right\rfloor$$ Thanks.
0
votes
4answers
91 views

I'm having trouble with induction. Prove $1 + 2^3 + 3^3 + … + n^3 = \frac{((n^2)(n+1)^2)}4$ [duplicate]

I started a new course and I'm expected to know this stuff, and I'm having trouble learning some on my own. I'm stuck with this problem: Prove $1 + 2^3 + 3^3 + ... + n^3 = \frac{((n^2)(n+1)^2)}4$ ...
0
votes
2answers
83 views

Is (a^b)/(b^a) greater than 1 for b > a?

Is it correct to say that: $(a ^ b)/(b^a) > 1$, for all values of a and b given that $b > a$? Can this be easily proven and if so, how?
1
vote
1answer
75 views

Find all nonnegative integers

Determine all nonnegative integers $x$ and $y$ so that $$3^x + 7^y$$ is a perfect square and $y$ is even. Without trial-and-error of course. $$3^x + 7^y = a^2$$ For some integer $a$. ...
-2
votes
3answers
60 views

stuck on Algebra [closed]

We have a function $f(x) =3-2x$. Given that $f(w)=19$, find $w$.
1
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1answer
20 views

Races(time speed and distance)

In a $2000$ m race between $A$ and $B$. $A$ gives $B$ a start of a minute but still beats him by $200$ m .When he increases the head start to $80$ seconds , the race ends in dead heat.Find ...
2
votes
1answer
86 views

Posed in regional mathematics Olympiad 2005

Let a, b, c be three positive real numbers such that $a+b+c = 1$. Let $\Delta= \min( a^{3} + a^{2}bc, b^{3}+ab^{2}c, c^{3}+abc^{2} )$. Prove that the roots of the equation $x^{2} + x + 4 \Delta = 0$ ...
3
votes
0answers
45 views

Probability of another 3 integers with same sum and product as the first 3 integers

Let us suppose $3$ integers are selected at random from a large range, say $$-1000\leq x\leq y\leq z\leq 1000$$ Now, we define the sum and product: $$\begin{align*}s&=x+y+z ...
0
votes
0answers
20 views

How to calculate the max home purchase price based on a maximum Debt to Income ratio?

I'm trying to calculate the maximum home purchase price a home buyer can afford given their annual salary, monthly debt, and a maximum debt to income ratio of 44%. I am not super familiar with ...
0
votes
1answer
28 views

Find the solution to the system (not linear)

Find all $(x, y, z) \in \mathbb{R^3}$ satisfying: $$x^2 + 4y^2 = 4xz \tag1$$ $$y^2 + 4z^2 = 4xy \tag2$$ $$z^2 + 4x^2 = 4yz \tag3$$ This is a very difficult problem. I added $-4(1) + (3)$ to ...