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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votes
1answer
35 views

Calculate how much ethanol to add to petrol to get a desired blend

Take the following problem: Data: I have $20$ liters of petrol in a tank: Assume that e85 is defined as $85\%$ of ethanol and $15\%$ of petrol; Assume that petrol does not contain ...
1
vote
1answer
20 views

Rate of convergence of a solution to an equation with a free parameter

Suppose that $\epsilon>0$ is a solution to the following equation: $$\epsilon^2-a(m)\ln\epsilon-b(m)=0,$$ where $a(m)\to 0^+$ and $b(m)\to 0$ as $m\to\infty$. Suppose that a solution ...
1
vote
1answer
43 views

Can this be simplified?

Does it make sense to reduce this expression: $\sqrt{a^2-b^2}$ I know that this does not equal square root of a squared minus square root of b squared. I also know that a has to be bigger than b. Is ...
-2
votes
1answer
17 views

Mixtures and Percentages [closed]

Found this in Lang's "Baisc Mathematics" book: A solution contains 35% alcohol and 65% water. If you start with 12 kg of solution, how much water must be added to make the percentage of alcohol equal ...
2
votes
2answers
33 views

Eliminating Sine

How is the $3.22502$ derived? $-0.0834301$ is $\arcsin(-0.0833333)$, but I can't figure out where $3.22502$ comes from. It's been bugging me all day! Thanks in advance!
-1
votes
2answers
55 views

Help me simplify: $\cos(−\theta) + \tan(−\theta) \sin(−\theta)$ [closed]

Simplify $$\cos(−\theta) + \tan(−\theta) \sin(−\theta)$$ to one term with no negative thetas.
6
votes
3answers
859 views

A basic inequality: $a-b\leq |a|+|b|$

Do we have the following inequality: $$a-b\leq |a|+|b|$$ I have considered $4$ cases: $a\leq0,b\leq0$ $a\leq0,b>0$ $a>0,b\leq0$ $a>0,b>0$ and see this inequality is true. However I ...
1
vote
1answer
37 views

An inequality with exponents, factorials and nth roots!

Problem: Prove for natural numbers $n > 2$, $$(\sqrt{2!}-1)((3!)^{\frac{1}{3}}-\sqrt{2!})\cdots(((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}}) < \frac{n!}{(n+1)^n}$$. I am unable to do this one. ...
0
votes
2answers
58 views

Integrate $\sin(3x)\cos(3x)$

Integrate $\sin(3x)\cos(3x)$ I looked at various answers on different sites but still do not understand how to use the u-substitution method in this question or the double angle rule.
0
votes
3answers
50 views

Intersecting circles and the sine and cosine rules

So I wrote a question using three numbers $r_1$, $r_2$ and $l$. I am struggling to solve it "in general" while playing by certain rules. The rules are: no calculator and no half- or double-angle ...
6
votes
3answers
94 views

Solve the equation $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$

Solve the following equation: $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$ Unfortunately I have no idea.
3
votes
3answers
71 views

Why does equating one of the brackets in $(x+1)(x+3)=0$ to zero valid?

When we want to solve an equation like the one given above, we set either $(x+1)$ or $(x+3)$ equal to $0$ to get $x = -1$ or $x = -3$. However, when we put one of those values in the equation, what we ...
4
votes
2answers
65 views

If $x+y+z=0$, prove that $\frac{x^2}{2x^2+yz}+\frac{y^2}{2y^2+zx}+\frac{z^2}{2z^2+xy}=1$

A problem in my homework had asked me: When $x+y+z=0$, evaluate$$\frac{x^2}{2x^2+yz}+\frac{y^2}{2y^2+zx}+\frac{z^2}{2z^2+xy}$$ Without too much difficulty, one can see that the value should be ...
0
votes
2answers
24 views

Suppose $f(x)=x(x-2)$ and $h(x)=e^{-2x}$. If $f(a)+h( \ln a ) =0$, show that $a^3-a^2-a-1=0$

Suppose $f(x)=x(x-2)$ and $h(x)=e^{-2x}$. If $f(a)+h(\ln a ) =0$, show that $$a^3-a^2-a-1=0$$ My attempt: $a(a-2) + \dfrac{1}{a^2}=0 \Rightarrow a^4-2a^3+1=0 \Rightarrow a^3 = \dfrac{1}{2-a}$ ...
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votes
3answers
64 views

Find minimum of $a+b$ under the condition $\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$ where $m,n$ are fixed arguments

Assume $m,n \in \mathbb{R}$ is fixed. And $a,b(a>b>0)$ satisfied the equation $$\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$$ Find $\min\{a+b\}$
0
votes
2answers
33 views

How to find value of $x$ in this equation?

Let's say I'm trying to solve for x in terms of a. Is there any way to simplify the following? $$a = x + \sqrt{ x^2+x} $$ I've been staring at this for a while and nothing immediate comes to ...
-1
votes
3answers
50 views

Solve without using quadratic formula: $\frac{4}{3x+3} = \frac{12}{x^2 - 1}$. [closed]

Solve without using quadratic formula: $\frac{4}{3x+3} = \frac{12}{x^2 - 1}$. Is there a way to solve this without using the quadratic formula? The quadratic formula is one of my biggest weaknesses, ...
2
votes
2answers
65 views

What's the point of “trigonometric proofs/identities” in introductory calculus/pre-calculus?

I remember back in high school at some point delving into worksheet after worksheet of trigonometric "identities", the vast majority of which are basically restatements of $\sin^2(x) + \cos^2(x) = 1$ ...
0
votes
1answer
26 views

Solving nonlinear system algebraically

I have the system of equations: $$2x(1+\lambda)=0$$$$2y(1+\lambda)=0$$$$2z(1-\lambda)=0$$$$x^2+y^2-(z^2+1)=0$$ It's easy to plug in a few values and see that the solution is $x^2+y^2=1$, $z=0$, and ...
2
votes
3answers
48 views

$(x^2 + 1)(x - 6) = 0$ How is the solution equal to 6?

The answer for this equation $(x^2 + 1)(x - 6) = 0$ is $x=6.$ May I know the technique? I am new here, please teach me how to style the equation too. Thanks.
0
votes
2answers
113 views

If $e^{i\pi}=-1$, then what does $e^{2i\pi}$ equal?

As the question says. As according to Euler's formula, $e^{i\pi}+1=0$ thus $e^{i\pi}=-1$, what therefore does $e^{2i\pi} $ equal?
1
vote
1answer
25 views

Maximum value of arg z

On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $|z-2-i|\leq1$ and $|z-i|\leq|z-2|$. Calculate the maximum value of arg $z$ for ...
0
votes
2answers
50 views

curve of $(x^2+y^2)^2=2(x^2-y^2)$

The diagram shows the curve $(x^2+y^2)^2=2(x^2-y^2)$ and one of its maximum points $M$. Find the coordinates of $M$. My attempt. Differentiate the equation and I got ...
1
vote
2answers
26 views

Equation arrangement.

How to arrange $x=\frac{1}{2} \ln({1+\frac{10}{x}})$ to $x=\frac{10}{e^{2x}-1}$ ? Can anyone give me step by step explanation? Thanks in advance.
1
vote
2answers
47 views

Rates of change between $x = 2$ and $x = 2 + h$ for $x^2$ and $1/x$

What is the simplified average rate of change between $x = 2$ and $x = 2 + h$ for the function: (Enter your expression as you would enter an equation in Winplot) a. $f(x) = x^2= 4+h$ b. $(x) = ...
-2
votes
1answer
29 views

How do you solve the equation $5e^4 = \log 3^t$? [closed]

How do you solve the equation $5e^4= \log 3^t$?
0
votes
0answers
38 views

A unbounded exponential function

Could you tell me how to show that the function $f(y)=xy-e^x$ is unbounded if $y <0$, where $x,y \in \mathbb{R}$?
0
votes
1answer
54 views

Taking natural log of $f(x) = 5^{x^2}$

$$f(x) = 5^{x^2}$$ I'm trying to figure out how to take the natural log of this problem. All rules that I can find for natural log don't explain what to do with multiple exponents. Can someone please ...
1
vote
1answer
34 views

How Are The Graphs Related?

We had this as a bonus question on my discrete math exam, and I have absolutely no idea how to even begin a question like this. Can someone explain this question to me, in case I see it as a bonus ...
-1
votes
1answer
32 views

Little arithmetic step in a proof

Uniqueness: let $a \in G$. Assume there's $b \in G$ s.t. $b^2 = a$. Then $(b^2)^{ord(a)} = a^{ord(a)} = e$. Then $ord(b) \mid 2ord(a)$. Since $ord(b)$ is odd, then $ord(b) \mid ord(a)$. So, ...
0
votes
3answers
23 views

decomposition of $-2x^2-3xy+2y^2$

I am trying to decompose $-2x^2-3xy+2y^2$ there are the following steps: $-2x^2-3xy+2y^2=0\Rightarrow (-2x^2-4xy)+(2y^2+xy)=0\Rightarrow -2x(x+2y)+y(2y+x)=0\Rightarrow (y-2x)(x+2y)=0$ is there a ...
2
votes
0answers
49 views

Is there a way to solve the exponential equation $a^x + b^x + c^x = d$ analytically?

So I came across this equation. $$a^x + b^x + c^x = d$$ where $a, b, c$ and $d$ are all constants. And I just wondered, is there any way to solve for x analytically?
1
vote
2answers
47 views

How can I solve this line & plane intersect question and verify the given answer? [closed]

Find an equation for the plane that passes through the point $(3,2,1)$ and contains the line of intersection of the planes with equations $x+y+z=3$ and $x+2y+3z=6$. The given answer from the key is: ...
0
votes
3answers
33 views

tricky 'simplification'

This popped up in a quantum mechanics assignment. I've committed to an hour of attempt and nothing good came out. I'd thought I'd seek help before being driven up the wall. I need to show that ...
-1
votes
2answers
115 views

Prove please that inequality ; if $x\geq 0,y\geq 0,z\geq 0$ $x+y+z\geq x^2y+y^2z+z^2x$

Prove please that inequality : if $$x\geq 0,y\geq 0,z\geq 0$$ and $$x^2+y^2+z^2=3,$$ then $$x+y+z\geq x^2y+y^2z+z^2x.$$
2
votes
0answers
52 views

Simplify a double root expression to least operations

How can I simplify this expression further so that it has the least amount of operations. WA is no help at all. $$\sqrt{\frac{\sqrt{8x+1}-1}2}$$ I'm not looking for complicated equations, all I want ...
7
votes
6answers
178 views

Find the sum of $-1^2-2^2+3^2+4^2-5^2-6^2+\cdots$

Find the sum of $$\sum_{k=1}^{4n}(-1)^{\frac{k(k+1)}{2}}k^2$$ By expanding the given summation, $$\sum_{k=1}^{4n}(-1)^{\frac{k(k+1)}{2}}k^2=-1^2-2^2+3^2+4^2-5^2-6^2+\cdots+(4n-1)^2+(4n)^2$$ ...
0
votes
2answers
19 views

How to give a bound e.g $\leq \epsilon$ in $e^{\frac{M^2 \log_e (\epsilon) + M^2 \log_e^2 (\epsilon)}{2\delta}}$

$\epsilon$ in $$e^{(M^2 \log_e (\epsilon) + M^2 \log_e^2 (\epsilon))/(2\delta)}$$ I know that $e^{M^2\times \log_e \epsilon}$ would result in $\epsilon^{M^2}$ But I am confused what to do with ...
-1
votes
0answers
30 views

Finding modulo inverse m

I'm trying to understand a solution to a problem I'm given. I'm told that: GCD of 23 and 118 = 1 through: $118 = 23(5)+3$ $23 = 3(7)+2$ $3 = 2(1) + 1$ So $1 = 3-2(1)$ $= ...
0
votes
1answer
41 views

Partial sum $\sum_{n=32}^{100} 5n$

I can't remember how to find the partial sum of $$\sum_{n=32}^{100} 5n$$ I know the formula is $\frac{k}{2}\Big(a_1+a_k\Big)$ but I can't remember how to apply the formula. Thank you!
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votes
1answer
28 views

Understanding interest

You have \$1 billion in a checking account earning .02 interest yearly. You spend \$1000 every day. How do you express the rate at which your money grows?
4
votes
2answers
39 views

prove $\max\{x,y\} = (x+y+|x - y|)/2$

Prove the following, $\max\{x,y\}=(x+y+|x-y|)/2$ Attempt at the proof: first off I started by separating the expression into the following, $(x+y)/2 + |x-y|/2 $ and noting that both of them are an ...
1
vote
3answers
114 views

Why does the same equation have different results?

I bring the equation in (1) in order to ilustrate what I mean. Since (1.1) $12 - 6$ (1.2) $(4*3) - (2*3)$ (1.3) $(4-2) * (3)$ (1.4) $(2) * (3)$ (1.5) $6$ So... (2.1) $9.999... - 0.999...$ ...
-1
votes
2answers
41 views

If g$ (x) = \max |y^2-xy|$ then minimum value of $g (x)$ is? [closed]

If g$ (x) = \max |y^2-xy|$ where (0 <=y <= 1) then minimum value of $g (x)$ (for real x ) is ? Any suggestions for this sum ? I couldn't solve
2
votes
1answer
45 views

A conditional inequality which itself implies a sharper version of it [duplicate]

Problem: Given that $m, n$ are positive integers such that $\sqrt{7} -\frac{m}{n} > 0$. Then show that $\sqrt{7}-\frac{m}{n} > \frac{1}{mn}$. I have failed to do this fascinating problem. My ...
1
vote
1answer
27 views

Can this be done without substitution of values?

If $${ \left( 1-{ x }^{ 3 } \right) }^{ n }=\sum _{ r=0 }^{ n }{ { a }_{ r }{ x }^{ r }{ \left( 1-x \right) }^{ 3n-2r } } $$ then find $a_r$. My first attempt: I wrote the above equation as: ...
0
votes
0answers
36 views

tricky completing the square

$$\frac{4k_{0}p_{1}+p_{0}\left ( k_{0}^{2}-2k_{0}k_{1}+k_{1}^{2} \right )}{\left ( k_{0}+k_{1} \right )^{2}p_{0}}$$ I need to show that this is equal to $1$ but for my life I can't figure how to ...
2
votes
2answers
73 views

What can we do to solve the following equation with $6$ variables with some information provided?

Q) There are unique integers $a_2, a_3, a_4, a_5, a_6, a_7$ such that $$\frac{a_2}{2!}+\frac{a_3}{3!}+\frac{a_4}{4!}+\frac{a_5}{5!}+\frac{a_6}{6!}+\frac{a_7}{7!}=\frac 57$$,where $0\le a_i < i$. ...
0
votes
1answer
57 views

How to prove that $a^x-b^x\leq(a-b)^x$ [closed]

It is also known that $a>0$, $b>0$, $a>b$ and $0< x <1$.
-1
votes
6answers
96 views

Can I turn 2 numbers to one number?? [closed]

I have two different numbers A and B, each one range from 0 to 255, I want to do some math and add the data of the two numbers to one number C, then if need the numbers A and B back again I can ...