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1. ### MHB Justify each step using commutativity and associativity in each step. (a-b)+(c-d) = (a+c)+(-b-d)

Looks good to me.
2. ### MHB Does anyone know the answer (s) to this?

Here's the derivative for the function of power, P=Av^2+\frac{B}{v} (\text{Where }A\text{ and }B\text{ are positive constants)} that they give in the introduction to the problem: P'(v)=2Av-\frac{B}{v^2} To answer \text{(a) What speed }vP\text{ minimizes power?} we set this expression...
3. ### MHB How to show this result?

That depends on what a, b and c are.
4. ### MHB Determine the area of a region between two curves defined by algebraic functions

yeah should have zoomed the page... lol... :)
5. ### MHB Need help on these two questions

In terms of getting useful replies, it would probably be best if you showed us where in the two problems you are having difficulty and how we can help. If you need to refer directly to the math in either of the two problems, please be quote it directly in your post. (clicking on the .pdf link...
6. ### MHB Summing Series Limit as N→∞

$\lim_{N\rightarrow\infty}\frac{x}{2^n}\sin^2\left(\frac{x}{2^n}\right)\rightarrow0\times0^2\rightarrow0$ but as the limit is taken over positive $x$ the limit tends to infinity.
7. ### MHB How do i improve my Prealgebra ?

There are a lot of online tutoring sites and which ones you may be interested in are are likely to be the ones concerning mathematics you want to study. https://khanacademy.org is one for elementary topics up to college/university topics.. https://desmos.com has a good online graphing calculator.
8. ### MHB Determine the area of a region between two curves defined by algebraic functions

You need the roots $a,b$ of the equation $3\sqrt{x}-4=3x\sqrt{5}-\frac{8}{5}$ Once you have established these roots use them as endpoints in $\int_{a}^{b}\left(3\sqrt{x}-4-3x\sqrt{5}+\frac{8}{5}\right)dx$ The result is the area $A$ of $R$.
9. ### MHB Sin(y) - y = x√(2)

In general, periodic functions are of interest due to their frequent occurrence in natural phenomenon. As speculation, this particular function may be of interest due the times and places it occurs.
10. ### MHB Quadratic With No Real Zeros

Note the concavity of $f(x)$. What does that tell you about lowering the graph of $f(x)$? (the line $y=0$ does not concern us presently).
11. ### MHB Quadratic With No Real Zeros

Note the range of $f(x)$. A and B are vertical shifts, C and D are horizontal shifts. Which one of the given shifts would result in the graph of $f(x)$ not crossing the $x-\text{axis}$?
12. ### MHB Find Value of c

What do you get when you expand the LHS and equate the resulting terms with the corresponding terms in the RHS of the given equation
13. ### MHB Solve Situational Problems Involving Trigonometric Identities

First, we need to establish $\sin\theta$ and $\cos\theta$. $9^2+(-5)^2=106$ (Pythagorean theorem) so $\sin\theta$ is $\sqrt{\frac{|-5|}{106}}, \text{that is}, \left(\frac{opp}{hyp}\right)$ and $\cos\theta$ is $\frac{3}{\sqrt{106}}, \text{that is}, \left(\frac{adj}{hyp}\right)$ (recall that...
14. ### MHB Equation of Normal to Curve at (1,5): Solved?

Country Boy Is correct in stating that the slope of the tangent line is 2. So the tangent line equation is y = 2x + 3 and the equation of the normal is then y = -x/2 + 11/2

Haha!
16. ### MHB Precalculus: What is the value of this sigma notation?

Yes, I agree. My error was missing i = 30 and assuming i = 1 .
17. ### MHB Precalculus: What is the value of this sigma notation?

Oh! Those definitions are implied... :poop: f(i)=1.8. and g(i)=1.2 both suffice as definitions for f,\,g if I am not mistaken... After a few basic calculations we may arrive at: \frac{420-90+600}{2}=465. Do you see that too? Hint: use the fact that summation is associative and sum each addend...
18. ### MHB Precalculus: What is the value of this sigma notation?

Do you have definitions for f and g?
19. ### MHB Square feet calculations assistance

Actually, I've never seen it before...and not everyone here is necessarily as "smart" as you might be assuming they are. I've edited my initial post in this thread. Please review it and post accordingly, if you will. :)
20. ### MHB Square feet calculations assistance

Hi riffwraith. Please use accurate terminology in your thread titles. The use of "ftg" is ambiguous as, as far as I know, it is not standard terminology. In fact, please try to avoid abbreviations altogether. I've edited the title for you. .

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22. ### MHB 1.6.365 AP Calculus Exam Limits

You need to look at left-hand and right-hand limits. I don't see where "$\pm$" comes into it.
23. ### MHB [ASK] Proving Trigonometry 2sinα+2sinβ+2sinγ=4sinαsinβsinγ

\begin{align*}2(\sin2x+\sin2y+\sin2z)&=\sin x\cos(y-z)+\sin y\cos(x-z)+\sin z\cos(x-y) \\ &=6\sin x\sin y\sin z+\sin x\cos y\cos z+\sin y\cos x\cos z+\sin z\cos x\cos y \\ &=6\sin x\sin y\sin z+\sin x\cos x+\sin y\cos y+\sin z\cos z \\ \frac32(\sin2x+\sin2y+\sin2z)&=6\sin x\sin y\sin z \\...
24. ### MHB [ASK] Proving Trigonometry 2sinα+2sinβ+2sinγ=4sinαsinβsinγ

If $x+y+z=\pi$ then \sin2x+\sin2y+\sin2z=4\sin x\sin y\sin z
25. ### MHB Euler/Riemann Point of Departure in Riemann's 1859 paper containing RH

In his 1859 paper entitled "On the Number of Primes Less than a Given Magnitude", Riemann gives as his point of departure the equation \prod\frac{1}{1-\frac{1}{p^s}}=\sum\frac{1}{n^s} where $p$ is all primes and $n$ is all natural numbers. The function of the complex variable $s$, wherever...
26. ### MHB 2.1.2 AP calculus Exan particle move along the x-axis

$p'(t)=v(t),\quad v'(t)=a(t)$
27. ### MHB 2.1.2 AP calculus Exan particle move along the x-axis

Hmmm... $p(t)=3t^2-\frac{t^3}{3}+C$ (Wondering)
28. ### MHB Solve x^3-7x^2+14x-8=0

(a-b)^3=a^3-3a^2b+3ab^2-b^3 a^3-b^3=3a^2b-3ab^2+(a-b)^3 a^3-b^3=3ab(a-b)+(a-b)^3 a^3-b^3=(a-b)(3ab+(a-b)^2) a^3-b^3=(a-b)(3ab+a^2-2ab+b^2) a^3-b^3=(a-b)(a^2+ab+b^2)
29. ### MHB Solve x^3-7x^2+14x-8=0

\begin{align*}x^3-7x^2+14x-8&=x^3-8-7x(x-2) \\ &=(x-2)(x^2+2x+4)-7x(x-2) \\ &=(x-2)(x^2+2x+4-7x) \\ &=(x-2)(x^2-5x+4) \\ &=(x-2)(x-1)(x-4)\end{align*}
30. ### MHB AP calculus exam tikx graph of e (tan x ) -2

Observing that $y=e^{\tan x} - 2$ has a root in [0, 1] at $\text{atan}(\log2)$, we need to evaluate $2\sec^2(\text{atan}(\log2))$. That is approximately 2.961, hence choice D is correct.
31. ### MHB Golden section and yin-yang symbol proportions

What angle are you referring to? What do you mean by an "exact" proof?

33. ### MHB De value of second derivative

Re: 231 value of second dirivative See https://mathhelpboards.com/calculus-10/297-ap-calculus-exam-2nd-derivative-26690.html.
34. ### MHB 2.1.312 AP Calculus Exam Int of half circle

u^2=25-x^2 -u\,du=x\,dx \int_0^5u^2\,du=\frac{125}{3}-0=\frac{125}{3}
35. ### MHB 3.3.04 AP Calculus Exam 2nd derivative

(x+2y)\frac{dy}{dx}=2x-y x\frac{dy}{dx}+2y\frac{dy}{dx}=2x-y \frac{dy}{dx}+x\frac{d^2y}{dx^2}+2\left(\frac{dy}{dx}\right)^2+2y\frac{d^2y}{dx^2}=2-\frac{dy}{dx} 2+3\frac{d^2y}{dx^2}+8=0 \frac{d^2y}{dx^2}=-\frac{10}{3}
36. ### MHB Need solution for limit

Maybe the Dirac delta function?
37. ### MHB Value of b, y-intercept of Quadratic graph

y=(x - 2)(x - 4)=x^2-6x+8

Hey, it's not an equation it's an expression! Here's what an online CAS thinks.
39. ### MHB 6.6.60 limiit possible L'H

The limit is with respect to $x$, so $a$ and $b$ are treated as constants. Getting things into a form where we can apply L'Hopital's rule, \]\exp\left(\frac{\log\left(1+\frac ax\right)}{\frac{1}{bx}}\right) Now *differentiate and simplify* inside the brackets; you'll end up with $e^{ab}$.
40. ### MHB 6.6 inverse functions

\sin^2x+\cos^2x=1 1+\cot^2x=\csc^2x \cot^2x-\csc^2x=-1
41. ### MHB 4.2.236 AP calculus Exam integral with u substitution

u=e^{\sqrt x} 2\,du=\frac{e^{\sqrt x}}{\sqrt x}\,dx 2\int_e^{e^2}\,du=2e(e-1)
42. ### MHB Solving Algebra equation 3x=15

3x = 15 x = 15/3 x = 5
43. ### MHB Year 10 Maths Find the length and width that will maximize the area of rectangle

W=\frac AL \frac{5A}{L}+2L=550 5A+2L^2=550L A=110L-\frac{2L^2}{5} $A$ has a maximum at the vertex of this inverted parabola, so $L=\frac{275}{2}$. Finding $A$ and $W$ from here should be straightforward.
44. ### MHB Year 10 Maths Find the length and width that will maximize the area of rectangle

Here's a start: Let $W$ be width, $L$ be length an $A$ be the desired area. Then, 5W+2L=550 LW=A Can you make any progress from there?

46. ### MHB 2.1.207 AP calculus practice exam problem Lim with tan(4X)/6x

\frac{4x}{4x}\cdot\frac{\tan4x}{6x}=\frac{4x}{4x}\cdot\frac{\sin4x}{6x\cdot\cos{4x}}=\frac{4x}{6x}\cdot\frac{\sin4x}{\cos4x\cdot4x} \frac23\lim_{x\to0}\frac{\sin4x}{\cos4x\cdot4x}=\frac23
47. ### MHB Simplify Surds 10/√5 + √20

Hello Mooija and welcome to MHB. :) √20 = √4√5 = 2√5 10/√5 = 10√5/5 = 2√5
48. ### MHB 1.1.205 AP calculus exam practice question

See stuff in red - otherwise looks good. :) Use ... tags.
49. ### MHB 2.2.206 AP Calculus Practice question derivative of a composite sine

g(x)=\log(2x)=\log(2)+\log(x),\quad g'(x)=\frac1x or, by the chain rule, g'(x)=\frac{2}{2x}=\frac1x
50. ### MHB 85 find circle from 3 points

(-1-h)^2+(3-k)^2=(6-h)^2+(2-k)^2=(-2-h)^2+(-4-k)^2 (-1-h)^2-(6-h)^2+(3-k)^2-(2-k)^2=0 -7(5-2h)+(5-2k)=0 -35+14h+5-2k=0 -30+14h-2k=0\quad (-1-h)^2-(-2-h)^2+(3-k)^2-(-4-k)^2=0 (-3-2h)+7(-1-2k)=0 -10-2h-14k=0\quad +7*\implies k=-1\implies h=2 (x-2)^2+(y+1)^2=25