Answers
First, I'll make f(x) = sin(px) + cos(px) because this expression shows up quite a lot, and such a substitution makes life a bit easier for us.
Let's apply the first derivative of this f(x) function.
[tex]f(x) = \sin(px)+\cos(px)\\\\f'(x) = \frac{d}{dx}[f(x)]\\\\f'(x) = \frac{d}{dx}[\sin(px)+\cos(px)]\\\\f'(x) = \frac{d}{dx}[\sin(px)]+\frac{d}{dx}[\cos(px)]\\\\f'(x) = p\cos(px)-p\sin(px)\\\\ f'(x) = p(\cos(px)-\sin(px))\\\\[/tex]
Now apply the derivative to that to get the second derivative
[tex]f''(x) = \frac{d}{dx}[f'(x)]\\\\f''(x) = \frac{d}{dx}[p(\cos(px)-\sin(px))]\\\\ f''(x) = p*\left(\frac{d}{dx}[\cos(px)]-\frac{d}{dx}[\sin(px)]\right)\\\\ f''(x) = p*\left(-p\sin(px)-p\cos(px)\right)\\\\ f''(x) = -p^2*\left(\sin(px)+\cos(px)\right)\\\\ f''(x) = -p^2*f(x)\\\\[/tex]
We can see that f '' (x) is just a scalar multiple of f(x). That multiple of course being -p^2.
Keep in mind that we haven't actually found dy/dx yet, or its second derivative counterpart either.
-----------------------------------
Let's compute dy/dx. We'll use f(x) as defined earlier.
[tex]y = \ln\left(\sin(px)+\cos(px)\right)\\\\y = \ln\left(f(x)\right)\\\\\frac{dy}{dx} = \frac{d}{dx}\left[y\right]\\\\\frac{dy}{dx} = \frac{d}{dx}\left[\ln\left(f(x)\right)\right]\\\\\frac{dy}{dx} = \frac{1}{f(x)}*\frac{d}{dx}\left[f(x)\right]\\\\\frac{dy}{dx} = \frac{f'(x)}{f(x)}\\\\[/tex]
Use the chain rule here.
There's no need to plug in the expressions f(x) or f ' (x) as you'll see in the last section below.
Now use the quotient rule to find the second derivative of y
[tex]\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right]\\\\\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{f'(x)}{f(x)}\right]\\\\\frac{d^2y}{dx^2} = \frac{f''(x)*f(x)-f'(x)*f'(x)}{(f(x))^2}\\\\\frac{d^2y}{dx^2} = \frac{f''(x)*f(x)-(f'(x))^2}{(f(x))^2}\\\\[/tex]
If you need a refresher on the quotient rule, then
[tex]\frac{d}{dx}\left[\frac{P}{Q}\right] = \frac{P'*Q - P*Q'}{Q^2}\\\\[/tex]
where P and Q are functions of x.
-----------------------------------
This then means
[tex]\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + p^2\\\\\frac{f''(x)*f(x)-(f'(x))^2}{(f(x))^2} + \left(\frac{f'(x)}{f(x)}\right)^2 + p^2\\\\\frac{f''(x)*f(x)-(f'(x))^2}{(f(x))^2} +\frac{(f'(x))^2}{(f(x))^2} + p^2\\\\\frac{f''(x)*f(x)-(f'(x))^2+(f'(x))^2}{(f(x))^2} + p^2\\\\\frac{f''(x)*f(x)}{(f(x))^2} + p^2\\\\[/tex]
Note the cancellation of -(f ' (x))^2 with (f ' (x))^2
------------------------------------
Let's then replace f '' (x) with -p^2*f(x)
This allows us to form ( f(x) )^2 in the numerator to cancel out with the denominator.
[tex]\frac{f''(x)*f(x)}{(f(x))^2} + p^2\\\\\frac{-p^2*f(x)*f(x)}{(f(x))^2} + p^2\\\\\frac{-p^2*(f(x))^2}{(f(x))^2} + p^2\\\\-p^2 + p^2\\\\0\\\\[/tex]
So this concludes the proof that [tex]\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + p^2 = 0\\\\[/tex] when [tex]y = \ln\left(\sin(px)+\cos(px)\right)\\\\[/tex]
Side note: This is an example of showing that the given y function is a solution to the given second order linear differential equation.
Related Questions
The price of the cinemas meal deal has increased from £7.25 to £9.28.
What is the percentage change in price?
Answers
ejheheksjshdhdhgdgege is a irtel of the main thing is mute the main 22sec and jen best friends movie is mute the main thing for kids are vaccine safe to make a diy boat 6main thing for kids are vaccine the main thing for kids are vaccine safe 6to make a diy boat main 6thing for kids are vaccine safe 66G PHONE is a diy plane to the 7main thing for kids are vaccine
Answer:
8.28% increase i believe
Step-by-step explanation:
formula: final amount - initial ÷ initial amount.
9.28-7.25= 2.03. now divide by the initial amount. 2.03÷7.25=8.28.
so, an 8.28% increase to the price
Solve 9(7x-2) = 5 (10x - 1)
Answers
Answer:
[tex]9(7x - 2) = 5(10x - 1) \\ \\ 63x - 18 = 50x - 5 \\ \\ 63x - 50x = - 5 + 18 \\ \\ 13x = - 5 + 18 \\ \\ 13x = 13 \\ \\ x = \frac{13}{13} \\ \\ x = 1[/tex]
Answer:
[tex]9(7x - 2) = 5(10x - 1) \\ 9 \times 7x - 9 \times 2 = 5 \times 10x - 5 \\ 63x - 18 = 50x - 5 \\ 63x - 50x = - 5 + 18 \\ 13x = 13 \\ x = 1 \\ thank \: you[/tex]
Write the equation of a line in slope intercept form that passes through the two points. (-1,3) and (2,9)
Answers
Answer:
[tex]y = 2x +5[/tex]
Step-by-step explanation:
Finding the Slope:
m = rise/run
[tex]m=\frac{9-3}{2-(-1)}=\frac{6}{3}=\boxed{2}[/tex]
The slope is 2.
Finding the y-intercept:
[tex]y = 2x + b\\\\9 = 2(2) + b\\\\9 = 4 + b\\\\9 - 4 = 4 - 4 + b\\\\\boxed{ 5 = b}[/tex]
The y-intercept is (0,5).
The equation should be: [tex]y = 2x +5[/tex].
Hope this helps.
Answer:
y = 2x+5
Step-by-step explanation:
To find the equation in slope intercept form, we first have to find the slope of the two points.
The formula for finding the slope is:
[tex]\frac{y2-y1}{x2-x1}[/tex]
We are given the points:
(-1, 3) and (2, 9)
x1, y1 and x2, y2.
[tex]\frac{9-3}{2-(-1)}[/tex]
[tex]\frac{6}{3}[/tex] or 2 (the slope).
The slope intercept form is written as:
y= mx + b
y=2x+b
To find b, we can plug in either of the points given. Personally, I like to work with positive numbers. I'll be using the point (2, 9), however, either point will get you to the same answer.
y=2x+b
9=2(2)+b
9=4+b
Now subtract 4 from both sides.
5=b
Which then leaves us with the equation:
y = 2x+5
There is a language with 12 distinct letters. How many words that are less than four letters long can possibly be formed? (Assume any combination of letters is valid and letters can be repeated). Can you help me out with this one?
Answers
Answer:
1884
Step-by-step explanation:
1)If you hear the condition "less than four letters" it means that each word can consist of 3,2 or 1 letter. Firstly, consider the easiest situation with the word from one letter. There are 12 letters, so there are 12 words from one letter.
Then the words from two letters. The first letter of such a word can be chosen in 12 means(because 12 letters are available), then the second letter can be chosen in 12 means too. 12*12=144 words with 2 letters. (Multiplying, not adding, because each two words form one combination).
Then for the words from three letters, use the same rule, 12 means for the first letter, 12 means for the second letter, 12 means for the third letter. 12*12*12=1728 words.
Having particular words from 1,2,3 letters add them to get total quantity of words: 12+144+1728=1884words,that is the answer.
If y varies directly as x, and x = 4 when y = 20, find y when x = 18.
Answers
Step-by-step explanation:
y=Kx
20=4k
k=5
when x=18
y=18 x 5
y=90
I need help with just 1 and 2
Answers
1. I think in Brooklyn, Manhattan, and the other different cities in New Jersey, the rents are overpriced.
2. Some factors are if it looks nice, there are no bugs or rodents, and if it's not a rip off.
**I can't answer the second part bcuz i don't see the tables.
What is the Square root of 30 ,12,36
Answers
Answer:
the square root of:
30: 5.477225575
12: 3.464101615
36: 6 <-- explanation of square root: any number that when multiplied by itself equals your wanted number (in this case 36) will be the square root of your wanted number.
Graph 5 and it’s opposite on the number line
Answers
Answer:
See pic below.
Step-by-step explanation:
The opposite of 5 is -5.
help i’ll give brainliest
Answers
explination: the x-intercept is where the line touches the X line kn the graph, the line intercepts at -4
Answer:
(0,-4)
Step-by-step explanation:
The plot for x intercept is at -4,0 and the y intercept is at 0,5
type it in as (0,-4), some require the (#,#) format
which shape is both a parallelogoram and a rhombus
Answers
Answer:
A square can be defined as a rhombus which is also a rectangle and a parellelogram with four congruent sides and four right angles
I hope this helps
Mrs Lee had $7500 in her bank account. The bank paid 4% interest at the
end of each year. How much did she have in the bank at the end of 1 year?
Answers
Answer:
$7800
Step-by-step explanation:
1. Principal x interest x rate
So: $7500 + 4% (0.04) x 1 year = $300
2. Interest + Principal
So: $7500 + $300
Mrs Lee had $7800 in the bank.
NO LINKS OR ANSWERING QUESTIONS YOU DON'T KNOW. Find each measurement. Round your answers to the nearest tenth. Part 2dd
Answers
Answer:
see explanation
Step-by-step explanation:
Using the Sine rule in all 3 questions
[tex]\frac{a}{sinA}[/tex] = [tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex]
(2)
[tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex] , substitute values
[tex]\frac{45}{sin133}[/tex] = [tex]\frac{c}{sin26}[/tex] ( cross- multiply )
c × sin133° = 45 × sin26° ( divide both sides by sin133° )
c = [tex]\frac{45sin26}{sin133}[/tex] ≈ 27.0 ( to the nearest tenth )
(4)
[tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex] , substitute values
[tex]\frac{19}{sinB}[/tex] = [tex]\frac{30}{sin97}[/tex] ( cross- multiply )
30 sinB = 19 sin97° ( divide both sides by 30 )
sinB = [tex]\frac{19sin97}{30}[/tex] , then
∠ B = [tex]sin^{-1}[/tex] ( [tex]\frac{19sin37}{30}[/tex] ) ≈ 38.9° ( to the nearest tenth )
(6)
[tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex], substitute values
[tex]\frac{18}{sin102}[/tex] = [tex]\frac{xAB}{sin45}[/tex] ( cross- multiply )
AB sin102° = 18 sin45° ( divide both sides by sin102° )
AB = [tex]\frac{18sin45}{sin102}[/tex] ≈ 13.0 ( to the nearest tenth )
The graph below represents which of the following functions?
Answers
Answer:
hiuu ui9i io9
Step-by-step explanation:
iu9uj k
Một công ty sản xuất một thùng hình trụ tròn không có nắp và có thể tích 8x ft Chi phí sản xuất đáy là 2 USD / ft ?, chi phí sản xuất mặt bên là 4,8 USD / ft . Hỏi chi phí tối ưu nhất để sản xuất cái thằng này là bao nhiêu USD ?,
Answers
Answer:
nsjdjdnajakla ,bxn. dnsksksvhdjeke dkekdkbdjdls d
a store sells pencils pens and markers that sells two times as many markers as pencils and three times as many pens as pencils is the store sells a total of 1950 pencils and pens and markers in a week how many of each were sold
Answers
Answer:
Pencils = 325 ; Pens = 975 ; Markers = 650
Step-by-step explanation:
Let :
Number of Pencils = x
Number of pens = y
Number of markers = z
2 times as many markers as pencils
z = 2x
3 times as many pens as pencils
y = 3x
x + y + z = 1950
Write z and y in terms of x in the equation :
x + 3x + 2x = 1950
6x = 1950
Divide both sides by 6
6x / 6 = 1950 / 6
x = 325
Number of pencils = 325
Pens = 3 * 325 = 975
Markers = 2 * 325 = 650
Pencils = 325 ; Pens = 975 ; Markers = 650
190 of 7
6 7 8 9 10
-3
4
5
6
The slope of the line shown in the graph is
and the intercept of the line is
Answers
Answer:slope 2/3
Y-int 6
Step-by-step explanation:
n eight-sided die, which may or may not be a fair die, has four colors on it; you have been tossing the die for an hour and have recorded the color rolled for each toss. What is the probability you will roll a blue on your next toss of the die
Answers
Answer:
The answer is "21%".
Step-by-step explanation:
[tex]\to 26 + 20 + 29 + 49 = 124\\\\=\frac{26\ \text{brown times}}{124 \ \text{total times}} \\\\= \frac{26}{ 124} \\\\ = 0.209677 \\\\[/tex]
Calculating the percentage:
[tex]= 20.9677 \times 100\\\\=20.9677\% \approx 21\%[/tex]
HELP PLEASE! INEEED HELPPP RIGHT NOW The graph of y=x−2−−−−√ is is transformed to become y=x+3−−−−√−2. Which of the following statements best describes the effect this transformation has on the graph of y=x−−√?
Answers
Answer:
I think it's D but I don't know how to do this, but I ask my Friend and he gave me this graph I hope it helps.
Option (C) the graph is translated 5 units left and 2 units down is the correct answer.
What is graph of a function?A graph can be defined as a pictorial representation or a diagram that represents data or values in an organized manner. The graph of a function f is the set of all points in the plane of the form (x, f(x)).
For the given situation,
The function is [tex]y=\sqrt{x-2}[/tex]
This function is transformed to become the function as [tex]y=\sqrt{x+3}-2[/tex]
Now compare this transformation with the function [tex]y=\sqrt{x}[/tex].
These functions are plotted in the graph below.
On comparing this transformation,
The graph is translated 5 units left and 2 units down.
Hence we can conclude that option (C) the graph is translated 5 units left and 2 units down is the correct answer.
Learn more about graph of a function here
https://brainly.com/question/2275360
#SPJ2
Volunteering: The General Social Survey asked 1309 people whether they performed any volunteer work during the past year. A total of 518 people said they did. Part: 0 / 30 of 3 Parts Complete Part 1 of 3 (a) Find a point estimate for the population proportion of people who performed volunteer work in the past year. Round the answer to at least three decimal places. The point estimate for the proportion of people who performed volunteer work in the past year is .
Answers
Answer:
The point estimate for the population proportion of people who performed volunteer work in the past year is 0.396.
Step-by-step explanation:
Point estimate of a proportion:
Proportion is the number of desired outcomes divided by the number of total outcomes.
518 out of 1309 people performed volunteer work:
This means that:
[tex]p = \frac{518}{1309} = 0.396[/tex]
The point estimate for the population proportion of people who performed volunteer work in the past year is 0.396.
Let f(x, y) = 2xy − 3xy + y + 1. Find the derivative of f with respect to y.
Answers
Answer:
2x-3x+1
Step-by-step explanation:
f'(x,y)=2x-3x+1
If an 8" X 8" square cake serves 8 people, how many 24" X 24" square cakes are needed for a
wedding attended by 144 people?
Answers
Answer:
2 24" X 24" square cakes are needed.
Step-by-step explanation:
This question is solved by proportions.
Area of a square:
The area of a square of side l is given by:
[tex]A = l^2[/tex]
8" X 8" square cake serves 8 people
Thus:
[tex]A = 8*8 = 64[/tex]
Area of 64 serves 8 people, so for 1 people, an area of 64/8 = 8 is needed.
144 people
The total area needed is:
[tex]A = 144*8 = 1152[/tex]
How many 24" X 24" square cakes?
24*24 = 576
1 cake - 576
x cakes - 1152
Applying cross multiplication:
1152/576 = 2
2 24" X 24" square cakes are needed.
Determine the remaining sides and angles of the triangle ABC.
c=6 mi, B = 38.71°, C = 32.51°
Find the measure of angle A.
A=°
(Type an integer or a decimal.)
Find the length of side a.
а:
mi
(Round to the nearest mile as needed.)
Find the length of side b.
b=mi
(Round to the nearest mile as needed.)
Answers
9514 1404 393
Answer:
A = 108.78°
a = 11 mi
b = 7 mi
Step-by-step explanation:
The sum of angles in a triangle is 180°, so the third angle is ...
A = 180° -38.71° -32.51°
A = 108.78°
__
The remaining sides can be found from the law of sines.
a/sin(A) = c/sin(C)
a = sin(A)·c/sin(C) ≈ 0.946762 × 11.163896
a ≈ 11 mi
b = sin(B)·11.163896 ≈ 0.625379 × 11.163896
b ≈ 7 mi
solve the following equations
Answers
Answer:
x=0,5/2
Step-by-step explanation:
how do you change 79/8 to a decimal
Answers
Answer: 79/8=9.875
Answer:
Divide 79 by 8
Step-by-step explanation:
79/8
= 9.875
20 POINTS! The answer is x=4 and x= -5, how should I word it, saying how he is wrong?
Answers
Answer:
The student took the numbers for the factors 5, -4 for the zeros instead of solving the equation
The zeros are -5, 4
Step-by-step explanation:
x^2 +x - 20=0
What 2 number multiply to -20 and add to 1
5*-4 = -20
5+-4 = 1
(x+5)(x-4) =0
Using the zero product property
x+5 = 0 x-4 =0
x=-5 x=4
Answer:
Solution given:
equation is:
x²+x-20=0
doing middle term factorisation
note:we need to get 1 while subtracting factor of the product of constant and coefficient of x².
20*1=20=2*5*2*1
we get 1 while subtracting 5-2*2=5-4
now substitute value 5-4 at coefficient of x
we get
x²+(5-4)x-20=0
now
distribute
x²+5x-4x-20=0
taking common from each two term
x(x+5)-4(x+5)=0
again taking common (x+5) and keeping remaining at another bracket
(x+5)(x-4)=0
either
x+5=0
x=-5
or
x-4=0
x=4
Error is:
x= 4 not -4
x=-5 not 5.
(b) In a company, out of 20 candidates 14 men and 6 women apply for two vacancies. What is the probability that (1) both men are selected ? (ii) both women are selected ? (iii) one man and one woman are selected ?
Answers
i hope my answer helps you and you will give me the best answer
solve 3x-4=√(2x^2-2x+2)
Answers
Answer:
Step-by-step explanation:
Begin the solution by squaring both sides of the given equation. We get:
(3x - 4)^2 = 2x^2 - 2x + 2, or:
9x^2 - 24x + 16 = 2x ^2 - 2x + 2
Combining like terms results in:
7x^2 - 22x + 14 = 0
and the coefficients are a = 7, b = -22, c = 14, so that the discriminant of the quadratic formula, b^2 - 4ac becomes (-22)^2 - 4(7)(14) = 92
According to the quadratic formula, the solutions are
-b ± √discriminant -(-22) ± √92 22 ± √92
x = ------------------------------- = ----------------------- = ------------------------
2a 14 14
Write an equation for the graph below in terms of x
Answers
Answer:
x = -2
Step-by-step explanation:
the line goes through -2 on the x-axis
the line also goes through -3 on the y-axis
what is the square root of 24
Answers
Answer:
the square root of 24 is 4.89897948557
Answer:
4.89
Step-by-step explanation:
2root 6........***
g U is the set of in the United States. A is the set of in the United States that have the in their name. B is the set of in the United States that have the in their name. Describe the set in words. Choose the correct answer below. A. is the set of in the United States that have the . B. is the set of in the United States that have the . C. is the set of in the United States that have the . D. is the set of in the United States that have the .
Answers
Question isn't well formatted picture if actual question is attached below :
Answer:
A' set of colleges in the United States which Don not have the word community in their name.
Step-by-step explanation:
If a universal set U is defined as :
U =set of colleges in the United States.
A = the set of colleges in the United States that have the word community in their name.
B is the set of colleges in the United States that have the name of an entertainer in their name.
The set A' is called the A complement. The complement of a particular refers to elements in the universal set which are not in A ;
A' = 1 - A'
Therefore, A' set of colleges in the United States which Don not have the word community in their name.
