Answer:
90% confidence the Population mean number of texts per day
(42.2561 ,47.1439)
Step-by-step explanation:
Step(i):-
Given sample size 'n' = 147
mean of the sample size x⁻ = 44.7
standard deviation of the sample 'S' = 17.9
90% confidence the Population mean number of texts per day
[tex](x^{-} - t_{\alpha } \frac{S}{\sqrt{n} } ,(x^{-} + t_{\alpha } \frac{S}{\sqrt{n} })[/tex]
Step(ii):-
Degrees of freedom
ν=n-1=147-1=146
t₀.₁₀ = 1.6554
[tex](x^{-} - t_{\alpha } \frac{S}{\sqrt{n} } ,(x^{-} + t_{\alpha } \frac{S}{\sqrt{n} })[/tex]
[tex](44.7 - 1.6554 \frac{17.9}{\sqrt{147} } ,(44.7 + 1.6554 \frac{17.9}{\sqrt{147} })[/tex]
(44.7 - 2.4439 ,44.7 + 2.4439 )
(42.2561 ,47.1439)
Conclusion:-
90% confidence the Population mean number of texts per day
(42.2561 ,47.1439)
Find the minimum and maximum values of 3 sin^2x – 2 cos^2x + 9
The cost in dollars y of producing x computer
desks is given by y = 40x + 4000
X
100
200
300
a. Complete the table
y
b. Find the number of computer desks that can be produced for $6200. (Hint: Find x when y = 6200.)
a. Complete the table.
х
100
200
300
y
b. For $6200,_ computer desks can be produced
Answer:
a.
y= 40x +4000
x= 100 --> y= 40(100)+4000= 4000+4000=8000
x=200 --> y= 40(200)+4000= 6000+4000= 10000
x=300 --> y= 40(300)+4000= 12000+4000= 16000
(in $)
b.
y= 40x+4000
6200= 40x+4000
6200-4000= 40x
2200= 40x
2200/40= x
55= x
(in unit)
Step-by-step explanation:
I hope this helps
if u have question let me know in comments ^_^
Which point is located at (5, –2)?
Explanation:
The origin is the center of the grid. This is where the x and y axis meet. The location of this point is (0,0).
Start at the origin and move 5 places to the right. Note how the x coordinate is 5 which tells us how to move left/right. Positive x values mean we go right.
Then we go down 2 spots to arrive at point D. We move down because the y coordinate is negative.
You could also start at (0,0) and go down 2 first, then to the right 5 to also arrive at point D. Convention usually has x going first as (x,y) has x listed first.
Answer:
Point D is located at (5, -2)
Step-by-step explanation:
The coordinates are in the form of (x,y) so that means the point has the x value of 5 and the y value of -2
Help! Marking as brainlyest
What is the effect on the graph of the function () = 1/ when () is replaced with 1/2() + 7? A) compressed vertically and shifted 7 units up B) stretched vertically and shifted 7 units down C) compressed vertically and shifted 7 units left D) stretched vertically and shifted 7 units right
Answer:
Step-by-step explanation:
I used x instead of ()
The initial function is:
● x = 1
The function after the changes is
● (1/2)x + 7
The function was shifted 15 unit to the left
Annie has 3/2 pounds of cookie dough. If she needs 1/16 of a pound of cookie dough to make one cookie, how many cookies can she make
Answer:
[tex]\boxed{\sf 24\ cookies}[/tex]
Step-by-step explanation:
1 cookie = 1/16 of a pound of cookie
If we want to find how many cookies can be made by 3/2 pounds ( 1.5 pounds) then we need to divide 3/2 pounds by 1/16
=> [tex]\frac{3}{2} / \frac{1}{16}[/tex]
=> [tex]\frac{3}{2} * 16[/tex]
=> 3*8
=> 24 cookies
Answer:
24 cookies
Step-by-step explanation:
3/2= 1.5 and 1/16= 0.0625
if you divide the amount of dough you have by the amount needed for each cookie you will have 24
1.5/0.0625=24
Explain the difference between using the sine ratio to solve for a missing angle in a right triangle versus using the cosecant ratio. You must use complete sentences and any evidence needed (such as an example) to prove your point of view. (10 points)
Answer:
The sine ratio is the ratio between the opposite side over hypotenuse. The cosecant ratio is the ratio between the hypotenuse over the opposite side, therefore cosecant is the reciprocal of sine.
To find a missing angle using sine, you would need to use the inverse of sine. For example, if the sine was [tex]\frac{30}{40}[/tex], to find the angle you would need to find sin⁻¹ of [tex]\frac{30}{40}[/tex] which is x = sin⁻¹ (0.75). Therefore x equals approximately 49°.
3. Solve for x2=81 C. 10
Answer:
9
Step-by-step explanation:
9 x 9 = 81
Answer:
x = ±9
Step-by-step explanation:
x^2 = 81
Take the square root of each side
sqrt(x^2 ) = ±sqrt(81)
x = ±9
Use mathematical induction to prove the statement is true for all positive integers n. The integer n3 + 2n is divisible by 3 for every positive integer n.
Answer:
Prove:
Using 1
n³+2n = (1)³+2(1) = 1+2= 3 ---> 3/3= 1 ✔
Using 2
n³+2n = (2)³+2(2)= 8+4=12 --> 12/3=4✔
Using 3
n³+2n= (3)³+2(3)= 27+6= 33 --> 33/3=11✔
So it is proven that n³+2n is divisible by 3 for every positive integer.
I hope this helps
if u have question let me know in comments
Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.) ln(x) = 1 /x − 3
Answer:
x ≈ {0.653059729092, 3.75570086464}
Step-by-step explanation:
A graphing calculator can tell you the roots of ...
f(x) = ln(x) -1/(x -3)
are near 0.653 and 3.756. These values are sufficiently close that Newton's method iteration can find solutions to full calculator precision in a few iterations.
In the attachment, we use g(x) as the iteration function. Since its value is shown even as its argument is being typed, we can start typing with the graphical solution value, then simply copy the digits of the iterated value as they appear. After about 6 or 8 input digits, the output stops changing, so that is our solution.
Rounded to 6 decimal places, the solutions are {0.653060, 3.755701}.
_____
A similar method can be used on a calculator such as the TI-84. One function can be defined a.s f(x) is above. Another can be defined as g(x) is in the attachment, by making use of the calculator's derivative function. After the first g(0.653) value is found, for example, remaining iterations can be g(Ans) until the result stops changing,
I don’t understand this, may I get some help?
●✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎❀✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎●
Hi my lil bunny!
❧⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯☙
80,023 written in scientific notation is [tex]8,0023[/tex] x [tex]10^4[/tex].
Step 1
To find a, take the number and move a decimal place to the right one position.
Original Number: 80,023
New Number: 8.0023
Step 2
New Number: 8 . 0 0 2 3
Decimal Count: 1 2 3 4
Now, to find b, count how many places to the right of the decimal.
There are 4 places to the right of the decimal place.
Step 3
Building upon what we know above, we can now reconstruct the number into scientific notation.
Remember, the notation is: a x 10^b
a = 8.0023
b = 4
Now the whole thing:
8.0023 x 104
Step 4
Check your work:
10^4 = 10,000 x 8.0023 = 80,023
❧⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯☙
●✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎❀✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎●
Have a great day/night!
❀*May*❀
Suppose that prices of a certain model of new homes are normally distributed with a mean of $150,000. Use the 68-95-99.7 rule to find the percentage of buyers who paid: between $150,000 and $152,400 if the standard deviation is $1200.
A. 68%
B. 99.7%
C. 47.5%
D. 34%
Answer:
C. 47.5%
Step-by-step explanation:
The summary of the given statistics include:
mean =150000
standard deviation: 1200
The objective is to use tributed with a mean of $150,000. Use the 68-95-99.7 rule to find the percentage of buyers who paid: between $150,000 and $152,400
The z score formula can be use to calculate the percentage of the buyer who paid.
[tex]z = \dfrac{X - \mu}{\sigma}[/tex]
For the sample mean x = 150000
[tex]z = \dfrac{150000 - 150000}{1200}[/tex]
[tex]z = \dfrac{0}{1200}[/tex]
z = 0
For the sample mean x = 152400
[tex]z = \dfrac{152400 - 150000}{1200}[/tex]
[tex]z = \dfrac{2400 }{1200}[/tex]
z = 2
From the standard normal distribution tables
P(150000 < X < 152400) = P(0 < z < 2 )
P(150000 < X < 152400) =P(z<2) -P(z<0)
P(150000 < X < 152400) =0.9772 -0.5
P(150000 < X < 152400) = 0.4772
P(150000 < X < 152400) = 47.7% which is close to 47.5% therefore option C is correct
This question is based on concept of statistics. Therefore, correct option is C i.e. 47.5% of buyers who paid: between $150,000 and $152,400 if the standard deviation is $1200.
Given:
Mean is $150,000, and
Standard deviation is $1200.
We need to determined the percentage of buyers who paid: between $150,000 and $152,400 as per given mean and standard deviation.
By using z score formula can be use to calculate the percentage of the buyer who paid,
[tex]\bold{z=\dfrac{x-\mu }{\sigma}}[/tex]
As given in question sample mean i.e. X= 150,000
[tex]z=\dfrac{150000-150000}{1200} \\\\z= \dfrac{0}{1200}\\\\z=0[/tex]
Now for the sample mean X = 152,400 ,
[tex]z=\dfrac{152400-150000}{1200} \\\\\\z= \dfrac{24000}{1200}\\\\\\z=2[/tex]
By using standard normal distribution table,
P(150000 < X < 152400) = P(0 < z < 2 )
P(150000 < X < 152400) =P(z<2) -P(z<0)
P(150000 < X < 152400) =0.9772 -0.5
P(150000 < X < 152400) = 0.4772
P(150000 < X < 152400) = 47.7% which is close to 47.5%
Therefore, correct option is C that is 47.5%.
For further details, please prefer this link:
https://brainly.com/question/23907081
What is the relationship between factorising and expanding?
Answer:
The relation ship is both are opposites
Step-by-step explanation:
so what is factorising ???
factorizing is like this example : 4x+32 = 4(x+8)
so u take the expression make it factorized or shorter or in a way that you multiply them .
what is expanding well its the opposite
suck as 4(x+8)=4x+32
Express the product of z1 and z2 in standard form given that [tex]z_{1} = -3[cos(\frac{-\pi }{4} )+isin(\frac{-\pi }{4} )][/tex] and [tex]z_{2} = 2\sqrt{2} [cos(\frac{-\pi }{2} )+isin(\frac{-\pi }{2} )][/tex]
Answer:
Solution : 6 + 6i
Step-by-step explanation:
[tex]-3\left[\cos \left(\frac{-\pi }{4})\right+i\sin \left(\frac{-\pi }{4}\right)\right]\cdot \:2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi }{2}\right)\right][/tex]
This is the expression we have to solve for. Now normally we could directly apply trivial identities and convert this into standard complex form, but as the expression is too large, it would be easier to convert into trigonometric form first ----- ( 1 )
( Multiply both expressions )
[tex]-6\sqrt{2}\left[\cos \left(\frac{-\pi }{4}+\frac{-\pi \:\:\:}{2}\right)+i\sin \left(\frac{-\pi \:}{4}+\frac{-\pi \:\:}{2}\right)\right][/tex]
( Simplify [tex]\left(\frac{-\pi }{4}+\frac{-\pi }{2}\right)[/tex] for both [tex]\cos \left(\frac{-\pi }{4}+\frac{-\pi }{2}\right)[/tex] and [tex]i\sin \left(\frac{-\pi }{4}+\frac{-\pi }{2}\right)[/tex] )
[tex]\left(\frac{-\pi }{4}+\frac{-\pi }{2}\right)[/tex] = [tex]\left(-\frac{3\pi }{4}\right)[/tex]
( Substitute )
[tex]-6\sqrt{2}\left(\cos \left(-\frac{3\pi }{4}\right)+i\sin \left(-\frac{3\pi }{4}\right)\right)[/tex]
Now that we have this in trigonometric form, let's convert into standard form by applying the following identities ----- ( 2 )
sin(π / 4) = √2 / 2 = cos(π / 4)
( Substitute )
[tex]-6\sqrt{2}\left(-\sqrt{2} / 2 -i\sqrt{2} / 2 )[/tex]
= [tex]-6\sqrt{2}\left(-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i\right)[/tex] = [tex]-\frac{\left(-\sqrt{2}-\sqrt{2}i\right)\cdot \:6\sqrt{2}}{2}[/tex]
= [tex]-3\sqrt{2}\left(-\sqrt{2}-\sqrt{2}i\right)[/tex] = [tex]-3\sqrt{2}\left(-\sqrt{2}\right)-\left(-3\sqrt{2}\right)\sqrt{2}i[/tex]
= [tex]3\sqrt{2}\sqrt{2}+3\sqrt{2}\sqrt{2}i:\quad 6+6i[/tex] - Therefore our solution is option a.
60feet to meters plaese with work
Answer:
60 Feet = 18.288 Meters
Step-by-step explanation:
foot = 12 inch = 0.3048 m
0.3047 × 60
18.288 meters
If f(x)=2x-6and g(x)=3x+9 find (f+g)(x)
Answer:
(f+g)(x) = 5x + 3
Step-by-step explanation:
(f+g)(x) is the sum (term by term) of f(x) and g(x):
(f+g)(x) = 2x - 6 + 3x + 9
Combining like terms, we get
(f+g)(x) = 5x + 3
Answer:
(f+g)(x)= 5x+3
Step-by-step explanation:
The question asks us to find (f+g)(x). In other words, the sum of f(x) and g(x).
f(x) + g(x)
We know that f(x)= 2x-6 and g(x)=3x+9. Therefore, we can substitute the expressions in.
(2x-6) + (3x+9)
Now, simplify by combining like terms. Add the terms with variables, then the terms without variables.
(2x+3x) + (-6+9)
Add 2x and 3x.
5x + (-6 + 9)
Add -6 and 9.
5x + 3
If f(x)=2x-6and g(x)=3x+9, then (f+g)(x) is 5x+3
Simply. Who ever answers this will be marked Brainlist.
Answer:
Step-by-step explanation:
Hello,
[tex]r^3s^{-2}\cdot 8r^{-3}s^4\cdot 4rs^5\\\\=r^{3-3+1}s^{-2+4+5}\cdot 8\cdot 4\\\\\boxed{=32\cdot r\cdot s^7}[/tex]
Hope this helps.
Do not hesitate if you need further explanation.
Thank you
Will mark Brainliest! A stick has a length of $5$ units. The stick is then broken at two points, chosen at random. What is the probability that all three resulting pieces are longer than $1$ unit?
Answer:
0.16
Step-by-step explanation:
Length = 5 unitsNumber of broken sticks= 3Equal lengths = 5 units/3See the picture attached for reference.
As you see the best points are the green areas which covers 2 out of 5 zones.
Since it is same for both broken points, the probability of this is:
2/5*2/5 = 4/ 25 = 0.16Answer is 0.16
2. You are going to produce tennis shoes
that come in 3 different colors. In order to
decide how many to make in each color,
you conduct a survey. Of the 300 people
you survey, 75 said that they would
purchase the yellow shoes. If you are
going to make 10,000 pairs of shoes, how
many should be yellow?
Please help thank you
Answer:
Hey there!
[tex]\frac{75}{300}[/tex]=[tex]\frac{x}{10000}[/tex]
750000=300x
x=2500
They should make 2500 yellow shoes.
Hope this helps :)
It takes amy 8 minutes to mow 1/6 of her backyard. At that rate how many more minutes will it take her to finish mowing her backyard
Answer:
40 minutes
Step-by-step explanation:
If it takes her 8 minutes to mow 1/6 of it, we can find the total amount of time it will take by multiplying 8 by 6, since 1/6 times 6 is 1 (1 represents the whole lawn mowed)
8(6) = 48
The question asks for how many more minutes it will take, so subtract 48 by 8.
48 - 8 = 40
= 40 minutes
Answer:
40 minutes
Step-by-step explanation:
We can use ratios to solve
8 minutes x minutes
------------------- = ----------------
1/6 yard 1 yard
Using cross products
8 * 1 = 1/6 x
Multiply each side by 6
8*6 = 1/6 * x * 6
48 = x
48 minutes total
She has already done 8 minutes
48-8 = 40 minutes
What is the correct answer and how can this be solved?
Answer:
[tex]$\mathbf{\frac{1}{19} }[/tex]
Step-by-step explanation:
[tex]$$\bullet \Nth \ Term;\\$$$\frac{n+2}{2n^{2} +3n-2}[/tex]
[tex]$$\bullet U_{10} \ Term;\\\\$$\boxed{\frac{(10+2) }{2*10^{2} +3*10-2}= \frac{1}{19} }[/tex]
Answer:
[tex]\boxed{\displaystyle \frac{1}{19}}[/tex]
Step-by-step explanation:
[tex]\displaystyle \frac{n+2}{2n^2 +3n-2}[/tex]
Replace n with 10 to find the 10th term.
[tex]\displaystyle \frac{10+2}{2(10)^2 +3(10)-2}[/tex]
Evaluate.
[tex]\displaystyle \frac{12}{2(100) +30-2}[/tex]
[tex]\displaystyle \frac{12}{200 +30-2}[/tex]
[tex]\displaystyle \frac{12}{228}[/tex]
Simplify.
[tex]\displaystyle \frac{1}{19}[/tex]
What are the slope and y-intercept of the equation 2x - 5y = -10?
Answer:
Step-by-step explanation:
y=2/5x+2
x= 5/2y-5
hopefully this works
The domain of the following relation has how many elements?
[(1/2, 3.14/6), (1/2, 3.14/4), (1/2, 3.14/3), (1/2,3.14/2)]
a. 0
b. 1
c. 4
Answer:
b. 1
Step-by-step explanation:
All first coordinates are 1/2.
Answer: b. 1
For some postive value of Z, the probability that a standardized normal variable is between 0 and Z is 0.3770. The value of Z is
Answer:
1.16
Step-by-step explanation:
Given that;
For some positive value of Z, the probability that a standardized normal variable is between 0 and Z is 0.3770.
This implies that:
P(0<Z<z) = 0.3770
P(Z < z)-P(Z < 0) = 0.3770
P(Z < z) = 0.3770 + P(Z < 0)
From the standard normal tables , P(Z < 0) =0.5
P(Z < z) = 0.3770 + 0.5
P(Z < z) = 0.877
SO to determine the value of z for which it is equal to 0.877, we look at the
table of standard normal distribution and locate the probability value of 0.8770. we advance to the left until the first column is reached, we see that the value was 1.1. similarly, we did the same in the upward direction until the top row is reached, the value was 0.06. The intersection of the row and column values gives the area to the two tail of z. (i.e 1.1 + 0.06 =1.16)
therefore, P(Z ≤ 1.16 ) = 0.877
The area of a rectangular garden if 6045 ft2. If the length of the garden is 93 feet, what is its width?
Answer:
65 ft
Step-by-step explanation:
The area of a rectangle is
A = lw
6045 = 93*w
Divide each side by 93
6045/93 = 93w/93
65 =w
Answer:
[tex]\huge \boxed{\mathrm{65 \ feet}}[/tex]
Step-by-step explanation:
The area of a rectangle formula is given as,
[tex]\mathrm{area = length \times width}[/tex]
The area and length are given.
[tex]6045=93 \times w[/tex]
Solve for w.
Divide both sides by 93.
[tex]65=w[/tex]
The width of the rectangular garden is 65 feet.
A population has a standard deviation of 16. If a sample of size 64 is selected from this population, what is the probability that the sample mean will be within 2 of the population mean?
a. Since the mean is not given, there is no answer to this question.
b. -0.6826
c. 0.3413
d. 0.6826
e. -0.3413
Answer:
The correct option is D
Step-by-step explanation:
From the question we are told that
The standard deviation is [tex]\sigma = 16[/tex]
The sample size is n = 64
The standard error of mean is mathematically evaluated as
[tex]\sigma _{\= x } = \frac{\sigma }{\sqrt{n} }[/tex]
substituting values
[tex]\sigma _{\= x } = \frac{16 }{\sqrt{64} }[/tex]
[tex]\sigma _{\= x } = 2[/tex]
Generally the probability that the sample mean will be within 2 of the population mean is mathematically represented as
[tex]P( \mu - 2 < \= x < \mu + 2) = P(\frac{( \mu - 2 ) - \mu }{\sigma_{\= x }} < \frac{ \= x - \mu }{\sigma_{\= x }} < \frac{( \mu +2 ) - \mu }{\sigma_{\= x }} )[/tex]
Generally [tex]\frac{ \= x - \mu }{\sigma_{\= x }} = Z (The \ standardized \ value \ of \ \= x )[/tex]
So
[tex]P( \mu - 2 < \= x < \mu + 2) = P(\frac{( \mu - 2 ) - \mu }{\sigma_{\= x }} < Z< \frac{( \mu +2 ) - \mu }{\sigma_{\= x }} )[/tex]
[tex]P( \mu - 2 < \= x < \mu + 2) = P(\frac{( -2 }{\sigma_{\= x }} < Z< \frac{ 2 }{\sigma_{\= x }} )[/tex]
substituting values
[tex]P( \mu - 2 < \= x < \mu + 2) = P(\frac{-2 }{2} < Z< \frac{ 2 }{2} )[/tex]
[tex]P( \mu - 2 < \= x < \mu + 2) = P(-1< Z< 1 )[/tex]
=> [tex]P( \mu - 2 < \= x < \mu + 2) = P(Z < 1) - P(Z < -1)[/tex]
From the normal distribution table [tex]P(Z < 1 ) = 0.84134[/tex]
[tex]P(Z < - 1) = 0.15866[/tex]
=> [tex]P( \mu - 2 < \= x < \mu + 2) = 0.84134 - 0.15866[/tex]
=> [tex]P( \mu - 2 < \= x < \mu + 2) = 0.6826[/tex]
An observer standing on a cliff 320 feet above the ocean measured angles of depression of the near and far sides of an island to be 16.5 and 10.5 respectively. How long is the island ?
Answer:
154.10 Feets
Step-by-step explanation:
Given the following :
Height (h) of cliff = 320 feet
Angle of depression of near side = 16.5°
Angle of depression of far side = 10.5°
Using trigonometry :
We can obtain x and y as shown in the attached picture :
Tanθ = opposite / Adjacent
Adjacent = height of cliff = 320 Feets
For the near side :
Tanθ = opposite / Adjacent
Tan (16.5°) = x / 320
0.2962134 = x / 320
x = 0.2962134 * 320
x = 94.788318 Feets
For the far side :
Tanθ = opposite / Adjacent
Tan (10.5°) = x / 320
0.1853390 = x / 320
x = 0.1853390 * 320
x = 59.308494 Feets
Length of island = (59.308494 + 94.788318) feet
= 154.10 Feets
16.50 and pays 20.00 in cash the change due is
Answer:
Change due is 3.50
Step-by-step explanation:
20.00-16.50 is 3.50
Answer: $3.50
Step-by-step explanation:
You subtract 20 from 16.50.
If m(x) =x+5/x-1 and n(x) = x - 3, which function has the same domain as (mºn)(x)?
We have
M(X) = (X + 5)/(X - 1)
N(X) = X - 3
So,
M(N(X)) = [(X - 3) + 5]/[(X - 3) - 1]
M(N(X)) = [X + 2]/[X - 4]The M(N(X)) domain will be:
D = {X / X ≠ 4}
4 ∉ to the M(N(X)) domain, otherwise we would have a/0, which is not possible (a denominator with zero). An equivalent function would be
H(X) = 1/(X - 4)
you write a short story, but you want to make sure your work is protected before you post it online. what should you do to help protect your copyright?
Answer:
Hey there!
Here are a few steps:
Make sure your work is properly marked, because then it will be protected under law.
Register your work.
Keep or register supporting evidence.
Let me know if this helps :)
EXAMPLE 5 If F(x, y, z) = 4y2i + (8xy + 4e4z)j + 16ye4zk, find a function f such that ∇f = F. SOLUTION If there is such a function f, then
If there is such a scalar function f, then
[tex]\dfrac{\partial f}{\partial x}=4y^2[/tex]
[tex]\dfrac{\partial f}{\partial y}=8xy+4e^{4z}[/tex]
[tex]\dfrac{\partial f}{\partial z}=16ye^{4z}[/tex]
Integrate both sides of the first equation with respect to x :
[tex]f(x,y,z)=4xy^2+g(y,z)[/tex]
Differentiate both sides with respect to y :
[tex]\dfrac{\partial f}{\partial y}=8xy+4e^{4z}=8xy+\dfrac{\partial g}{\partial y}[/tex]
[tex]\implies\dfrac{\partial g}{\partial y}=4e^{4z}[/tex]
Integrate both sides with respect to y :
[tex]g(y,z)=4ye^{4z}+h(z)[/tex]
Plug this into the equation above with f , then differentiate both sides with respect to z :
[tex]f(x,y,z)=4xy^2+4ye^{4z}+h(z)[/tex]
[tex]\dfrac{\partial f}{\partial z}=16ye^{4z}=16ye^{4z}+\dfrac{\mathrm dh}{\mathrm dz}[/tex]
[tex]\implies\dfrac{\mathrm dh}{\mathrm dz}=0[/tex]
Integrate both sides with respect to z :
[tex]h(z)=C[/tex]
So we end up with
[tex]\boxed{f(x,y,z)=4xy^2+4ye^{4z}+C}[/tex]
