(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of −∇f(x). Let θ be the angle between ∇f(x) and unit vector u. Then Du f = |∇f| cos θ _________ . Since the minimum value of cos θ _________ is -1 ________ occurring, for 0 ≤ θ < 2π, when θ = ________ , the minimum value of Du f is −|∇f|, occurring when the direction of u is the opposite of _______ the direction of ∇f (assuming ∇f is not zero). (b) Use the result of part (a) to find the direction in which the function f(x, y) = x4y − x2y3 decreases fastest at the point (2, −5)._________
Answer:
Step-by-step explanation:
[tex]\text{Show that a differentiable function f decreases most rapidly at x in the }[/tex]
[tex]\text{direction opposite the gradient vector, that is, in the direction of}[/tex] [tex]-\bigtriangledown f(x)[/tex][tex]\text{. Let}\ \theta \ \text{be the angle between} \bigtriangledown f(x) \ \text{and unit vector u. Then } D_u f = \mathbf{|\bigtriangledown f| \ cos \ \theta }}[/tex]
[tex]\text{Since the minimum value of} \ \ \mathbf{cos \ \theta} \ \ is \mathbf{-1} \ \text{occuring \ for \ 0} \le \ \theta \ < 2x, \\ \\ when \ \theta = \mathbf{\pi} , \text{the mnimum value of} \ D_uf \ is} -|\bigtriangledown f|, \text{occuring when the direction of u is } \\ \\ \ \mathbf{the \ opposite \ of} \ \text{the direction of } \ \bigtriangledown f (assuming \ \bigtriangledown f\ is \ not \ zero)[/tex]
b) [tex]\text{From part A:}[/tex]
[tex]If \ f(x,y) = x^4y -x^2y^2 \ \ decreases \ fastest \ at \ the \point \ (2,-5)\\ \\ F(x,y) = x^4y -x^2y^3 \\ \\ f_x = \dfrac{df}{dx}= \dfrac{d}{dx}(x^4y-x^2y^3) \\ \\ f_x = \dfrac{df}{dx}= y4x^3 -2y^3x \\ \\ For(2,-5) \\ \\ f_x = (-5)4(2)^3 -2(-5)^3(2) \\ \\ \mathbf{ f_x = 340}[/tex]
[tex]However; f_y = \dfrac{df}{dy} = \dfrac{d}{dy}(x^4y - x^2y^3) \\ \\ f_y = x^4 -3x^2y^2 \\ \\ Now, for (2, -5)\\ \\f_y = (2)^4 -3(2)^2(-5)^2 \\ \\ f_y = -284[/tex]
[tex]So; \bigtriangledown = < 340,-284> \text{this is the direction of fastest decrease}[/tex]
The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 50 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.3 pounds, what is the probability that the sample mean will be each of the following? a. More than 59 pounds b. More than 56 pounds c. Between 56 and 57 pounds d. Less than 53 pounds e. Less than 49 pounds *Round the values of z to 2 decimal places. Round your answer to 4 decimal places, the tolerance is +/-0.0001. **Round the values of z to 2 decimal places. Round your answer to 4 decimal places. a. * b. * c. * d. * e. **
Answer:
a. 0.1038 = 10.38% probability that the sample mean is more than 59 pounds.
b. 0.6772 = 67.72% probability that the sample mean is more than 56 pounds.
c. 0.2210 = 22.10% probability that the sample mean is between 56 and 57 pounds.
d. 0.0146 = 1.46% probability that the sample mean is less than 53 pounds.
e. 0% probability that the sample mean is less than 49 pounds.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. The population standard deviation of annual usage is 12.3 pounds.
This means that [tex]\mu = 56.8, \sigma = 12.3[/tex]
Sample of 50:
This means that [tex]n = 50, s = \frac{12.3}{\sqrt{50}} = 1.74[/tex]
a. More than 59 pounds
This is 1 subtracted by the pvalue of Z when X = 59.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{59 - 56.8}{1.74}[/tex]
[tex]Z = 1.26[/tex]
[tex]Z = 1.26[/tex] has a pvalue of 0.8962
1 - 0.8962 = 0.1038
0.1038 = 10.38% probability that the sample mean is more than 59 pounds.
b. More than 56 pounds
This is 1 subtracted by the pvalue of Z when X = 56. So
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{56 - 56.8}{1.74}[/tex]
[tex]Z = -0.46[/tex]
[tex]Z = -0.46[/tex] has a pvalue of 0.3228
1 - 0.3228 = 0.6772
0.6772 = 67.72% probability that the sample mean is more than 56 pounds.
c. Between 56 and 57 pounds
This is the pvalue of Z when X = 57 subtracted by the pvalue of Z when X = 56.
X = 57
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{57 - 56.8}{1.74}[/tex]
[tex]Z = 0.11[/tex]
[tex]Z = 0.11[/tex] has a pvalue of 0.5438
X = 56
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{56 - 56.8}{1.74}[/tex]
[tex]Z = -0.46[/tex]
[tex]Z = -0.46[/tex] has a pvalue of 0.3228
0.5438 - 0.3228 = 0.2210
0.2210 = 22.10% probability that the sample mean is between 56 and 57 pounds.
d. Less than 53 pounds
This is the pvalue of Z when X = 53.
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{53 - 56.8}{1.74}[/tex]
[tex]Z = -2.18[/tex]
[tex]Z = -2.18[/tex] has a pvalue of 0.0146
0.0146 = 1.46% probability that the sample mean is less than 53 pounds.
e. Less than 49 pounds
This is the pvalue of Z when X = 49.
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{49 - 56.8}{1.74}[/tex]
[tex]Z = -4.48[/tex]
[tex]Z = -4.48[/tex] has a pvalue of 0
0% probability that the sample mean is less than 49 pounds.
9+6+4+4+4+4+4+4+44+444+44444+444+4+4+4+4+4+4+4+44+44+4=?
Answer:
45535
Step-by-step explanation:
Answer:
45415
Step-by-step explanation:
PLEASE PLEASE!! HELP
93 POINTS
Answer:
a) -150 feet per minute
b)-11,250 feet
c)240 minutes
Step-by-step explanation:
-3000/20 = -150 ft per minute
b) -150 times 75
c)-36000= -150x
5.
The Luxor Hotel in Las Vegas, Nevada,
is shaped like a square pyramid whose
surface is covered in dark glass. The
sides of the building are 606 feet in
length. The slant height of the building
is 463 feet. Find the lateral surface area
to determine the square feet of glass
needed to cover the surface of the Luxor
Hotel.
Answer:
= 869,516 sq feet
i think will you mark me brainlest
=))
Step-by-step explanation:
Which fractions are less than 2/3? *
Answer:
I dont see any options....
But 1/4, 1/5, 1/6, 1/7, 2/7, etc
Step-by-step explanation:
Answer:
a lot but here are a few
Step-by-step explanation:
1/3 3/6 2/6 1/6 7/12 6/12 5/12 4/12 3/12 2/12 1/12 16/24 15/24 14/24 13/24 12/24 11/24 10/24 9/24 8/24 7/24 6/24 5/24 4/24 3/24 2/24 and 1/24
What are the coordinates of the x-intercept of the line 2x-y=6
a warehouse received a shipment of 700 cartons of raspberries, 900 cartons of blueberries, and 1200 cartons of strawberries. If the produce manager wants to check the quality of the fruit which method would yield the best results?
Answer:
2,800
Step-by-step explanation:
700+900+1200=2800
Using the coordinate plane shown below, a third point is plotted 5 units to the right of (-4, 4). Where should a fourth point be plotted in order to create a rectangle?
(1, -4)
(1, 4)
(4, -1)
(-4, 1)
Answer:The answer is (1,-4).
Step-by-step explanation: Please mark this brainliest.
WHAT IS X ? PLEASE HELP
Answer:
x = 44
Step-by-step explanation:
x + x + 14 + 2x - 10 = 180
4x + 4 = 180
4x = 180 - 4
4x = 176
x = 176 / 4
x = 44
Hope that helps!
Which number line represents the solution 6x<42 ?
Answer:
7
Step-by-step explanation:
+2($($;$)2:
hope it helps
find the solution(s)
y= x + 1
y= x² - 4x + 5
Answer:
1. Many possible solutions, two solutions would be: (-1,0) and (1,0)
2. x = −
2
± i
Step-by-step explanation:
For the first equation, since its linear, there are many possible solutions!
two solutions would be: (-1,0) and (1,0)
For the second one, replace y with 0 and solve
x = −2 ± i
Hope i helped, brainliest would be appreciated
Have a nice day!
-Aadi x
Please help! NO FAKE ANSWERS PLEASE!!!!!
There are 20 cookies in a bowl and 19 of them are chocolate. What percentages of the cookies are chocolate?
A student earned a grade of 80% on a math test that had 35 problems. How
many problems on this test did the student get correct?
Round your answer to the nearest whole.
Answer:
28
Step-by-step explanation:
Answer:
The student got 28 questions correct and 7 wrong.
Graph the following features:
Y-intercept = 4
Slope = 2
Answer:
y=2x+4
Step-by-step explanation:
Tim bought 4 packs of football cards for $3.97 each, and a deck of
Digimon cards for $6.11. How much did Tim spend on cards?
Answer: $21.99
Step-by-step explanation:
First, we need to solve for the total cost of 4 packs of football cards
We can either do: $3.97+$3.97+$3.97+$3.97
or
$3.97 x 4
This will equal: $15.88
Now we need to add $6.11 to $15.88
$6.11+$15.88
$21.99
The length off a swimming pool cover is (2x+1), the width is (x+8). Find the area of the swimming pool as a polynomial in standard form
How many permutations are there of the letters in the following word, if all the letters are used without repetition?
ADMIRES
Give the answer using permutation notation, factorial notation, or other operations. Then evaluate.
ADMIRES is the word
Answer:
5040 permutations
Step-by-step explanation:
In AMNO, m = 56 cm, n = 19 cm and o=57 cm. Find the measure of Zo to the nearest
10th of a degree.
Answer:83.3∘
Step-by-step explanation:
The measure of angle O from the triangle MNO is 156.92°.
What is cosine rule?In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
The formula for the cosine rule is c=√(a²+b²-2ab cosC)
here, we have,
Given that,
in triangle MNO,
m = 720 cm, n = 160 cm and o=870 cm.
Now, 870=√(720²+160²-2×720×160 cosO)
870²=720²+160²-2×720×160 cosO
756900=544000-230400 cosO
212900=-230400 cosO
cosO= -212900/230400
cosO= -0.92
∠O=156.92°
Therefore, the measure of angle O from the triangle MNO is 156.92°.
Learn more about the cosine rule here:
brainly.com/question/26952873.
#SPJ7
i’m confused and need help fast please.
Answer:
C) 48.3
Step-by-step explanation:
The lines are parallel, meaning that when the third line is intersecting them both, the angles formed will be exactly the same. The top angle that is to the right is 25°, meaning that he bottom angle to the left is 25°. All that is left is to find the last two angles in order to find out what the answer to the equation is. The 25° angle is next to a triangle causing the lines to be flat, or 180°.
180 - 25 = 155
That means the bottom angle to the right is 155. In order to find x, subtract 10 and divide by 3.
3x + 10 - 10 = 155 - 10
3x = 135
3x/3 = 135/3
x = 48.3
Use the Graphical Method to solve the following linear programming problems.Mr. Trenga plans to start a new business called Rivers Explorers, which will rent canoes & kayaks to people to travel 10 miles down the Clarion River in Cook Forest State Park. He has $45,000 to purchase new boats (canoes & kayaks). He can buy canoes for $600 each & kayaks for $750 each. His facility can hold up to 65 boats. The canoes will rent for $25 per day & the kayaks will rent for $30 per day. How many canoes & how many kayaks should he buy to earn the most revenue (rental income)?
Answer:
To maximize his earnings, Mr. Trenga must purchase 40 kayaks and 25 canoes.
Step-by-step explanation:
Since Mr. Trenga plans to start a new business called Rivers Explorers, which will rent canoes & kayaks to people to travel 10 miles down the Clarion River in Cook Forest State Park, and he has $ 45,000 to purchase new boats (canoes & kayaks) , and he can buy canoes for $ 600 each & kayaks for $ 750 each, and his facility can hold up to 65 boats, if the canoes will rent for $ 25 per day & the kayaks will rent for $ 30 per day, to determine how many canoes & how many kayaks should he buy to earn the most revenue, the following calculation should be performed:
65 x 750 = 48,750
48,750 - 45,000 = 3,750
750 - 600 = 150
3,750 / 150 = 25
25 x 600 + 40 x 750 = 45,000
25 x 25 + 30 x 40 = 1,825
600 x 5 = 3,000
750 x 4 = 3,000
40 x 30 + 25 x 25 = 1,825
60 x 30 + 0 x 25 = 1,800
56 x 30 + 5 x 25 = 1,805
52 x 30 + 10 x 25 = 1,810
48 x 30 + 15 x 25 = 1,815
44 x 30 + 20 x 25 = 1,820
Thus, to maximize his earnings, Mr. Trenga must purchase 40 kayaks and 25 canoes.
I would really appreciate it if someone could answer this I will give a brainlist:)
Answer:
Hypotenuse= 97
Step-by-step explanation:
H² = B² + P²
H² = (65)² + (72)²
H² = 4225 + 5184
H²= 9409
Taking square root on both sides,
H= 97
Pls help!! I’ve been trining to get the answer but I can’t pls help!!
Answer:
$5740
Step-by-step explanation:
get 2.5% of 5600
5600/x=100/2.5
(5600/x)*x=(100/2.5)*x
5600=40*x
5600/40=x
140=x
x=140 <----- 2.5% of 5600
5600 + 140 = $5740
Find the missing side of this right
triangle.
12
x
8
Answer:
The missing side is 14.4
x=14.4
Answer:
80Step-by-step explanation:
Please help giving brainliest
PLZ HELP !!!!!!!Solve for the value of x and y
68
(4x - 2)
30
Answer: 10
Step-by-step explanation:
Simplifying
68 = 4x + -2 + 30
Reorder the terms:
68 = -2 + 30 + 4x
Combine like terms: -2 + 30 = 28
68 = 28 + 4x
Solving
68 = 28 + 4x
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-4x' to each side of the equation.
68 + -4x = 28 + 4x + -4x
Combine like terms: 4x + -4x = 0
68 + -4x = 28 + 0
68 + -4x = 28
Add '-68' to each side of the equation.
68 + -68 + -4x = 28 + -68
Combine like terms: 68 + -68 = 0
0 + -4x = 28 + -68
-4x = 28 + -68
Combine like terms: 28 + -68 = -40
-4x = -40
Divide each side by '-4'.
x = 10
Simplifying
x = 10
Answer:
x = 10, y = 112
Step-by-step explanation:
4x - 2 + 30 and 68 are vertical angles and are congruent , then
4x - 2 + 30 = 68
4x + 28 = 68 ( subtract 28 from both sides )
4x = 40 ( divide both sides by 4 )
x = 10
------------------------
y and 68 lie on a straight line and sum to 180° , then
y + 68 = 180 ( subtract 68 from both sides )
y = 112
PLEASEEE HELP I WILL MARK YOU BRAINIEST
Answer:
12a
3/4
34.8
a = 3.2
Step-by-step explanation:
9a + 6 = 34.8
9a = 28.8
28.8/9 = 3.2
If closing costs of $1,900 are associated with the refinance of a mortgage that would reduce the monthly payment from $1,140 to $1,072 refinance, it would take approximately ____ months to cover these costs. (Round your answer to the nearest full month.)
Answer:
28 Months
Step-by-step explanation:
(1140-1072)= 68
(1900/68)=27.9 ~ 28
-179=-9-10m can anyone solve this?
immediately
Answer:
m=17
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
m
=
17
Evaluate the expression x (3) – (3 + x)2 for x = 4.
Answer:
if x=4, the answer is -2.
Step-by-step explanation:
4x3-(3+4)2
12-7x2
12-14
= -2
i hope this helps :)