(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]
HELP ILL MAKE BRAINLIESTT
A seamstress charges a $5 fee per garment, plus $15 for each hour of work that the garment requires. The graph below shows the cost of having a single garment fixed by the seamstress, according to the number of hours of work that are needed.
image
Which linear function models the data?
A. c(t) = 5t
B. c(t) = 15t
C. c(t) = 5 + 15t
D. c(t) = 15 + 5t
Answer:
where is the graph though?
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
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:
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
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 :)
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
Triangle PQR is shown. The triangle has a base of 4 in and a height a 2.5 in.
Which measurement is closest to the area of triangle PQR in square inches?
5 in2.
8 3/4in.2
6 1/2in.2
11in.2
Answer:
A
Step-by-step explanation:
Area = 1/2 * b * h
b = 4 in
h = 2.5 inches
Area = 1/2 * 4 * 2.5
Area = 2 * 2.5
Area = 5 in^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
Triangle B = 10cm
H = 4cm
area =
Answer:
I think it's 20
Step-by-step explanation:
10x4÷2 because vase times height,the divide by 1/2 which it 2 so answer is 20
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
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
-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
What is measurement angle B?
Answer: 80 degrees
Step-by-step explanation:
Answer:
B = 54 to the nearest degree.
Step-by-step explanation:
Remark
This is a sine law question. They can be rather nasty because no answer or 2 answers are possible. This one is well behaved because B, you are told, is acute. The two answer kind needs the supplement of an acute angle.
Sorry this blabs on and shouldn't. You are looking for <B
Formula
Sin(<B)/the side opposite = Sin(<A) / side opposite it
Simplified
Sin(<B) / 9 = Sin(A)/11
Sin(<B) / 9 = Sin(80) / 11 Multiply both sides by 9
9*Sin(B) / 9 = 9 * Sin(80) /11 The 9s Cancel
Now all you need to do is put it in your calculator. Do you need the steps?
Sin(B) = 9*sin(80) / 11
Sin(B) = 9 * 0.9848 /11
Sin(B)= 0.8058
B = Sin-1(0.8058)
B = 53.68
B = 54
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 = $5740
a warehouse received a shipment of 700 cartons of raspberries, 900 cartons of blueberries, and 1200 cartons of strawberries. If the produce manager wants to check the quality of the fruit which method would yield the best results?
Answer:
2,800
Step-by-step explanation:
700+900+1200=2800
9+6+4+4+4+4+4+4+44+444+44444+444+4+4+4+4+4+4+4+44+44+4=?
Answer:
45535
Step-by-step explanation:
Answer:
45415
Step-by-step explanation:
Using the coordinate plane shown below, a third point is plotted 5 units to the right of (-4, 4). Where should a fourth point be plotted in order to create a rectangle?
(1, -4)
(1, 4)
(4, -1)
(-4, 1)
Answer:The answer is (1,-4).
Step-by-step explanation: Please mark this brainliest.
What type of number is 0?
Select each correct answer.
integer
I rational
irrational
real
O natural
whole
Answer:
0 is a rational number, whole number, integer, and a real number.
Step-by-step explanation:
Which number line represents the solution 6x<42 ?
Answer:
7
Step-by-step explanation:
+2($($;$)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
Graph the following features:
Y-intercept = 4
Slope = 2
Answer:
y=2x+4
Step-by-step explanation:
the sum of two numbers is 61 the smaller number is 21 less than the larger number what are the numbers
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.
Please help giving brainliest
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?
Sarah took clothes to the cleaners three times last month. First, she brought 2 shirts and 1 of slacks and paid $9.97. Then she brought 5 shirts, 3 pairs of slacks, and 1 sports coat and paid $34.91. Finally, she brought 3 shirts and 2 sports coats and paid $22.45.
How much was she charged for each shirt, each pair of slacks, and each sports coat?
Answer:
Each Shirt costs: $1.49
Each Slack costs: $4.49
Each Sport coat costs: $7.99
three times the quotient of 3 and 5
Three times the quotient of 3 and 5 is equivalent to 1.8
The question asked us to convert the word problem to equation and solve.
The quotient of 3 and 5 is expressed as 3/5
Taking the product of the result and 3 will give;
Three times the quotient of 3 and 5 = 3 * 3/5
Three times the quotient of 3 and 5 = 9/5 = 1.8
Hence three times the quotient of 3 and 5 is equivalent to 1.8
Learn more on expressions here: https://brainly.com/question/723406
#SPJ6
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.
10
Favorite Book Types
3%
Romance
Comedy
16%
Fantasy
14%
Science
Fiction
1896
Mystery
10%
Historical
Fiction
If 300 people were surveyed. How many people like comedy as their favorite book? *
(10 Points)
O 39
O 183
O 117
O 54
Answer:
117
Step-by-step explanation:
By adding the percentages given: 3+16+14+18+10
you get 61%
multiply 300 by .61 or 61%
you get 183
subtract 183 from 300
you are left with 117 which is the answer