1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
oksian1 [2.3K]
2 years ago
11

Let F = ∇f, where

Mathematics
1 answer:
SIZIF [17.4K]2 years ago
3 0

Use the gradient theorem. Since F is the gradient of the scalar function f(x, y) = sin(x - 8y), the line integral of F along any path C is equal to

\displaystyle \int_C \vec F \cdot d\vec r = \int_{(a,b)}^{(c,d)} \nabla f(x, y) \cdot d\vec r = f(c,d) - f(a,b)

where (a, b) and (c, d) are any distinct points. (They have to be distinct, otherwise the path would be a closed loop or a single point.)

(a) You just need to pick two points (a, b) and (c, d) from the (x, y)-plane such that

f(c, d) - f(a, b) = 0

where the path C₁ starts at (a, b) and ends at (c, d).

Let (a, b) be the origin, (0, 0). Then we want to find c and d such that

sin(c - 8d) - sin(0 - 8•0) = sin(c - 8d) = 0

⇒   c - 8d = arcsin(0) + 2nπ   or   c - 8d = π - arcsin(0) + 2nπ

(where n is any integer)

⇒   c - 8d = 2nπ   or   c - 8d = (2n - 1)π

If we also fix d = 0, then we have infinitely many choices for c, other than c = 0 since that would make (a, b) and (c, d) the same point. So let's pick n = 1 for the first solution set, so that c = 2π.

Our choice for C₁ can be any curve we wish, but let's just take the simplest one: the line segment joining (0, 0) and (2π, 0).

(b) Same as part (a), but this time we want

f(c, d) - f(a, b) = 0

We again take (a, b) to be the origin. Then we want c and d such that

sin(c - 8d) = 1

⇒   c - 8d = arcsin(1) + 2nπ   or   c - 8d = π - arcsin(1) + 2nπ

⇒   c - 8d = π/2 + 2nπ

Again, we're free to choose d = 0. Then if n = 0, we get c = π/2, and the path C₂ could be the line segment joining (0, 0) and (π/2, 0).

You might be interested in
12. If a manufacturer conducted a survey among randomly selected target market households and wanted to be 95% confident that th
atroni [7]

Answer:

n=\frac{0.5(1-0.5)}{(\frac{0.03}{1.96})^2}=1067.11  

And rounded up we have that n=1068

Step-by-step explanation:

We have the following info given:

Confidence= 0.95 the confidence level desired

ME =0.03 represent the margin of error desired

The margin of error for the proportion interval is given by this formula:  

ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}    (a)  

The confidence level is 95% or 0.95, the significance is \alpha=0.05 and the critical value for this case using the normal standard distribution would be z_{\alpha/2}=1.96

Since we don't have prior information we can use \hat p= 0.5 as an unbiased estimator

Also we know that ME =\pm 0.03 and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}   (b)  

And replacing into equation (b) the values from part a we got:

n=\frac{0.5(1-0.5)}{(\frac{0.03}{1.96})^2}=1067.11  

And rounded up we have that n=1068

6 0
3 years ago
A rectangles width is 5 feet less than its length. Write a quadratic function that expresses the rectangles area in terms of its
Ira Lisetskai [31]

Answer:

Area = L²-5L

Step-by-step explanation:

Area = length * width

Length = L

Width = L - 5

Area = L * (L-5) = L²-5L

4 0
3 years ago
HELP PLEASE!!! Which equation can be used to calculate the area of the shaded triangle in the figure below?
kykrilka [37]
It’s the first one 44 square feet
5 0
2 years ago
Read 2 more answers
Please help me please
ella [17]

Answer:

-4??

Step-by-step explanation:

4 0
2 years ago
Jean lives about 10 miles from the college where she plans to attend a 10-week summer class. There are two main routes she can t
jeka57 [31]

Answer:

We would advise her to choose Country Route because in this route time consistent between 15 and 18 minutes.

Step-by-step explanation:

We are given that Jean lives about 10 miles from the college where she plans to attend a 10-week summer class. There are two main routes she can take to the school, one through the city and one through the countryside.

Jean sets up a randomized experiment where each day she tosses a coin to decide which route to take that day. She records the following data for 5 days of travel on each route.

Country Route - 17, 15, 17, 16, 18  (in minutes)

City Route - 18, 13, 20, 10, 16 (in minutes)

Now, we have to decide which route is better for Jean to go to college.

As we can see from the data that the Country route timings is consistent between 15 to 18 minutes which means most of the times she will reach college between these minutes only.

While on the other hand, we can observe that City route timings are very much consistent as it has a low value of 10 minutes and high value of 20 minutes which means Jean can't be sure that at which time she will reach college.

Hence, we would advise her to choose Country route.

7 0
3 years ago
Other questions:
  • 5 different mixed numbers that add up to 10 with only 3rds and 4ths
    13·1 answer
  • Name a pair of supplementary angles. Just any pair of supplementary angles. Plz
    9·1 answer
  • The currency in a certain country is called the sirktian. In this country's tax​ system, a person pays 0 sirktians on the first
    13·1 answer
  • I have $147 then I spent $42 on sneakers then i earned $45 by when I raise in my new sneakers
    6·2 answers
  • (t,-3) and (2,6); Slope = -1
    12·1 answer
  • Which value, when placed in the box, would result in a system of equations with infinitely many solutions?
    10·2 answers
  • Annora bought a box of cookies that matches the dimensions of the solid below. How many square units of cardboard were used to c
    7·1 answer
  • Make a conditional relative frequency table for the columns of movie type. Determine which statement has the strongest associati
    7·2 answers
  • After a dilation centered at the origin, the image of CD is C'D'.
    6·1 answer
  • 1. Write a short definition of each statistical term (mean, median, and mode).
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!