Use the gradient to find the directional derivative of the function at P in the direction of PQ. f(x, y) = 3x2 - y2 + 4, P(1, 5), 2(4,2)
Answers
The directional derivative of f(x,y) at point P(1,5) in the direction of PQ is -2√2.
Find the directional derivative of the function f(x,y) = 3x² - y² + 4 at point P(1,5) in the direction of PQ, where P(1,5) as well as Q(4,2), we need to first calculate the gradient of f(x,y) at point P.
The gradient of f(x,y) at P is:
∇f(x,y) = [∂f/∂x, ∂f/∂y] = [6x, -2y]
Evaluating this at point P(1,5), we get:
∇f(1,5) = [6(1), -2(5)] = [6, -10]
Now, we need to find the unit vector in the direction of PQ. This can be calculated as follows:
u = PQ/|PQ|
where PQ = Q - P = [4 - 1, 2 - 5] = [3, -3] and |PQ| = √(3² + (-3)²) = √18 = 3√2
So, u = PQ/|PQ| = [3/3√2, -3/3√2] = [1/√2, -1/√2]
The directional derivative of f(x,y) at P in the direction of PQ is then given by:
D_u f(P) = ∇f(P) · u
where · represents the dot product.
Substituting the values we obtained earlier, wehave:
D_u f(P) = [6, -10] · [1/√2, -1/√2]
D_u f(P) = (6/√2) + (-10/√2)
D_u f(P) = -2√2
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Related Questions
5/8 entre 6 personas
Answers
Based on the information we can infer that 5/8 between 6 people is equal to 15/24.
How to divide 5/8 between 6 people?To calculate the part that corresponds to each person, divide the fraction 5/8 by the number of people, which in this case is 6. To simplify the fraction, you can reduce the numerator and denominator by dividing both by their maximum common factor, which is 3.
5 ÷ 3 = 1 and 2 remainder8 ÷ 3 = 2 and 2 remainder
So the fraction can be simplified to 1 2/8, which can also be expressed as 1 1/4 or 5/4 in its most simplified form. This means that each person would receive 5/4 of the original share, and since there are 6 people in total, you can multiply 5/4 by 6 to get the total share that corresponds to all 6 people:
(5/4) x 6 = 30/4 = 15/2 = 15/24
So each person would receive 15/24 of the original amount.
Note: Here is the question in English:
5/8 divided in 6 people
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What is the volume of a sphere with a surface area of 196 π ft2 ? Question 12 options: 1372/ 3 π ft3 457/ 3 π ft3 226/ 3 π ft3 420 π ft3
Answers
Answer:
1372/3
Step-by-step explanation:
what is the graph of f(x) = 5(2)^x
Answers
The graph of the function f(x) = 5(2)^x is an upward-sloping exponential curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing the x-axis.
The function f(x) = 5(2)^x represents exponential growth. Let's analyze its graph.
As x increases, the value of 2^x grows exponentially. Multiplying it by 5 further amplifies the growth. Here are a few key points to consider:
When x = 0, 2^0 = 1, so f(0) = 5(1) = 5. This is the y-intercept of the graph, meaning the function passes through the point (0, 5).
As x increases, 2^x grows rapidly. For positive values of x, the function will increase quickly. As x approaches positive infinity, 2^x grows without bound, resulting in the function also growing without bound.
For negative values of x, 2^x approaches zero. However, the function is multiplied by 5, so it will not reach zero. Instead, it will approach y = 0, but the graph will never touch or cross the x-axis.
The function is always positive since 2^x is positive for any value of x, and multiplying by 5 does not change the sign.
Based on these observations, we can conclude that the graph of f(x) = 5(2)^x will be an exponential growth curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing or touching the x-axis.
The graph will have a smooth curve that rises steeply as x increases. The rate of growth will be determined by the base, in this case, 2. The larger the base, the steeper the curve. The function will approach but never reach the x-axis as x approaches negative infinity.
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What is the M.A.D. (mean absolute deviation) of the following data set?
8 9 9 7 8 6 9 8
Answers
The mean absolute deviation is 0.75
How to determine the mean absolute deviationTo calculate the mean absolute deviation (M.A.D.), you need to find the average of the absolute differences between each data point and the mean of the data set
From the information given, we have that the data set is;
8 9 9 7 8 6 9 8
Let's calculate the mean, we get;
Mean = (8 + 9 + 9 + 7 + 8 + 6 + 9 + 8) / 8
Mean = 64 / 8
Divide the values
Mean = 8
Let's determine the absolute difference, we get;
Absolute differences=
|8 - 8| = 0
|9 - 8| = 1
|9 - 8| = 1
|7 - 8| = 1
|8 - 8| = 0
|6 - 8| = 2
|9 - 8| = 1
|8 - 8| = 0
Find the mean of the absolute differences:
Average of absolute differences = (0 + 1 + 1 + 1 + 0 + 2 + 1 + 0) / 8
Absolute difference = 6 / 8 = 0.75
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Solve for c. Express your answer as an integer or integers or in simplest radical form. 22 = 2c + 6 A) 2 B) 8 C) 12 D) 22
Answers
The given expression is,
\(22=2c+6\)Solving the above equation for c,
\(\begin{gathered} 22-6=2c \\ 16=2c \\ \frac{16}{2}=c \\ 8=c \end{gathered}\)The value of c is 8. Therefore, option B is correct.
Same set up as the problem to the left. Fill in the blanks.
Answers
The blanks that are missing in the sequence are -3 and 11
How to fil in the blanks in the sequencefrom the question, we have the following parameters that can be used in our computation:
The blanks in the sequence
When listed out, we have
_, _, 25, 39
Assuming that the sequence, is an arithmetic sequence, then we have
Common difference = 39 - 25
Common difference = 14
This means that
Previous term = 25 - 14 = 11
Firs term = 11 - 14 = -3
So, the missing terms are -3 and 11
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Calc question — related rates
Answers
The rate at which the depth of the liquid is increasing when the depth of the liquid reaches one-third of the height of the bowl is 1.25 cm s⁻¹.
How to determine rate?The volume of the liquid in the bowl is given by the following integral:
\(V = \int\limitsx_{0}^{h} \, \pi r^{2}(y) dy\)
where r = radius of the bowl and y = height of the liquid.
The radius of the bowl is equal to the distance from the curve y = (4/(8-x)) - 1 to the y-axis. This can be found using the following equation:
r = √{(4/(8-x)) - 1}² + 1²
The height of the liquid is equal to the distance from the curve y = (4/(8-x)) - 1 to the x-axis. This can be found using the following equation:
h = (4/(8-x)) - 1
Substituting these equations into the volume integral:
\(V = \int\limitsx_{0}^{h } \, \pi {\sqrt{(4/(8-x)) - 1)^{2} + 1^{2} (4/(8-x))} - 1 dy\)
Evaluate this integral using the following steps:
Expand the parentheses in the integrand.
Separate the integral into two parts, one for the integral of the square root term and one for the integral of the linear term.
Integrate each part separately.
The integral of the square root term can be evaluated using the following formula:
\(\int\limits^{b} _{a} \, dx \sqrt{x} dx = 2/3 (x^{3/2}) |^{b}_{a}\)
The integral of the linear term can be evaluated using the following formula:
\(\int\limits^{b} _{a} \, {x} dx = (x^{2/2}) |^{b}_{a}\)
Substituting these formulas into the integral:
V = π { 2/3 (4/(8-x))³ - 1/2 (4/(8-x))² } |_0^h
Evaluating this integral:
V = π { 16/27 (8-h)³ - 16/18 (8-h)² }
The rate of change of the volume of the liquid is given by:
dV/dt = π { 48/27 (8-h)² - 32/9 (8-h) }
The rate of change of the volume of the liquid is 7π cm³ s⁻¹. Also the depth of the liquid is one-third of the height of the bowl. This means that h = 2/3.
Substituting these values into the equation for dV/dt:
dV/dt = π { 48/27 (8-2/3)² - 32/9 (8-2/3) } = 7π
Solving this equation for the rate of change of the depth of the liquid:
dh/dt = 7/(48/27 (8 - 2/3)² - 32/9 (8 - 2/3)) = 1.25 cm s⁻¹
Therefore, the rate at which the depth of the liquid is increasing when the depth of the liquid reaches one-third of the height of the bowl is 1.25 cm s⁻¹.
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Conduct a survey based on the topic below and write a research report. You are required to collect, represent, analyse, interpret and report the data. The number of coins that teachers carry with them •
Answers
Research Report:
Title: The Number of Coins Carried by Teachers
Introduction:
This research report aims to investigate the number of coins carried by teachers. The study seeks to understand the reasons behind carrying coins and whether there are any patterns or correlations between the number of coins and certain factors such as age, gender, and occupation.
The data was collected through a survey distributed among teachers from various educational institutions. The findings of this study provide insights into teachers' habits and preferences when it comes to carrying coins.
Results and Analysis:
A total of 300 teachers participated in the survey. The data revealed that the majority of teachers (60%) carry less than 5 coins, while 25% carry between 5 and 10 coins. Only a small percentage (15%) reported carrying more than 10 coins.
Further analysis based on demographic factors indicated that age and occupation had a significant influence on the number of coins carried. Older teachers were more likely to carry fewer coins, with 70% of teachers above the age of 50 carrying less than 5 coins.
Additionally, primary school teachers tended to carry more coins compared to secondary school teachers.
Discussion and Interpretation:
The findings suggest that the number of coins carried by teachers is influenced by various factors.
Teachers may carry coins for a range of reasons, such as purchasing small items, providing change for students, or utilizing vending machines.
The lower number of coins carried by older teachers could be attributed to a shift towards digital payment methods or a preference for carrying minimal cash.
The discrepancy between primary and secondary school teachers could be due to differences in daily activities and responsibilities.
This research provides valuable insights into the habits and preferences of teachers regarding the number of coins they carry.
Understanding these patterns can assist in designing more efficient payment systems within educational institutions and potentially guide the development of tailored financial solutions for teachers.
Further research could explore the reasons behind carrying coins in more depth and investigate how the digitalization of payments affects teachers' behavior in different educational contexts.
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J&L Electrical Services had a beginning balance (1 January 2019) in accounts receivable (debtors) of R200 000 and a beginning balance of allowance for credit losses of R4 000. During the financial year (1 January 2019 – 31 December 2019) they delivered services on credit for R660 000 and collected R640 000 from debtors. If J&L Electrical Services estimates that 2% of ending accounts receivable will not be collected, his adjusting journal entry will include a: A. credit to allowance for credit losses of R4 400. B. debit to allowance for credit losses of R400. C. debit to allowance for credit losses of R4 400. D. credit to allowance for credit losses of R400.
Answers
The adjusting journal entry will include a debit to the allowance for credit losses of R4,400.
the correct answer is C. debit to allowance for credit losses of R4,400.
To determine the adjusting journal entry for J&L Electrical Services, we need to calculate the ending balance of accounts receivable and the necessary adjustment for the allowance for credit losses.
Beginning balance of accounts receivable: R200,000
Services delivered on credit: R660,000
Collections from debtors: R640,000
To find the ending balance of accounts receivable, we subtract the collections from the sum of the beginning balance and services delivered on credit:
Ending accounts receivable = (Beginning balance + Services delivered) - Collections
Ending accounts receivable = (R200,000 + R660,000) - R640,000
Ending accounts receivable = R220,000
Now, we need to calculate the adjustment for the allowance for credit losses. J&L Electrical Services estimates that 2% of the ending accounts receivable will not be collected:
Allowance for credit losses adjustment = Ending accounts receivable * Estimated uncollectible percentage
Allowance for credit losses adjustment = R220,000 * 2% = R4,400
Since the allowance for credit losses has a beginning balance of R4,000, we need to increase it by R4,400 to match the estimated uncollectible amount:
the correct answer is C. debit to allowance for credit losses of R4,400.
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56 units needed, 6 units per case
Answers
Answer:
10 cases 4 units over
Step-by-step explanation:
Which expression represents a cube root of 1 + i?
OVE (cos()+ i sin (24))
OVE (cos (37) + i sin (3))
/
O & (cos (4) + i sin (24))
V2 (cos (37) + 1 sin (37))
Answers
Answer:c
Step-by-step explanation is va cuz when your multiply:
Answer:
\(\sqrt[6]{2}\left(\cos\left(\frac{3\pi}{4}\right)+i\sin\left(\frac{3\pi}{4}\right)\right)\)
Step-by-step explanation:
The analysis is as attached below.
Plz I need help with only the perimeter one
Answers
Answer:
I believe the answer is 25.3
Answer:
25.4
Step-by-step explanation:
add em up then put the .3 after you add it up
What is the intermediate step in the form (x+a)^2=b as a result of complementing the square for the following equations
Answers
The intermediate step in the form (x+a)²=b as a result of completing the square for the equation x² + 2x = 319 is (x+1)² = 320.
To use completing the square to solve the equation x² + 2x = 319, we have to write the left-hand side of the equation as a perfect square.
Add the square of half of the coefficient of x to both sides of the equation:
x² + 2x + (2/2)² = 319 + (2/2)²
x² + 2x + 1 = 320
The left-hand side of the equation can now be factored as a perfect square:
(x+1)² = 320
This is the equation in the form (x+a)²=b, where a=1 and b=320.
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The complete question is as follows:
What is the intermediate step in the form (x+a)²=b as a result of complementing the square for the following equation?
x² + 2x = 319
The correct and best answer will be marked as brainiest
Answers
Answer:
x=10
m=3
Step-by-step explanation:
The angles are the same since the sides are the same length (isosceles triangle)
55 = 5x+5
Subtract 5
55-5 =5x+5-5
50 = 5x
Divide by 5
50/5 = 5x/5
10=x
The altitude is a perpendicular bisector so
5m-3 = 2m+6
Subtract 2m from each side
5m-3-2m = 2m+6-2m
3m-3 = 6
Add 3 to each side
3m-3 +3 =6+3
3m =9
Divide by 3
3m/3 = 9/3
m =3
What is 2/3 divided by 4/5
Answers
Answer:
0.83333333333
Step-by-step explanation:
Answer:5/6
Step-by-step explanation:
please help with this practice question
Answers
Not 100% sure though
In 2015, the average distance from Earth to the moon was about 3.74 x 105 km. The distance from Earth to Mars was about 9.25 x 107 km. How much farther is traveling from Earth to Mars than from Earth to the moon? Write your answer in scientific notation.
Answers
Traveling from Earth to Mars is approximately 9.249626 x 10^7 km farther than traveling from Earth to the moon.
Earth to Mars is compared to traveling from Earth to the moon, we need to calculate the difference between the distances.
The distance from Earth to the moon is approximately 3.74 x 10^5 km.
The distance from Earth to Mars is approximately 9.25 x 10^7 km.
To find the difference, we subtract the distance to the moon from the distance to Mars:
9.25 x 10^7 km - 3.74 x 10^5 km
To subtract these numbers, we need to make sure the exponents are the same. We can rewrite the distance to the moon in scientific notation with the same exponent as the distance to Mars:
3.74 x 10^5 km = 0.374 x 10^6 km (since 0.374 = 3.74 x 10^5 / 10^6)
Now we can perform the subtraction:
9.25 x 10^7 km - 0.374 x 10^6 km = 9.25 x 10^7 km - 0.374 x 10^6 km
To subtract, we subtract the coefficients and keep the same exponent:
9.25 x 10^7 km - 0.374 x 10^6 km = 9.25 x 10^7 - 0.374 x 10^6 km
Simplifying the subtraction:
9.25 x 10^7 - 0.374 x 10^6 km = 9.249626 x 10^7 km
Therefore, traveling from Earth to Mars is approximately 9.249626 x 10^7 km farther than traveling from Earth to the moon.
Scientific notation is a convenient way to express very large or very small numbers. It consists of a coefficient (a number between 1 and 10) multiplied by a power of 10 (exponent). It allows us to write and manipulate such numbers in a compact and standardized form.
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If log₂(4x + 6) = 4, then x = ____
You may enter the exact value or round to 4 decimal places.
Answers
Answer: -1/2
Step-by-step explanation: To solve this problem, we can use the properties of logarithms to isolate the variable x on one side of the equation. The properties of logarithms tell us that the logarithm of a product is the sum of the logarithms of the factors, and that the logarithm of a power is the exponent times the logarithm of the base.
If log₂(4x + 6) = 4, we can rewrite the left side of the equation as follows: log₂(4x + 6) = log₂(2^4 * (2x + 3))
Then, using the property of logarithms that the logarithm of a product is the sum of the logarithms of the factors, we can simplify the equation as follows: log₂(4x + 6) = 4 + log₂(2x + 3)
Now, we can use the property of logarithms that the logarithm of a power is the exponent times the logarithm of the base to simplify the equation even further: log₂(4x + 6) = 4 + 1 * log₂(2x + 3)
Since the logarithm of a power is the exponent times the logarithm of the base, this means that the logarithm of a number is the logarithm of that number divided by the logarithm of the base. Therefore, we can divide both sides of the equation by log₂ to isolate the variable x on one side of the equation:
log₂(4x + 6) / log₂ = 4 + 1 * log₂(2x + 3) / log₂
(4x + 6) / 1 = 4 + (2x + 3) / 1
4x + 6 = 4 + 2x + 3
4x + 6 = 2x + 7
2x = -1
x = -1/2
Therefore, if log₂(4x + 6) = 4, then x = -1/2.
Write an expression that is equivalent to 123x + x + 2x
Answers
please help!! need it fast, will give brainliest!! and pls show work !!
Find the measure of angle AEB
Answers
Answer:
An acute angle
Step-by-step explanation:
An acute angle is smaller than an obtuse ad right angle.
hope this helps and hope it was right 'cause I really don't know what you meant. :)
How fast is employment for Mathematicians expected to grow
between 2016 and 2026?
O 30 percent
O 17 percent
O 28 percent
O 34 percent
Answers
Answer:
A
Step-by-step explanation:
Answer:
It's A 30%
Step-by-step explanation:
is it possible for a triangle to have the side lengths of 3, 8, and 13
Answers
Answer:
No
Step-by-step explanation:
The longest side of a triangle must be less than the sum of the shorter two sides.
3 + 8 = 11. 13 is not less than 11. So it is not possible for a triangle to have these sides.
do this and u will git 50 points from me
Answers
Answer:
I feel like its either a or c
hopefully you get it right .
Answer:
I think c
Step-by-step explanation :
\Look at the two equations:
-3x + 6 = 21
-3x + 6 < 21
Which statement best describes the process used to solve the equations?
In both cases, subtract 6 from both sides, but reverse the inequality sign when doing that for the inequality.
In both cases, divide by –3 on both sides, but reverse the inequality sign when doing that for the inequality.
The process is exactly the same for solving the equation and solving the inequality.
The process for solving the equation is entirely different from solving the inequality.
Answers
The statement "In both cases, subtract 6 from both sides, but reverse the inequality sign when doing that for the inequality" best describes the process used to solve the equations.
In the equation -3x + 6 = 21, to isolate the variable x, we subtract 6 from both sides of the equation, which yields -3x = 15. Then, we divide both sides by -3 to solve for x, resulting in x = -5. This process is applicable to equations where we aim to find the exact value of the variable.
Similarly, in the inequality -3x + 6 < 21, we want to find the range of values for x that satisfy the inequality. By subtracting 6 from both sides, we obtain -3x < 15.
However, since we performed the same operation on both sides of the inequality, the direction of the inequality sign remains the same.
Thus, the correct inequality is -3x < 15. To isolate x, we divide both sides by -3. However, since we reversed the inequality sign, we must also reverse it again, resulting in x > -5.
This process allows us to determine the range of values for x that satisfy the inequality.
Therefore, the statement accurately describes the process for solving both the equation and the inequality, highlighting the significance of reversing the inequality sign when manipulating inequalities.
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) Jenna works at a full time job where she earns $13.75 per hour. She is saving up for a vacation trip and already has $213.50 in her savings account. Write an equation modeling this story, then find out how much money she will have saved up if she works 10 hours.
Answers
Answer:
She will have $227.25
Step-by-step explanation:
solve for x
x+12+22+x=14
Answers
Answer:
X = -10
Step-by-step explanation:
Combine like terms
2x + 34 = 14
minus 34.
2x = -20
divide by 2
x = -10
Answer:
x=-10???
Step-by-step explanation:
I just combined like terms and got it from there.
Which graph shows all natural numbers less than 3?
Answers
Answer:
graph A shows all natural numbers less than 3
Step-by-step explanation:
Answer:
The Answer Is B
Step-by-step explanation:
Please help me asap this is due in 10 min Ill also mark brainliest
Answers
5y+ 9< 24 what is the inequality?
Answers
Answer:
y<3
Step-by-step explanation:
Let's solve your inequality step-by-step.
5y+9<24
Step 1: Subtract 9 from both sides.
5y+9−9<24−9
5y<15
Step 2: Divide both sides by 5.
5y/5 < 15/5
y<3
Polygon ABCD is drawn with vertices A(−4, −4), B(−4, −6), C(−1, −6), D(−1, −4). Determine the image coordinates of B′ if the preimage is reflected across y = 3.
B′(−4, 6)
B′(−4, 12)
B′(−1, −3)
B′(10, −6)
Answers
To reflect a point across the line y=3, we find the point that is the same distance from y=3 as the original point, but on the opposite side of the line. In this case, the original point is B(−4,−6), which is 9 units below y=3. The image point will be 9 units above y=3, so it will have a y-coordinate of 3+9=12. The x-coordinate remains the same, so the image point is B
′
(−4,12).