Use the Chain Rule to evaluate the partial derivative ϑf/ϑs at (r, s) = (1,0), where f(x, y) = ln (xy), x = 3r + 5s, y = 2r + 11s (Express numbers in exact form. Use symbolic notation and fraction where needed)
Answers
The partial derivative ∂f/∂s at (r, s) = (1, 0) is ∂f/∂s = (10 + 33)/6 = 43/6
To evaluate the partial derivative ∂f/∂s at (r, s) = (1, 0), we'll use the Chain Rule. First, let's compute the necessary partial derivatives.
1. ∂f/∂x = ∂(ln(xy))/∂x = (1/xy) ∂(xy)/∂x = y/(xy) = 1/x
2. ∂f/∂y = ∂(ln(xy))/∂y = (1/xy) ∂(xy)/∂y = x/(xy) = 1/y
3. ∂x/∂s = ∂(3r + 5s)/∂s = 5
4. ∂y/∂s = ∂(2r + 11s)/∂s = 11
Now we can apply the Chain Rule:
∂f/∂s = ∂f/∂x ∂x/∂s + ∂f/∂y ∂y/∂s
Substitute the computed partial derivatives:
∂f/∂s = (1/x)(5) + (1/y)(11)
Now, we need to find x and y at the point (r, s) = (1, 0):
x = 3r + 5s = 3(1) + 5(0) = 3
y = 2r + 11s = 2(1) + 11(0) = 2
Substitute the values of x and y:
∂f/∂s = (1/3)(5) + (1/2)(11) = 5/3 + 11/2
To find the partial derivative ∂f/∂s at (r, s) = (1, 0), we can now combine the fractions:
∂f/∂s = (10 + 33)/6 = 43/6
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Related Questions
HI I really need help with this its in the picture
Answers
Explanation:
The blue rectangle has area of 4a^2*6a^2 = 24a^4
The red rectangle has area 4a^2*9a = 36a^3
The green rectangle has area 4a^2*3 = 12a^2
The total area is 24a^4 + 36a^3 + 12a^2
Using ƒ(x) = 3x − 3 and g(x) = -x, find g(ƒ(x))
a. 3x+3
b. 3-3x
c.-3-3x
d. 2x-3
Answers
Work Shown:
\(g(x) = -x\\\\g(f(x)) = -f(x)\\\\g(f(x)) = -(3x-3)\\\\g(f(x)) = -3x+3\\\\g(f(x)) = \boldsymbol{3-3x}\\\\\)
Explanation: We start with the outer function g(x). In the second step, we replace every copy of x with f(x). From there we plug in f(x) = 3x-3 and simplify using the distribution rule.
The notation \(g(f(x))\) is the same as writing \((g \circ f)(x)\)
Please help it due soon and the answer is meant to be in kg
Answers
Answer: 20 kg
Step-by-step explanation:
You follow the line of best fit until 50cm
Then you trace across and look at the x-axis.
There you will find that the dog will be 20kg at 50cm using the line of best fit.
4) Choose the correct name for the set of numbers
(0,1,2,,3,4)
a. natural numbers
b.irrational numbers
c.whole numbers
d.integers
e.rational numbers
f real numbers
Answers
Reason:
In mathematics, the natural numbers are those used for counting and ordering.
Multiple Choice Answer: A, B, C, D
Hope this helps!
Answer is: A) natural numbers.
What is number?A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words.
here, we have,
given that,
the set of numbers (0,1,2,,3,4)
It is a.) natural numbers.
Reason:
In mathematics, the natural numbers are those used for counting and ordering.
hence, Answer is: A) natural numbers.
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Please help me I don't know how to do this look at the attachment for the question
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the answer is 95.
Step-by-step explanation:
180-85=95
Answer:
m<1= 95
Step-by-step explanation:
180-85=95
Pleaseeeeeeeeeeeeeeeeeeeeeeeee hellllllppppp meeeeeeee it would be super helpful right now!!!!
(Geometry) problems are in the photo
Answers
Answer:
14. -21 over 5, 1 over 4
Step-by-step explanation:
PQ: perform the slope formula: y2-y1 divided by x2-x1
-3 - 17 divided by 1 - (-4) = -21 divided by 5
RS: 4 - 3 divided by -5 - (-9) = 1 divided by 4
The sale price of every item in a store is 85 percent of its usual price a. The usual price of a backpack is $30, what is the sale price? B. The usual price of a sweatshirt is $18, what is it’s sale price? so they forgot to include C and that's the only one I'm confused on.
Answers
Answer:
A. $25.5
B. $15.3
Explanation:
To find the sale price of every item, we need to find 85% of the usual price.
So, 85% of $30 is calculated as:
\(30\times85\text{ \% = 30}\times\frac{85}{100}=25.5\)Therefore, the sale price of the backpack is $25.5
In the same way, 85% of $18 is calculated as:
\(18\times85\text{ \% = 18}\times\frac{85}{100}=15.3\)Therefore, the sale price of the sweatshirt is $15.3
When approaching the top of a hill, passing is not allowed Select one: a. within 500 feet before. b. within 750-1,000 feet before. c. within 1,100 feet before. d. None of the above
Answers
The correct answer is a. within 500 feet before. When approaching the top of a hill, passing is not allowed for safety reasons.
The reason for this restriction is to ensure the safety of drivers and prevent potential accidents or collisions that can occur due to limited visibility and potential hazards on the road.
Passing within 500 feet before the top of a hill is prohibited because at this point, the driver's line of sight may be compromised. As the driver ascends the hill, the road ahead becomes obscured, making it difficult to see oncoming vehicles, pedestrians, or any potential obstacles. Passing in such conditions significantly increases the risk of a collision or a dangerous situation.
The restriction on passing within 500 feet before the top of a hill is a common traffic regulation implemented in many jurisdictions. It is designed to ensure that drivers maintain a safe distance and have sufficient time to react to any unexpected situations that may arise.
By prohibiting passing within this specified distance, authorities aim to minimize the likelihood of accidents caused by reduced visibility and limited reaction time. It allows drivers to navigate the hill safely and be prepared for any potential hazards that may appear once the crest of the hill is reached.
It is important for drivers to be aware of and adhere to these regulations to promote safe driving practices. Failure to comply with the passing restrictions near the top of a hill can lead to serious consequences, including accidents, injuries, and even loss of life.
In conclusion, passing is not allowed within 500 feet before the top of a hill due to reduced visibility and the potential for hazardous conditions. This restriction ensures the safety of drivers and helps prevent accidents on hilly roads.
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So I need help on solving a problem that has 4 pieces of fruit and two of them are bananas then I have to write 2 fractions that describe the fraction of fruit that are bananas
Answers
Answer:
1/ 2 and 2/4
Step-by-step explanation:
Total pieces of fruits = 4
Number of fruits that are banana = 2
Fraction of fruit that are banana :
Number of banana / total number of fruits
= 2 / 4
Or in its simplest form:
= 2/4 = 1/2
e-intercept form.
2. (2, 4); slope
1
2
write the linear equation in slope-intercept form
Answers
to work this out you need to solve for your y-intercept. by using what you have.
once you do so you get the answer y=1/2x+3
help meeeeeeeeeeeee pleaseeee rnnn!!!
Answers
The object falling from a tower of 1503ft will take 9.69 seconds to reach the ground
FunctionsA function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and codomain or range. A function is generally denoted by f(x) where x is the input.
In the function s(t) = 16t²
The height of the object is at 1503ft, let's substitute this and solve for t
1503 = 16t²
t² = 1503/16
t = 9.69seconds
It will take the object 9.69 seconds to fall from a height of 1503 ft
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PLS HELP ME ILL GIVE U BRAINLIEST THXX Find (F-g)(2)
Answers
Answer:
-9 is the correct answer
A player tosses a die 6 times.If gets a number 6 Atleast two times he wins 2 dollars ,otherwise he looses 1 dollar.. Find the expectation of the amount if he wins
Answers
Answer:
\(E(x)=-0.2101\)
Step-by-step explanation:
The expected value for a discrete variable is calculated as:
\(E(x)=x_1*p(x_1)+x_2*p(x_2)\)
Where \(x_1\) and \(x_2\) are the values that the variable can take and \(p(x_1)\) and \(p(x_2)\) are their respective probabilities.
So, a player can win 2 dollars or looses 1 dollar, it means that \(x_1\) is equal to 2 and \(x_2\) is equal to -1.
Then, we need to calculated the probability that the player win 2 dollars and the probability that the player loses 1 dollar.
If there are n identical and independent events with a probability p of success and a probability (1-p) of fail, the probability that a events from the n are success are equal to:
\(P(a)=nCa*p^a*(1-p)^{n-a}\)
Where \(nCa=\frac{n!}{a!(n-a)!}\)
So, in this case, n is number of times that the player tosses a die and p is the probability to get a 6. n is equal to 6 and p is equal to 1/6.
Therefore, the probability \(p(x_1)\) that a player get at least two times number 6, is calculated as:
\(p(x_1)=P(x\geq2)=1-P(0)-P(1) \\\\P(0) =6C0*(1/6)^{0}*(5/6)^{6}=0.3349\\P(1)=6C1*(1/6)^{1}*(5/6)^{5}=0.4018\\\\p(x_1)=1-0.3349-0.4018\\p(x_1)=0.2633\)
On the other hand, the probability \(p(x_2)\) that a player don't get at least two times number 6, is calculated as:
\(p(x_2)=1-p(x_1)=1-0.2633=0.7367\)
Finally, the expected value of the amount that the player wins is:
\(E(x)=x_1*p(x_1)+x_2*p(x_2)\\E(x)=2*(0.2633)+ (-1)*0.7367\\E(x)=-0.2101\)
It means that he can expect to loses 0.2101 dollars.
Answer:
Expected value = -0.2101
Step-by-step explanation :
Givena discrete variable the expected value is given as:
E(x) = xp(x) + yp(y)
Where x and y are the values the variables can assume, and p(x) and p(y) are their respective probabilities.
Probability that a player gets 6 at least twice = p(x ≥ 2) = 1 - p(x≤ 2)
= 1 - p(0) - p(1)
But p(0) = 0.3349
p(1) = 0.4018
p(x ≥ 2) = 1 - 0.3349 - 0.4018
= 0.2633
p(y) = 1 - p(x)
= 1 - 0.2633
= 0.7367
Expected value = 2(0.2633) + (-1)(0.7367)
= -0.2101
A loss of $0.2101 is expected.
Define the term functions limits and continuity as used in
business calculus and use an example
Answers
In business calculus, the term "functions" refers to mathematical relationships that associate inputs (typically denoted as x) with corresponding outputs (typically denoted as y). Functions can represent various aspects of business operations, such as revenue, cost, profit, demand, and supply.
The concept of "limits" in calculus deals with the behavior of a function as the input approaches a particular value. It determines the value that the function approaches or tends to as the input gets arbitrarily close to a specified value. Limits are essential for analyzing the behavior of functions near certain points, understanding rates of change, and evaluating derivatives and integrals.
"Continuity" of a function refers to its smooth and unbroken nature, without any abrupt jumps, holes, or vertical asymptotes. A function is continuous if its graph can be drawn without lifting the pen from the paper. Continuity ensures that small changes in the input correspond to small changes in the output, which is crucial for reliable analysis and prediction.
For example, consider the function f(x) = 2x + 1. This linear function represents a business scenario where x represents the number of units sold, and f(x) represents the total revenue generated. The limit of f(x) as x approaches 2 is 5, indicating that as the number of units sold approaches 2, the total revenue approaches $5. Since f(x) = 2x + 1 is a linear function, it is continuous across its entire domain.
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Consider removing the point (130, -35) and calculating a new least-squares regression line.
What effect(s) would removing this point have?
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When a point is removed (i.e. \((130, -35)\)), both average and standard deviation are changed and a new least-squares line is created.
How removing a point may alter a least-squares regression line
Given a minimum and representative set of points in a Cartesian plane, there is a given average and a standard deviation, of which slope and intercept of the least-squares line are determined.
When a point is removed, both average and standard deviation are changed and a new least-squares line is created. \(\blacksquare\)
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What is the volume, in cubic ft, of a rectangular prism with a height of 8ft, a width of 8ft, and a length of 18ft?
Answers
The volume of the cuboid is 1152 ft³
What is volume of a cuboid?A cuboid is a solid shape or a three-dimensional shape. A convex polyhedron that is bounded by six rectangular faces with eight vertices and twelve edges is called a cuboid.
A rectangular prism is called a cuboid and the volume of a cuboid is expressed as;
V = base area × height
The base is rectangle, the area of a rectangle is expressed as;
A = l× w
A = 18 × 8
A = 144 ft²
V = base area × height
V = 144 × 8
V = 1152 ft³
Therefore, the volume of the cuboid is 1152 ft³
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Can sombody pleas help me
Answers
Answer:
232
Step-by-step explanation:
(8)/(12)=(x)/(348)
x=232
Can some tell me what does x equal?
Answers
Answer:
x=10
Step-by-step explanation:
A straight line has an angle of 180 degrees. To solve for x we can add all the given angles and set them equal to 180.
\(180= 5x-1+9x+8+2x+13\)
Then combine all the common factors
\(180=16x+20\)
And then you can solve for x
\(180= 16x+20\\160=16x\\10=x\)
To check if x is correct you can plug in 10 to each equation. If all the equations add up to 180, then x is correct
\(5(10)-1= 49\\9(10)+8= 98\\2(10)+13= 33\\\\49+98+33=180\)
Please help if your skilled at math and evaluating functions because I’m not PLZ HELP!!!!
Answers
Hope this helped :)
Pls halp due today >_<. Thank you!
Answers
It is constant because otherwise it wouldn't be proportional
a rectangular parking lot is 67.5 ft wide and 148 ft long. what is the area of the parking lot in square meters?
Answers
The area of the rectangular parking lot is 929.03 square meters.
Use the formula for the area of a rectangle to calculate the area of the rectangular parking lot, which is given as:
Area = length × width
We know that the parking lot is 67.5 ft wide and 148 ft long, the area can be calculated as follows:
Area = 67.5 ft × 148 f
t= 9990 sq. ft
However, the question asks for the area in square meters, so we need to convert square feet to square meters. 1 square foot is equal to 0.092903 square meters, so we can use this conversion factor to convert square feet to square meters.
Area in square meters = Area in square feet × 0.092903
= 9990 sq. ft × 0.092903
= 929.03 sq meters
Therefore, the area is 929.03 square meters.
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An article is marked to sell at Rs.1300;if 10% discount is allowed,find the discount amount and selling price.
Answers
Answer:
discount amount=Rs. 130
selling price=Rs. 1170
Step-by-step explanation:
discount amount=10%of Rs.1300
=10/100*1300
=1300/10
=Rs. 130
selling price=MRP-discount amount
=1300-130
=Rs. 1170
Picture shown!
If
f(x) = x^2+ 2x - 4
and
g(x) = 3x + 1
Find
g(f(x)) = [ ? ]x^2+ [ ? ] + [ ? ]
![Picture shown!Iff(x) = x^2+ 2x - 4andg(x) = 3x + 1Findg(f(x)) = [ ? ]x^2+ [ ? ] + [ ? ]](https://i5t5.c14.e2-1.dev/c-qa-images/contents/attachments/eLFyoOWQyv86CRnGBG33n4lGzWAIdVcV.png)
Answers
Answer:
3x^2+6x-11
Step-by-step explanation:
g(f(x)) = g(x^2+2x-4) = 3(x^2+2x-4)+1 = 3x^2+6x-11
Multiply x(x^2+2x+4) and -2(x^2+2x+4)
Combine like terms:
Final answer:
Answers
x^3-8
hope this helps!!
Maureen is making a 3/10 scale drawing of a doll house if a line on the drawing represents 15 inches how long should the line be ?
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We have the following ratio scale:
\(\frac{\text{REAL}}{\text{DRAWING}}=\frac{10}{3}\)If we move the DRAWING term to the rght hand side, we get
\(\text{REAL}=\frac{10}{3}\cdot\text{DRAWING}\)In our case, the line is 15 inches long, then we have
\(\text{REAL}=\frac{10}{3}\cdot(15\text{inches)}\)which gives
\(\begin{gathered} \text{REAL}=\frac{10\times15}{3}\text{ inches} \\ \text{REAL}=10\times5\text{ inches} \\ \text{REAL}=50\text{ inches} \end{gathered}\)Therefore, the answer is 50 inches.
Short answer please ASAP
Answers
The lateral surface area of the rectangular prism given in the picture is 36 cm ²
How to find the lateral surface area of a rectangular prism ?The lateral surface area of a rectangular prism can be found by the formula :
= 2 x height x ( Length + Breadth )
Height = 2 cm
Length = 3 cm
Breadth = 6 cm
The lateral surface area is therefore:
= 2 x 2 x ( 3 + 6 )
= 4 x 9
= 36 cm ²
In conclusion, the lateral surface area is 36 cm ².
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Substitute *
y=5
3x - 2y = 11
What is the value of x?
Answers
3x -2(5) = 11
3x - 10= 11
You have to add 10 to 11 (inverse operation)
3x= 21
X=7
PLS HELP ASAP 4 BRAINLIEST
Answers
Answer:
The answer is A
Step-by-step explanation:
105+90=5x-70
195+70=5x-70+70
\( \frac{265}{5} = \frac{5x}{5} \)
53=x and then when you insert to in the formula 5(53)-70 you get 195
and 105+90 is also equal to 195
Youre welcome!
2. A fish tank that holds 80 gallons of water is 55% full. Create a line diagram (like the one
above) to represent this situation.
a. How many gallons are in the tank now? How many more gallons are needed to
fill the tank?
Answers
The tank currently contains 44 gallons of water, and 36 more gallons are needed to fill it completely.
How to calculate many more gallons are needed to fill the tanka. The fish tank that holds 80 gallons of water is 55% full, which means it contains:
44 gallons = 80 gallons × 0.55
b. To find out how many more gallons are needed to fill the tank, we can subtract the current amount of water in the tank from its full capacity:
80 gallons - 44 gallons = 36 gallons
Therefore, the tank currently contains 44 gallons of water, and 36 more gallons are needed to fill it completely.
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If g(x)= 1-x/x^2, evaluate g(2).
A) -1/4
B) 1/4
C) -1/2
D) 1/2
Answers
g(2) is -1/4 for function g(x)= 1-x/x².
What is a function?A relation is a function if it has only One y-value for each x-value.
The given function is g(x)= 1-x/x²
g of x equal to one minus x divided by x square
g(x)= 1-x/x²
Now we need to find the value of g(2).
Replace the x with 2
g(2)=1-2/2²
When two is subtracted from one we get minus one and two square is four
g(2)=-1/4
Hence, g(2) is -1/4 for function g(x)= 1-x/x².
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