If z=(x+y)e^y and x=5t and y=1−t^2, find the following derivative using the chain rule. Enter your answer as a function of t.
dz/dt= ?
Answers
By the chain rule, if \(z=f(x,y)=f(x(t),y(t))\), then
\(\dfrac{\mathrm dz}{\mathrm dt} = \dfrac{\partial f}{\partial x}\dfrac{\mathrm dx}{\mathrm dt} + \dfrac{\partial f}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}\)
We have \(f(x,y)=(x+y)e^y\), for which
• \(\dfrac{\partial f}{\partial x} = e^y\)
• \(\dfrac{\partial f}{\partial y} = e^y + (x+y)e^y = (x+y+1)e^y\)
and \(x(t)=5t\) and \(y(t)=1-t^2\), so that
• \(\dfrac{\mathrm dx}{\mathrm dt} = 5\)
• \(\dfrac{\mathrm dy}{\mathrm dt} = -2t\)
Putting everything together, we get
\(\dfrac{\mathrm dz}{\mathrm dt} = 5e^y - 2t(x+y+1)e^y \\\\ \dfrac{\mathrm dz}{\mathrm dt} = 5e^{1-t^2} - 2t(5t-t^2+2)e^{1-t^2} = \boxed{\left(2t^3-10t^2-4t+5\right)e^{1-t^2}}\)
The solution of the derivative using the chain rule is \(\dfrac{dz}{dt}=(2t^3-10t^2-4t+5)e^{1-t^2}\).
What is the Chain rule?The chain rule explains how to calculate a composite function's derivative.
What is a composite function?If you may write \(f(g(x))\) , a function is composite. It's a function within a function, or a function of a function, in other words.
By the chain rule, we can write if \(z=f(x,y)=f(x(t),y(t))\), then,
\(\dfrac{dz}{dt}=\dfrac{\partial f}{\partial x} \dfrac{dx}{dt}+\dfrac{\partial f}{\partial y} \dfrac{dy}{dt}\)
As it is given to us, \(f(x,y)=(x+y)e^y\), therefore, we can write,
\(\dfrac{\partial f}{\partial x}=e^y\)\(\dfrac{\partial f}{\partial y}=e^y+(x+y)e^y=(x+y+1)e^y\)And x(t)=5t, while y(t)=1-t², So,
\(\dfrac{dx}{dt}=5\)\(\dfrac{dy}{dt}=-2t\)Now, Substitute all the values together, in the equation,
\(\dfrac{dz}{dt}=\dfrac{\partial f}{\partial x} \dfrac{dx}{dt}+\dfrac{\partial f}{\partial y} \dfrac{dy}{dt}\)
Substituting the values we will get,
\(\dfrac{dz}{dt} &=(5 \times e^y)+[-2t(x+y+1)e^y]\\\\\)
\(=5e^{1-t^2}-2t(5t-t^2+2)e^{1-t^2}\\\\=(2t^3-10t^2-4t+5)e^{1-t^2}\)
Hence, the solution of the derivative using the chain rule is \(\dfrac{dz}{dt}=(2t^3-10t^2-4t+5)e^{1-t^2}\).
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Related Questions
The equation for the axis of symmetry is
y=k
O True
O False
REAL ANSWERS ONLY PLS HELP
Answers
Answer:
false
Step-by-step explanation:
pretty sure the equation is x=-b2a
In a certain town in 90 minutes 1/2 inch of rain falls continues at the same rate for a total of 24 hours which of the following statements are true about the amount of rain in the 24 hour.
Answers
Answer:
A. 16 times of 1/2 inch of rain
Step-by-step explanation:
24 hours = 1440 minutes
90 minutes = 1/2 inch of rain
1440 divided by 90 = 16
16 times of 1/2 inch of rain
2. Find the missing variable
(2x + 5)
(x + 9)
(3x - 2)
Answers
q(3x)=5q(x)+8solve for x
Answers
we have that
q(x)=7x-1
solve the equation
q(3x)=5q(x)+8
step 1
Find q(3x)
q(3x)=7(3x)-1
q(3x)=21x-1
step 2
substitute in the original expression
21x-1=5(7x-1)+8
21x-1=35x-5+8
21x-1=35x+3
35x-21x=-1-3
14x=-4
x=-4/14
x=-2/7Amy and Jade bought packages of gum. Amy bought a total of 48 pieces. Jade bought a total of 32 pieces. If each package had the same number of pieces, what would be the number of pieces in each package of gum?
Answers
Answer:
16
Step-by-step explanation:
16x2=32
16x3=48
Please help !!!!!! 20 points
Answers
The value of x in the polygon will be 13.25 degrees.
The value of x is 10.
We can find the value of x by plugging in the number of sides of the regular polygon into the formula x = (n-2)*15° - 1.
How to calculate the valueThe sum of the interior angles of a regular polygon with n sides is (n-2) x 180 degrees.
Sum of angles = (24-2) x 180 = 22 x 180 = 3960 degrees
Since all the angles in a regular polygon are congruent, we can divide the sum of the angles by the number of angles to find the measure of one angle:
Measure of one angle = 3960/24 = 165 degrees
165 = 12x + 6
159 = 12x
x = 13.25
Therefore, the value of x is 13.25 degrees.
Each of the triangles in our decomposition has one angle equal to (17x+2)°, so the sum of all the angles in the triangles is 43 × (17x+2)° = 731x+86°.
Therefore, we have:
731x+86° = 7380°
Solving for x, we get:
731x = 7294°
x = 10
Therefore, the value of x is 10.
The equation that can be used to find the value of x is:
(9x+48)° + (15x-24)° = (n-2)*180°
24x + 24 = (n-2)*180°
Dividing both sides by 24, we get:
x + 1 = (n-2)*15°
Subtracting 1 from both sides, we get:
x = (n-2)*15° - 1
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A hollow metal pipe of internal diameter 7cm and external diameter 8cm is 6.16m long calculate the volume of metal making the pipe in cubic centimetres
Answers
Answer:
The volume of the hollow metal pipe is approximately 72.571 cubic centimeters.
Step-by-step explanation:
The volume of the metal is the space occupied by the hollow pipe. Since it is a hollow cylinder, the volume (\(V\)), in cubic centimetres, is defined by following expression:
\(V = \frac{\pi}{4} \cdot (D_{e}^{2}-D_{i}^{2})\cdot l\) (1)
Where:
\(D_{e}, D_{i}\) - External and internal diameters, in centimetres.
\(l\) - Length, in centimetres.
If we know that \(D_{i} = 7\,cm\), \(D_{e} = 8\,cm\) and \(l = 6.16\,cm\), then the volume of the hollow metal pipe is:
\(V = \frac{\pi}{4} \cdot (D_{e}^{2}-D_{i}^{2})\cdot l\)
\(V\approx 72.571\,cm^{3}\)
The volume of the hollow metal pipe is approximately 72.571 cubic centimeters.
(+5)+(-5)+(-10)+(+20) find the sum
Answers
Answer:
10
Step-by-step explanation:
5-5=0
0-10=-10
-10+20=10
Answer:
positive(+) + negative (-) = negative
5 + (-5) =0
0 + (-10) = (-10)
(-10) + 20 = 10
if my medical expenses are $30,000 per year for 35 years with inflation at 5.13% how much money would I have to put in an interest bearing account with a 5% return to cover all medical expenses and the account be $0 at the end of 35 years
Answers
Answer:
$1,021,337.13
Step-by-step explanation:
The sum of present values of medical expenses is ...
30,000/1.05 +30,000(1.0513)/1.05^2 +30,000(1.0513^2)/1.05^3 +...
So, the series has an initial value of 30,000/1.05 and a common ratio of 1.0513/1.05. Its sum is given by ...
S = a(r^n -1)/(r -1)
where a = 30,000/1.05, n = 35, r = 1.0513/1.05.
Filling in these values and doing the arithmetic, we get ...
S = $1,021,337.13
You need an initial deposit of $1,021,337.13 to cover rising expenses for 35 years.
You cut out a piece of fabric in the shape of a kite so that the congruent angles of the kite are each
100°. Of the remaining two angles, one is 4 times larger than the other. What is the measure of the
largest angle in the kite?
Answers
The measure of the largest angle in the kite is 26.67 (approx)°.
The given problem states that you cut out a piece of fabric in the shape of a kite so that the congruent angles of the kite are each 100°. The other two angles of the kite are not equal. If one of the angles is 4 times the size of the other, then the larger angle is 4x, and the smaller one is x.
We need to determine the measure of the largest angle in the kite. In order to find out the measure of the largest angle in the kite, we need to know the sum of all the angles in the kite. Let's recall the sum of angles in a kite. A kite is a quadrilateral whose four sides can be grouped into two pairs of adjacent sides that are equal in length. A kite has two opposite pairs of equal angles. The sum of the angles of a kite is 360°.In a kite, the angles can be represented as a, a, b, and c where a is the congruent angle and b and c are non-congruent angles.
Then, 2a + 2b = 360°2a + 2x + 4x = 360°2a + 6x = 360°2a = 360° - 6x2a = 180° - 3xa = (180° - 3x)/2
Since a = 100°, we can substitute it in the equation above, a = (180° - 3x)/2 and find the value of x.
Substituting a = 100° in the above equation, we get:
100 = (180 - 3x)/2100 × 2 = 180 - 3x200 = 180 - 3x20 = -3x/-3x = -20/(-3)x = 6.67 (approx)
Now we can substitute this value of x in the expression for the larger angle to find its measure. The larger angle is 4x, so it is equal to 4(6.67) = 26.67 (approx)°.
Therefore, the measure of the largest angle in the kite is 26.67 (approx)°.
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Please help find the area for numbers 5 and 6
Answers
On solving the provided question, we can say that form an equation = area of the figure = x = 1/2*9*9 + 6*18 = 40.5 + 48 = 88.5
What is equation?A mathematical equation is a formula that joins two statements and uses the equal symbol (=) to indicate equality. A mathematical statement that establishes the equality of two mathematical expressions is known as an equation in algebra. For instance, in the equation 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart. The relationship between the two sentences on either side of a letter is described by a mathematical formula. Often, there is only one variable, which also serves as the symbol. for instance, 2x – 4 = 2.
here,
to find the area
we have to form an equation =
area of the figure = x = 1/2*9*9 + 6*18 = 40.5 + 48 = 88.5
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SOMEONE PLEASE help me answer this
Answers
Answer:
5.7 m is the answer
Step-by-step explanation:
a = 7m
b = ?
c = 9m
According to the Pythagorean theorem,
a² + b² = c²
7² + b² = 9²
49 + b² = 81
b² = 81 - 49
b² = 32
b = 5.656
Round to the nearest tenth,
b = 5.7 m
Can you please find the domain and range of the relation? And please explain if it is or is not a function.
Answers
The function f(x) = -(x-3)² + 9 can be used to represent the area of a rectangle with a perimeter of 12 units, as a
function of the length of the rectangle, x. What is the maximum area of the rectangle?
O3 square units
O 6 square units
O9 square units
O 12 square units
Answers
The maximum area of the rectangle is A = 9 units²
What is the Area of a Rectangle?The area of the rectangle is given by the product of the length of the rectangle and the width of the rectangle
Area of Rectangle = Length x Width
Given data ,
Let the function for area be f ( x )
where f ( x ) = - ( x - 3 )² + 9
Let the width of the rectangle be W.
The perimeter of a rectangle is given by:
Perimeter = 2(length + width) = 12 units
12 = 2(x + width)
Simplifying this equation, we get:
6 = x + width
width = 6 - x
Now, the area of the rectangle is given by:
Area = Length x Width
Substituting the value of width in terms of x, we get:
Area = x(6 - x)
Area of rectangle A = 6x - x²
This is a quadratic function of x, which has a maximum value at the vertex of the parabola. The vertex of the parabola can be found by completing the square or by using the formula:
x = -b/(2a)
In this case, a = -1 and b = 6, so we have:
x = -6/(2(-1)) = 3
Therefore, the maximum area of the rectangle occurs when x = 3, and the value of the maximum area is:
Area of rectangle A = 3(6 - 3) = 9 square units
Hence , area of rectangle is A = 9 units²
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Desperate Need Of Help
Answers
The domain and range of the graph above in interval notation include the following:
Domain = [-6, 3]
Range = [-3, 3]
What is a domain?In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.
In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = [-6, 3] or -6 ≤ x < 3.
Range = [-3, 3] or -3 < y < 3
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A report on the income of American households says that the median income of households was about $70,000 in 2021. The mean income of these same households was about $100,000 in 2021. What explains the difference between these two measures of center?
Answers
The median is the middle value in a sorted list of data. Half of the values are greater than the median and half are less than the median. In the case of income data, the median income of households is about $70,000, which means that half of the households have incomes greater than $70,000 and half have incomes less than $70,000.
On the other hand, the mean is the average of all the values in a dataset. It is calculated by summing all the values and then dividing by the number of values. In the case of income data, the mean income of households is about $100,000, which means that the total income of all households is divided by the number of households to get an average income of $100,000.
The difference between the median and mean income in this case can be explained by the presence of high-income households that can skew the average (mean) income. If there are a few households with extremely high incomes, the mean will be much larger than the median. This is because the high incomes of those households pull up the mean income of all households. In contrast, the median is not affected by outliers or extreme values, so it can be a better representation of the typical income of households in the dataset.
Therefore, in this case, the mean income of about $100,000 is higher than the median income of about $70,000, indicating the presence of high-income households in the dataset that increased the mean income
The difference between the median and mean income can be explained by the distribution of the data.
What are the mean and median of data?
A data set's mean (average) is calculated by summing all of the numbers in the set, then dividing by the total number of values in the set. When data collection is ranked from least to greatest, the median is the midpoint. The mode is the number that occurs most often in a data set.
The median and mean are both measures of center used to describe a distribution of data. In this case, the median income of American households in 2021 is reported as $70,000, which means that half of the households have an income below $70,000 and a half have an income above $70,000.
On the other hand, the mean income of these same households in 2021 is reported as $100,000. This means that if we were to add up the income of all households and divide by the total number of households, we would get an average of $100,000 per household.
The difference between the median and mean income can be explained by the distribution of the data. If the data is symmetrically distributed around the center (i.e., the median), the mean and median will be similar. However, if the data is skewed, meaning it is not evenly distributed, then the mean and median will be different.
In this case, it is possible that a small number of households have very high incomes, which would pull up the mean income while not affecting the median income as much. This could explain why the mean income is higher than the median income.
It's also worth noting that while the median and mean income are both useful measures, they do not tell the whole story about income distribution.
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What is the volume of this figure?
Enter your answer in the box.
ft³
Three-dimensional figure formed by placing two rectangular prisms together to form an L shape where the bottom of the L is on the left and the top of the L extends to the right. The wider part of the L has a width of 3 feet. The other part of the L has a width of 2 feet. The total length of the L is 7 feet, and the length of the narrower part of the L is 6 feet. The height connecting the L shaped bases is 5 feet.
Answers
The volume of the three-dimensional figure that forms an L-shape is: volume of the two rectangular prisms.
Volume of Rectangular Prism?Volume = length × width × height
The volume of the figure = volume of both rectangular prisms that formed the L-shape.
To find the volume of the figure, first find the volume of each of the rectangular prisms and add both together.
Therefore, the volume of the three-dimensional figure that forms an L-shape is: volume of the two rectangular prisms.
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Rewrite the expression using a single, positive exponent: 14^{-3}\times 14^{12}
Answers
1/2-1/4= We have to make sure the denominators are equal.
Since 2 is evenly divided by 4, we can multiply just one term to get a common denominator.
Multiply 1 by 2, and get 2, then we multiply 2 by 2 and get 4. So now our fractions look like this:
2
4
+
1
4
Step 2
Since our denominators match, we can add the numerators.
2 + 1 = 3
So the answer is:
3/4
Answers
Answer:
i have an idea what it is but i dont really know sorry
Step-by-step explanation:
A line includes the points (4,1) and (8, 2). What is its equation in slope-intercept form?
Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answers
Answer: y = 4x
Step-by-step explanation: Slope intercept form is y = mx + b where m is the slop and b is the x intercept
We can solve for slope using the given points
(2-1)/(8-4) = 1/4, thus m = 1/4
y = 1/4x +b
Now to solve for b, we can just plug in one of the points to see what makes b work
1 = 4*(1/4) + b
1 = 1 +b
b = 0
Central limit theorem: Sample proportion
The control department of a light bulb manufacturer randomly picks 4400 light bulbs from the production lot every week. The records show that, when there is
no malfunction, the defect rate inthe manufacturing process (due to imperfections in the material used) is 1%. When 1.5% or more of the light bulbs in the
sample of 4400 are defective, the control unit calls repair technicians for service.
Answer the following. (If necessary, consult a list of formulas.)
(a) Find the mean of P, where p is the proportion of defective light bulbs in a sample of 4400 when there is no malfunction.
(b) Find the standard devition
(c) Compute an approximation for PCP >0.015), which is the probability that the service technicians will be called even though the system is functioning
properly. Round your answer to four decimal places.
Answers
6 let A= (1,4,9,16) and B=(2,3,5,7). Find AS
Prove that A-B=( )
Answers
zip file, and phrase, angle of applicable. where are u doing now I need to make a sentence, but it was not the same time for changing the full range. uses the nation, angle and phrase, angle. uses a
X
-1
0
1
2
3
y
1
10
1252
25
2
125
2
52:41
What is the rate of change of the function described in
the table?
ㅇ12/5
O5
ㅇ25/2
○ 25
Answers
Based on the figures given in the table, the rate of change of the function is b. 5.
What is the function's rate of change?This can be found as:
= (Y₂ - Y₁) / (X₂ - X₁)
Use points:
(0, 1/2) and (1, 5/2)
Solving gives:
= (5/2 - 1/2) / (1 - 0)
= 5
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If you pls help lol I only have 52 mins left
Answers
Step-by-step explanation:
\(2y + 106 = 180\)
\(y = 106 \div 2 = 53\)
\(y = 53\)
\(4z + 16 = 180\)
\(4z = 180 - 16\)
\(z = 164 \div 4 = 41\)
and the answer for x is 106
Using the present value approach, solve the following:
Tom has $100 in a bank account that pays a guaranteed 5% interest rate each year. How much would Tom have at the end of Year 3?
Answers
Answer:
Step-by-step explanation:
$100x0.5x1=$5
The following table shows the breakdows of opinions for both faculty and students in a recent survey about the new restructuring of the campus to include an HonomCollege. Find the probability that a randomly selected person is either a student opposed to the change or a faculty member who has an opinion either for or againstExpress your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth,
Answers
Given,
The number of total students and faculty is 213.
From the table,
The number of students opposed to the change is, n(student oppose) = 44.
The number of faculty members opposed to the change is, n(faculty oppose) = 20.
The number of faculty members favor to the change is, n(faculty favour) = 11.
The probability of elected person is a student oppose to the change or faculty member either for or against is,
\(\begin{gathered} \text{Probability}=\frac{n\mleft(\text{student oppose}\mright)+n\mleft(faculty\text{ oppose}\mright)+n\mleft(faculty\text{ favour}\mright)}{\text{total members}} \\ =\frac{44+20+11}{213} \\ =\frac{75}{213} \\ =\frac{25}{71} \\ =0.352113 \end{gathered}\)Hence, the probability is 25/71.
Find AC, round to the nearest whole number
Answers
20
Triangles are 180 total so just add 118 and 22 then subtract it from 180 for your answer
Aaron is considering two health insurance plans. The Get Well Soon Company charges an annual fee of 100
plus 10
per doctor visit, given y=100+10x
. The Mega Coverage Company charges $35 per doctor, with no annual fee, represented by the equation: y=35x
.
Which insurance company has the LOWER rate and what is the rate?
Answers
The insurance company has 4 doctor visits y=35x is lower rate and Rate 35 $ per doctor visit for more than 4 doctor visits y=100+10x.
What is the system of equation?A system of equations is a collection of two or more equations with a same set of unknowns.
In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.
Aaron is considering two health insurance plans. The Get Well Soon Company charges an annual fee of 100 plus 10 per doctor visit, given y=100+10x
The Mega Coverage Company charges $35 per doctor, with no annual fee, represented by the equation: y=35x.
On solving both the equations
\(\rm 35x=100+10x\\\\35x-10x=100\\\\25x=100\\\\x=\dfrac{100}{25}\\\\x=4\)
For 4 doctor visits charges are same and Rate 35 $ per doctor visit.
For less than 4 doctor visits y=35x is lower rate and Rate 35 $ per doctor visit for more than 4 doctor visits y=100+10x.
Hence, The insurance company has 4 doctor visits y=35x is lower rate and Rate 35 $ per doctor visit for more than 4 doctor visits y=100+10x.
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You created a playlist, and 100 of your friends listened to it and shared if they
liked it or not. Mario said the ratio of the number of people who liked the playlist
to the number of people who did not like the playlist is 75:25. Lila said that for
every three people who liked the playlist, one person did not. Do they agree? *
Answers
Answer:
its yes
Step-by-step explanation:
given sin = 4/5, use the pythagorean Theorem to find the value of cos
Answers
You have that sin x = 4/5, which means that the hypotenuse of a triangle is 5 and the opposite side to x has a length of 4.
By using the Pythagorean Theorem you have:
h² = c1² + c2²
where h = 5 and c1 = 4. Solve the previous equation for c1:
c2² = h² - c1²
c2² = 5² - 4²
c2² = 25 - 16
c2² = 9
c2 = 3
next, take into account that cos x is given by the quotient of the adjacent side c2 over the hypotenuse h:
cos x = c2/h = 3/5
Hence, cos x = 3/5