Answer:
[tex]\displaystyle \boxed{ \frac{dz}{du} = \frac{e^\big{\frac{u(2u + v)}{v}}}{v} \Bigg[ 4u + v \Bigg] }[/tex]
[tex]\displaystyle \boxed{ \frac{dz}{dv} = \frac{ue^\big{\frac{u(2u + v)}{v}}}{v} \Bigg[ 1 - \frac{2u + v}{v} \Bigg] }[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Multivariable Calculus
Partial Derivatives
Partial Derivative Rule [Chain Rule]:
[tex]\displaystyle \frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial t} + \frac{\partial w}{\partial z} \frac{\partial z}{\partial t}[/tex]
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle x = 2u + v[/tex]
[tex]\displaystyle y = \frac{u}{v}[/tex]
[tex]\displaystyle z = e^{xy}[/tex]
Step 2: Find Derivatives
[tex]\displaystyle \frac{dz}{du}[/tex]:
[Derivative] Rewrite [Partial Derivative Rule - Chain Rule]:We can find the partial derivatives by differentiating using basic differentiation rules found under "Calculus":
[tex]\displaystyle\begin{aligned}\frac{\partial z}{\partial x} & = ye^{xy} \\\frac{\partial x}{\partial u} & = 2 \\\frac{\partial z}{\partial y} & = xe^{xy} \\\frac{\partial y}{\partial u} & = \frac{1}{v} \\\end{aligned}[/tex]
Substituting in our partial derivatives and our variables x and y, we can obtain a final derivative:
[tex]\displaystyle\begin{aligned}\frac{dz}{du} & = 2ye^{xy} + \frac{xe^{xy}}{v} \\& = \frac{2ue^\big{\frac{u(2u + v)}{v}}}{v} + \frac{(2u + v)e^\big{\frac{u(2u + v)}{v}}}{v} \\& = \boxed{ \frac{e^\big{\frac{u(2u + v)}{v}}}{v} \Bigg[ 4u + v \Bigg] }\end{aligned}[/tex]
∴ we have found the derivative of z with respect to u.
[tex]\displaystyle \frac{dz}{dv}[/tex]:
[Derivative] Rewrite [Partial Derivative Rule - Chain Rule]:We can find the partial derivatives by using the same method(s):
[tex]\displaystyle\begin{aligned}\frac{\partial z}{\partial x} & = ye^{xy} \\\frac{\partial x}{\partial v} & = 1 \\ \frac{\partial z}{\partial y} & = xe^{xy} \\\frac{\partial x}{\partial v} & = \frac{-u}{v^2} \\ \end{aligned}[/tex]
Substituting in our partial derivatives and our variables x and y, we can obtain a final derivative:
[tex]\displaystyle\begin{aligned}\frac{dz}{dv} & = ye^{xy} + \frac{-uxe^{xy}}{v^2} \\& = \frac{ue^\big{\frac{u(2u + v)}{v}}}{v} + \frac{-u(2u + v)e^\big{\frac{u(2u + v)}{v}}}{v^2} \\& = \boxed{ \frac{ue^\big{\frac{u(2u + v)}{v}}}{v} \Bigg[ 1 - \frac{2u + v}{v} \Bigg] }\end{aligned}[/tex]
∴ we have found the derivative of z with respect to v.
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Topic: Multivariable Calculus
Unit: Partial Derivatives and Applications
Please help me! I would appreciate it a lot!
Answer:
d
Step-by-step explanation:
A function is shown below.
f(x) = 8x - 11
What value of x makes f(x) = 21?
The correct value in this function is f(21) = 157
This question is related to substitution of values. To solve it, just: change x by the indicated value and thus calculate normally.
– The given function being f(x) = 8x - 11, replace X by 21:
f(x) = 8x - 11
f(21) = 8 . 21 - 11
– Finally, just calculate the expression now, like this, multiplying and subtracting the numbers:
f(21) = 8 . 21 - 11
f(21) = 168 - 11
f(21) = 157
Therefore, the correct value of this function will be f(21) = 157
Which of the following best describes the graph shown below ?
B. This is the graph of a function, but it is not one-to-one
Step-by-step explanation:A function is a type of graph that has a specific relationship between the x and y values.
Vocabulary
First, let's define the words that were used in the answer choices.
Function - A function is a type of relationship where the x-values do not repeat. This means that each x-value has only one specific y-value.One-To-One - A one-to-one function still has no repeating x-values; however, y-values also do not repeat. This means that each y-value has only one x-value and vice versa.Finding a Function
There are different ways to find if a relationship is a function, but with a graph, the easiest way is the vertical line test. The vertical line test is when you draw vertical lines over your graph. If you are able to draw a vertical line everywhere without that line crossing the graph more than once, then the graph is a function.
In this graph, there is no spot in which a vertical line would intersect with the graph more than once. Since it passed the vertical line test, this must mean that x-values never repeat. Thus, this is a function.
Finding a One-To-One Function
Now that we know this is a function, we need to find if this is a one-to-one function. Once again, this can be done multiple ways, but one way is the horizontal line test. This test is similar to the method above; however, horizontal lines are used. In this test, you draw horizontal lines are drawn on the graph. If you can draw a horizontal line at every value without the line intersecting more than once, then it is a one-to-one function.
On this graph, if you were to draw a horizontal line at any of the y-values between 0 and 6, then the line would intersect multiple times. This means that y-values do repeat. Thus, this is not a one-to-one function.
Which of the following is the absolute value parent function?
Answer:
A. F(x) = |x|
Step-by-step explanation:
Any number that substitutes for x inside of the absolute value parent function will be converted to a positive. Example: |-43| = 43
Which of the following isn't equivalent to -1.2 + 6?
A. 6 - 1.2
B. -2 (0.6 - 3)
C. (-2.4 + 12) / 2
D. -1 (1.2 + 6)
E. I don't know.
As a member of the student council at your school, you want to determine if the seventh-grade students prefer to
have an after-school soccer program or an after-school movie club. There are 110 seventh-grade students at your
school, and you decide to take a sample of 20 students.
Describe in detail how you can randomly select 20 seventh-grade students for your sample.
Answer: I dont know
Step-by-step explanation: i need more information
Find the missing length
Apply Pythagorean theorem
H²=P²+B²x²=14²+10²x²=196+100x²=296x=17.8Is it SSS, SAS, ASA, AAS, or HL?
Triangle ABC is congruent to triangle DEF based on the SAS congruence theorem.
What is the SAS Theorem?The SAS theorem is a triangle congruence theorem that states that two triangles are congruent if they have two pairs of congruent sides that are corresponding to each other, and a pair of congruent included angles.
Triangles ABC and DEF have:
A pair of congruent included angles - ∠DEF ≅ ∠ABCTwo pairs of congruent sides - AB ≅ DE and CB ≅ FETherefore, they are congruent by SAS.
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A small plane travels at 450 miles in 3 hours. At this rate, how far can it travel in 7 hours?
Answer:
1050 miles
Step-by-step explanation:
velocity=distance/time
Distance=450 miles
Time=3 hours
Velocity=150 miles/hours
if time is 7 and velocity is 150 distance is 7x150=1050 miles
Estimate the cost of painting a concrete patio if it is a 14 foot by 8 foot rectangle and a quart of paint that covers 47 square feet costs $10.99
Answer:
Estimate the cost of painting a concrete patio if it is a 14 foot by 8 foot rectangle is $25.76
Step-by-step explanation:
Underline is the given:
a concrete patio if it is a 14 foot by 8 foot rectangle
area of the concrete patio: 14ft x 8ft = 112
quart of paint that covers 47 square feet costs $10.99
the cost per foot: 10.99 / 47 = $0.23
the area is 112 so multiply it by cost per foot:
112 x $0.23 = $25.76
The estimated cost of painting the concrete patio is approximately $32.97.
To estimate the cost of painting a concrete patio, we first need to calculate the total area of the patio. The patio is a rectangle with dimensions 14 feet by 8 feet, so the total area is 14 feet × 8 feet = 112 square feet.
Next, we determine how many quarts of paint are needed to cover the patio. Since each quart covers 47 square feet, we divide the total area by 47:
112 square feet ÷ 47 square feet/quart ≈ 2.38 quarts
Since we can't purchase a fraction of a quart, we'll round up to 3 quarts.
Finally, with each quart costing $10.99, the estimated cost of painting the concrete patio is:
3 quarts × $10.99/quart = $32.97
So, the estimated cost of painting the concrete patio is approximately $32.97.
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Can someone please help me with this? Thank you so much! <33
Answer:
Q63. B
Step-by-step explanation:
5 cars = 2 houses
1 house = 10 pools = 2.5 cars
1 car = 10÷2.5 = 4
James uses of a cup of sugar in his cookies, and Jade uses of a cup of sugar in her cookies. Noel takes the sum of the two amounts of sugar and divides it into two equal portions, planning to use only one portion of it in his cookies. How much sugar does Noel use in his cookies?
Answer:
5/8 of a cup
Step-by-step explanation:
Equation of the line passing through (7,5) and (1, 2)
Answer:
[tex]y = \frac{1}{2}x + \frac{3}{2}[/tex]
Step-by-step explanation:
Assuming this is a linear equation in the form of [tex]y = mx + b[/tex], we can first solve for the slope using [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex], for these points, the slope would be [tex]\frac{1}{2}[/tex]. Now we plug in the points to get the equation:
plugging in (7,5), we get [tex]5 = \frac{1}{2}[/tex] ×[tex]7[/tex] + [tex]b[/tex]
and solving for b, we get b = [tex]\frac{3}{2}[/tex]
therefore, the equation of the line passing through (7,5) and (1,2) is [tex]y = \frac{1}{2}x + \frac{3}{2}[/tex].
hope this helps!
A linear function has a y-intercept of -9 and a slope of 4/3 . What is the equation of the line
Answer:
[tex]y=\frac{4}{3} x-9[/tex]
Step-by-step explanation:
Here this is in standard form. Hope this helps!
Answer:
y=4/3x-9
Step-by-step explanation:
equasion for linear function is y=mx+b where y is m is slope and b is y intercept. just plug in your variables
Select all of the expressions that are equal to 8 × 35 . A. 8 × ( 30 + 5 ) B. 5 + 3 × 35 C. ( 8 × 30 ) + ( 8 × 5 ) D. ( 8 + 30 ) × ( 8 + 5 ) E. 8 × ( 20 + 15 )
The expression 8 × 35 is equal to 8 × (30 + 5), (8 × 30) + (8 × 5), and 8 × (20 + 15) option A, C, and E are correct.
What is an arithmetic operation?It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has a basic four operators that is +, -, ×, and ÷.
We have expression:
= 8 × 35 ⇒ 280
A. 8 × (30 + 5) ⇒ 280
B. 5 + 3 × 35 ⇒ 110
C. (8 × 30) + (8 × 5) ⇒ 280
D. (8 + 30) × (8 + 5) ⇒ 494
E. 8 × (20 + 15) ⇒ 280
Thus, the expression 8 × 35 is equal to 8 × (30 + 5), (8 × 30) + (8 × 5), and 8 × (20 + 15) option A, C, and E are correct.
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Colin invests £2900 into his bank account.
He receives 3% per year compound interest.
How much will Colin have after 4 years?
Give your answer to the nearest penny where appropriate.
Answer:
£3263.98 (nearest penny)
Step-by-step explanation:
Compound interest formula
[tex]\sf A=P(1+\frac{r}{n})^{nt}[/tex]
where:
A = final amountP = principalr = interest rate (in decimal form)n = number of times interest applied per time periodt = number of time periods elapsedGiven:
P = £2900r = 3% = 0.03n = 1t = 4Substituting given values into the formula and solving for A:
[tex]\implies \sf A=2900(1+\frac{0.03}{1})^{1 \times 4}[/tex]
[tex]\implies \sf A=2900(1.03)^{4}[/tex]
[tex]\implies \sf A=3263.975549[/tex]
Therefore, Colin will have £3263.98 after 4 years (to the nearest penny).
Answer:
$3264
Explanation:
[tex]\sf compound \ interest : P(1+\frac{r}{100} )^n[/tex]
"P" is principal, "r" is rate of interest, "n" is time (years)
Here given:
principal: £2900rate of interest: 3%time: 4 yearsAfter 4 years amount:
[tex]\rightarrow \sf 2900(1+\frac{3}{100})^4[/tex]
[tex]\rightarrow \sf 2900(1.03)^4[/tex]
[tex]\rightarrow \sf 3263.9755[/tex]
[tex]\rightarrow \sf 3264[/tex] (rounded to nearest whole number)
) Jeff needs to rent a car for one day, so he shops around to find a car rental company with the lowest overall price. He finds a national car rental company that charges a $15 daily fee plus $0.20 per mile. He also finds a local car rental company that charges a $14 daily fee plus $0.30 per mile. Which equation below represents a linear model for the cost of a rental from the national car rental?
*
1 point
Answer:
Step-by-step explanation:
National car rental equation is
C = 14d + 0.20m
- where C= total cost, d = the number of days and m = the total number of miles driven.
A recipe of chocolate chip cookies need 28 oz of oil for every 44 cups of flower. Approximately how many ounces of oil do you need for 89 cups of flower?
Type your answer to the nearest whole ounce
Answer:
57 ounces
Step-by-step explanation:
He needs 28 oz of oil for every 44 cups. Multiply both to get 56 oz of oil for 88. Since there is one more cup of flour, you will need one more ounce of oil.
Captain Ashley has a ship, the H.M.S Crimson
Lynx. The ship is two furlongs from the dread
pirate Umaima and her merciless band of
thieves.
If her ship hasn't already been hit, Captain
1
Ashley has probability of hitting the pirate
2
ship. If her ship has been hit, Captain Ashley will
always miss.
If her ship hasn't already been hit, dread pirate
1
Umaima has probability of hitting the
3
Captain's ship. If her ship has been hit, dread
pirate Umaima will always miss.
If the Captain and the pirate each shoot once,
and the pirate shoots first, what is the probability
that both the pirate and the Captain hit each
other's ships?
Triangle ABC is similar to triangle DEF.
a. Find the scale factor from triangle ABC to triangle DEF.
b. Find x.
c. Find y.
Answer: x=7.7, y=40
3/x = 15/38.5
38.5*3=115.5
115.5/15= 7.7
x=7.7
3/8 = 15/y
15*8=120
120/3= 40
y=40
orrrr you can just easily find the scale factor which is 5, multiply 8 by 5 and get 40. And 38.5/5 which is 7.7
What is the slope of a line that is parallel to the line shown?
-4/3
-4
-3/4
-2/3
Answer:
m = -4/3
Step-by-step explanation:
m = y2-y1/x2-x1
therefore at (-3,2) and (0-2)
m = -2-2/0 - (-3)
m = -4/3
Estimate the value of
Answer:
The given equation results in 1.98, so the answer is D) 2
✏ [tex]\sf Solve[/tex]
[tex]\sqrt{2} = 1.41\\\\1.41 * 3.14 = 4.4274\\\\\sqrt{5} = 2.23\\\\\frac{4.4274}{2.23} = 1.98[/tex]
Hellppp Im DYING TO KNOW THE ANSWER RNN-
Enter the number that belongs in the green box.
y=[?]
Answer:
60 degrees
Step-by-step explanation:
As shown, this is a equilateral triangle which has the same angle measures.
According to the Triangle Sum Theorem, the sum of all three angles in a triangle is 180 degrees.
Thus,
180 (The sum of all angles in a triangle) /3 (The amount of angles in a triangle) = 60 degrees.
60 because all of the side are equal so it would be 180 divided by 3.
In October 2017 a pound of apples cost $1.55, while oranges cost $1.19. Two years earlier the price of apples was only $1.34 a pound and that of oranges was $1.05 a pound. What was the annual compound rate of growth in the price of apples
The annual growth rate in the price of the apple will be 7.83%.
What is the growth rate?
The growth rate is defined as the increment of any quantity over a period of time.
Given that:-
In October 2017 a pound of apples cost $1.55, while oranges cost $1.19.Two years earlier the price of apples was only $1.34 a pound and that of oranges was $1.05 a pound.The growth rate will be calculated as::-
Growth rate = Cost in 2017 / Cost in 2015
Growth rate = 1.55 / 1.34 = 1.15
The growth rate will be 15.67% for the apple in 2 years so for 1 year it will be
15.67 / 2 = 7.8%
Therefore the annual growth rate in the price of the apple will be 7.83%.
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: Find the midpoint of the segment with the following endpoints.
(5,-5) and (2,8)
Answer:
(3.5; 1.5).
Step-by-step explanation:
1) formula for midpoint:
[tex]x=\frac{X_A-X_B}{2}; \ y=\frac{Y_A-Y_B}{2};[/tex]
2) according to the formula above the required midpoint is:
x=0.5(5+2)=3.5; y=0.5(8-5)=1.5
Find the surface area, to the nearest square centimeter, of a cylinder with diameter 6 cm and height 8 cm.
a. 48 cm²
C.
207 cm²
b.
452 cm²
d.
72 cm²
Answer:
[tex]\sf 207 \ cm^2[/tex]
Step-by-step explanation:
Surface area of cylinderdiameter = 6 cm
r = 6/2 = 3 cm
h = 8 cm
[tex]\sf \boxed{\text{\bf Surface area of cylinder = $2\pi r(h +r)$}}[/tex]
[tex]\sf = 2*3.14*3 * (3 +8)\\\\ = 2*3.14* 3 * 11\\\\= 207.24\\\\= 207 \ cm^2[/tex]
Translate and solve: 220% of what number is 3300?
Note: Enter your answer as a whole number.
The dot plot below shows the hourly rate of some babysitters in a city:
A number line is shown from 1 dollar to 7 dollars in increments of 1 dollar for each tick mark. The horizontal axis label is Dollar per hour. There are 4 dots above 4 dollars, 2 dots above 5 dollars, 3 dots above 6 dollars, and 1 dot above 7 dollars. The title of the line plot is Babysitting Rates.
Which statement best describes the shape of the data?
It is a cluster from $1.00 to $7.00 and has no gaps.
It is a cluster from $1.00 to $7.00 and has gaps.
It is not symmetric and has a peak at $4.00.
It is symmetric and has a peak at $7.00.
*will mark brainliest*
Answer:
A dot plot is like a histogram but it uses dots instead of bars. Like histogram, a dot plot also presents the frequency of data points. The dot plot as described by the problem is shown in the picture. A bell shape is drawn for a symmetrical data. From the choices, the answer is: It is symmetric and has no gaps. The other statements are wrong because the peak is at $5 or $6.
Answer:
A dot plot is like a histogram but it uses dots instead of bars. Like histogram, a dot plot also presents the frequency of data points. The dot plot as described by the problem is shown in the picture. A bell shape is drawn for a symmetrical data. From the choices, the answer is: It is symmetric and has no gaps. The other statements are wrong because the peak is at $5 or $6.
Step-by-step explanation:
A dot plot is like a histogram but it uses dots instead of bars. Like histogram, a dot plot also presents the frequency of data points. The dot plot as described by the problem is shown in the picture. A bell shape is drawn for a symmetrical data. From the choices, the answer is: It is symmetric and has no gaps. The other statements are wrong because the peak is at $5 or $6.
find which one is f f' f"
f - green (A)
f' - red (B)
f'' - blue (C)
The graph of f attains local extrema at the same points that f' = 0. There's evidence of this at x = 0 and between x = 3 and x = 3.5. (At both points, curve A attains a maximum, while curve B crosses the x-axis.)
Similarly, the graph of f' attains local extrema at the same points that f'' = 0. This is seen between x = 1.5 and x = 2. (B has a max, C crosses axis)
Also, the graph of A is continuous at x = 1.5 but exhibits a sharp turn, so at this point f is not differentiable and the graph of f' (and subsequently f'') is not continuous.
