The derivative[tex]\( \frac{dw}{dt} \) is \( \frac{7t^6 e^{4-t}}{2+9t} - \frac{t^7 e^{4-t}}{2+9t} - \frac{9t^7 e^{4-t}}{(2+9t)^2} \).[/tex]
To find [tex]\( \frac{dw}{dt} \)[/tex] using the chain rule for the given function[tex]\( w = \frac{x e^y}{z} \), where \( x = t^7 \), \( y = 4 - t \), and \( z = 2 + 9t \)[/tex], follow these steps:
1. **Express ( w ) in terms of ( t ):**
Substitute ( x ), ( y ), and ( z ) into ( w ):
[tex]\[ w = \frac{x e^y}{z} = \frac{(t^7) e^{(4 - t)}}{2 + 9t} \][/tex]
2. **Apply the chain rule:**
The chain rule states that for a function ( w(t) ) defined implicitly by ( w = f(x(t), y(t), z(t)) ), the derivative [tex]\( \frac{dw}{dt} \)[/tex] is given by:
[tex]\[ \frac{dw}{dt} = \frac{\partial w}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial w}{\partial y} \cdot \frac{dy}{dt} + \frac{\partial w}{\partial z} \cdot \frac{dz}{dt} \][/tex]
3. **Compute partial derivatives of ( w ) with respect to ( x ), ( y ), and ( z ):**
[tex]\( \frac{\partial w}{\partial x} = \frac{e^y}{z} \)[/tex]
[tex]\( \frac{\partial w}{\partial y} = \frac{x e^y}{z} \)[/tex]
[tex]\( \frac{\partial w}{\partial z} = -\frac{x e^y}{z^2} \)[/tex]
4. **Compute [tex]\( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \):**[/tex]
[tex]\( \frac{dx}{dt} = 7t^6 \)[/tex]
[tex]\( \frac{dy}{dt} = -1 \)[/tex]
[tex]\( \frac{dz}{dt} = 9 \)[/tex]
5. **Substitute these into the chain rule formula:**
[tex]\[ \frac{dw}{dt} = \frac{e^y}{z} \cdot 7t^6 + \frac{x e^y}{z} \cdot (-1) + \left(-\frac{x e^y}{z^2}\right) \cdot 9 \][/tex]
6. **Substitute[tex]\( x = t^7 \), \( y = 4 - t \), \( z = 2 + 9t \)[/tex] into the expression:**
[tex]\( e^y = e^{4 - t} \)[/tex]
Substitute these values into the formula for [tex]\( \frac{dw}{dt} \):[/tex]
[tex]\[ \frac{dw}{dt} = \frac{e^{4 - t}}{2 + 9t} \cdot 7t^6 - \frac{t^7 \cdot e^{4 - t}}{2 + 9t} - \frac{9t^7 \cdot e^{4 - t}}{(2 + 9t)^2} \][/tex]
Therefore, [tex]\( \frac{dw}{dt} \)[/tex] is:
[tex]{\frac{dw}{dt} = \frac{7t^6 e^{4 - t}}{2 + 9t} - \frac{t^7 e^{4 - t}}{2 + 9t} - \frac{9t^7 e^{4 - t}}{(2 + 9t)^2} } \][/tex]
Order these numbers from least to greatest. 14325 , 5.753 , 534 , 5.34
A= P(1+nr) ; solve for r
The table shows the number of each type of toy in the store. the toys will be placed on shelves so that each shelf has the same number of each type of toy how many shelves are needed for each type of toy so that it has the greatest number of toys?
Table says toy amount
dolls 45
footballs 105
small cars 75
Which BEST describes the construction of a triangle if given one side length of 10 cm and one right angle?
A)
Unique triangle
B)
More than one triangle
C)
Triangle not possible
D)
Cannot be determined
Answer:
b
Step-by-step explanation:
Carlos has 7 times as many red toy cars as blue toy cars. He has 56 total cars. How many more red cars does he have?
let x = blue cars
so 7x = red cars
x +7x = 56
8x = 56
x = 56/8 =7 blue cars
7*7 =49 red cars
49-7 =42 more red cars
Twenty different books are to be put on five book shelves, each of which holds at least twenty books.
Consider the following equation:
f′(a)=limh→0 (f(a+h)−f(a))/h
Let f(x)=3√x If a≠0, use the above formula to find f′(a)=
Show that f′(0) does not exist and that f has a vertical tangent line at (0,0)
Final answer:
To find the derivative of f(x) = 3√x at a nonzero point, use the definition of the derivative with limits. The derivative at x=0 does not exist because the function has an infinite slope at this point, demonstrated by the absence of a limit, signifying a vertical tangent line at (0,0).
Explanation:
To find f'(a) for f(x) = 3√x when a ≠ 0, we can use the given definition of the derivative:
f'(a) = limh→0 (f(a+h) – f(a)) / h.
To show that f'(0) does not exist and that f has a vertical tangent line at (0,0), we need to evaluate the limit as h tends to zero for the derivative definition applied at a = 0. Since the square root function is not differentiable at 0 (due to the function having an infinite slope at this point), the limit will not exist, indicating that the derivative at 0 does not exist, and the graph of f exhibits a vertical tangent at that point.
Today, you deposit $10,750 in a bank account that pays 3 percent simple interest. how much interest will you earn over the next 7 years?
a. $1,935.00
b. $2,086.06
c. $2,257.50
d. $2,471.14
e. $2,580.00
A sweater is on sale for
20% off the regular price.
The sale price is $60.
What is the regular price of the sweater?
Answer:
$48
Step-by-step explanation:
List price = $60.00
Discount = 20% .
Hence the new price is (100-20)% = 80% of original price .
=> price = $ 60* 80/100
=> price = $ 48
write the number 1,744,326 in expanded form
Given that the volume decreases at a rate of 3 cubic meters per minute, find the rate of change of an edge, in meters per minute, of the cube when the volume is exactly 64 cubic meters
Differentiate f and find the domain of f. (enter the domain in interval notation.)f(x) = ln(x2 − 12x)derivative f '(x) = 2x−12x2−12x correct: your answer is correct.domain
katie and carol bought toys for charity toy drive.katie bought 125 dolls for 13$ each. carol bought 65 toy boats for 32$ each.who spent more money altogher
mr. renato had $150 in his bank account. he deposited $50. acuple of days later he withdrew $20. a week later he deposited $75 .the next week he withdrew $100. what is mr. Renato balance after all the transactions
Find the AGI and taxable income
gross income $23670
Adjustments $0
1 exception $8200
Deduction 0
A G I = Adjustable Gross income,
1 exception = $ 8200,
A G I= Total Income (Gross Income + Amount earned through other means) - (Adjustments +Deduction)
= ($ 23670+$8200)-($ 0 + $0)
= $ 31,870
1 exception = $ 8200, there may be many reasons for exception.
So, Taxable Income = Gross Income - 1 Exception
= $ 23670 - $ 8200
= $ 15,470
A soccer coach purchased one goal and some soccer balls for the team. The expression 180 + 15x represents the cost before tax. What do the different parts of the expression model? Drag the parts of the expression into the boxes to match each description.
The expression [tex]180+15x[/tex] consists of ONE CONSTANT [180] and another VARIABLE EXPRESSION [15x].
Since the coach purchased ONE GOAL, this corresponds to the CONSTANT, which is 180. 180 is the price of the goal.
Since coach purchased several soccer balls, but doesn't say how many, it is the variable part, 15x, where
either 15 is price of ball and x is the number of balls, OR15 is the number of balls and x is the price of eachANSWER:
180 represents the price of the goal
15x represents the total price of the soccer balls
Answer:
180 represents the price of the goal
15x represents the total price of the soccer balls
Step-by-step explanation:
Don tracked the price of gas for several months. In January gas cost $2.65, in February the price had risen $0.56, in March decreased $0.23, in April risen $1.02, and in May decreased $0.48. By how many dollars has the price of gas changed from January to May?
The solution is, By $.87 dollars has the price of gas changed from January to May
What is Subtraction?Subtraction is the process of taking away a number from another. It is a primary arithmetic operation that is denoted by a subtraction symbol (-) and is the method of calculating the difference between two numbers.
here, we have,
given that,
Starting price is 2.65, increased by .56
2.65+.56=3.21
Price is 3.21, decreased by .23
3.21-.23=2.98
Price is 2.98, increased by 1.02
2.98+1.02= 4
Price is 4, decreased by .48
4-.48=3.52
Now find the difference from the starting price (January) from the end price (May)
3.52-2.65=.87
The difference from gas prices is $.87.
Hence, The solution is, By $.87 dollars has the price of gas changed from January to May.
To learn more on subtraction click:
brainly.com/question/2346316
#SPJ2
ABCD is a trapezoid. = 2.5 cm, and = 4 cm, and the area = 6.5 cm2. What is the altitude of the trapezoid?
Answer:
Altitude of the trapezoid is 2 cm.
Step-by-step explanation:
ABCD is a trapezoid. Since two bases of the trapezoid are
B1 = 2.5 cm and B2 = 4 cm.
Area given is 6.5 cm²
Then we have to calculated he altitude of the trapezoid.
Now we know the formula of area of the trapezoid = [tex]\frac{1}{2}[/tex] ( Base1 + Base2)
By putting the values in the formula
6.5 = [tex]\frac{1}{2}[/tex] ( 2.5 + 4 ) × h
6.5 = [tex]\frac{6.5}{2}[/tex] × h
h = \frac{2\times 6.5}{6.5} = 2 cm.
Answer is altitude of the trapezoid is 2 cm.
Bart bought a digital camera with a list price of $219 from an online store offering a 6 percent discount. He needs to pay $7.50 for shipping. What was Bart's total cost?
$205.86
$211.50
$213.36
During her first year of subscribing to a newspaper, Christie paid $48. During each subsequent year, the annual cost was 1.5 times the price paid the previous year. Which of the following equations may be used to calculate the total cost, C, of subscribing to the newspaper after n years?
4 is to 5 as 10 is to a
A.8
B.12
C.12.5
D.20
Which fraction is less than 1/2 a.3/8 b.5/8 c.5/7 d.9/16
You are enrolled in a wellness course at your college. You achieved grades of 70, 86, 81, and 83 on the first four exams. The fina exam counts the same as an exam given during the semester. A.) If x represents the grade on the final exam, write an expression that represents your course average (arithmetic mean). B.) If your average is greater than or equal to 80 and less than 90, you will earn a B in the course. Using the expression from part A for your course average, write a compound inequality that must be satisfied to earn a B.
The answer provides an expression for the course average and a compound inequality to earn a B in a wellness course.
Expression representing the course average: 1/5(70 + 86 + 81 + 83 + x)
Compound inequality for earning a B: 80 ≤ 1/5(70 + 86 + 81 + 83 + x) < 90
Distribution with a mean of 100 and standard deviation of 15...
What is the percentile score of
112?
82?
The percentile scores for a distribution with a mean of 100 and a standard deviation of 15, the z-scores for 112 and 82 are 0.8 and -1.2, respectively.
Calculating Percentile Scores
To determine percentile scores for the values given in the original student question, we need to calculate the z-scores for 112 and 82 from a distribution with a mean of 100 and a standard deviation of 15.
To find the z-score, use the formula:
Z = (X - mu) / sigma
Where X is the raw score, \\mu is the mean, and \\sigma is the standard deviation.
For 112:
Z = (112 - 100) / 15
Z = 12 / 15
Z = 0.8
For 82:
Z = (82 - 100) / 15
Z = -18 / 15
Z = -1.2
Once the z-scores are calculated, we can look up the corresponding percentiles in a standard normal distribution table.
A z-score of 0.8 corresponds to approximately the 78.81st percentile, meaning about 78.81% of the scores are below 112.
A z-score of -1.2 corresponds to approximately the 11.54th percentile, meaning about 11.54% of the scores are below 82.
What is the value of mc009-1.jpg?
–34
–2
10
34
the pic is the .jpg
The value of the expression -4²+(5-2)(-6) is -34 when following the standard rules of arithmetic.
Explanation:The value of the expression −4²+(5−2)(−6) can be determined by first calculating the power and then performing the operations inside the parentheses, followed by the multiplication and addition/subtraction.
The power of −4² is 16 because the negative sign is not inside the parentheses, so it is not squared, and 16 will be considered as a negative number here (-16).
Next, we calculate the value within the parentheses (5−2), which gives us 3.
Then we multiply 3 by −6, which equates to −18. Finally, we add the two results: −16 + (−18) = −16 −18 = −(16 + 18) = −(34), resulting in the value −34.
Shelly used exactly 1/ 3 of a cup of oil and 3/ 8 of a cup of water in a recipe. How many cups of liquid did Shelly use in all?
Answer: =[tex]\dfrac{17}{24}[/tex]
Step-by-step explanation:
Given : The amount of oil is used in a recipe. = [tex]\dfrac{1}{3}[/tex]
The amount of water used in a recipe = [tex]\dfrac{3}{8}[/tex]
The total number of cups of liquid used by Shelly:-
[tex]\dfrac{1}{3}+\dfrac{3}{8}=\dfrac{8+3\times3}{24}\\\\\=\dfrac{8+9}{24}=\dfrac{17}{24}[/tex]
Hence, the total number of cups of liquid used in all =[tex]\dfrac{17}{24}[/tex]
the GCF of the following numbers. 32 and 40
a homeowner has 200 feet of fence to enclose an area for a pet.
(a) if the area is given by A(x)=X(100-x), what dimension maximize the area inside the fence?
(b) what is the maximum area?
(c) determine the domain and range of A in this application.
(a) The function describes a parabola that opens downward. The value of A(x) is zero when x=0 and when x=100. The vertex (maximum) is halfway between those zeros, at x=50. The dimensions are 50 and 100-50 = 50. The maximum area pen will be 50 ft square.
(b) A(50) = (50 ft)² = 2500 ft² . . . the maximum area
(c) A(x) will be negative if x < 0 or x > 100, so the domain is 0 ≤ x ≤ 100.
The value of A(x) ranges from 0 to its maximum, 2500. Hence the range is 0 ≤ A(x) ≤ 2500.
For a rectangular fenced area, the dimensions that maximize the area are 50 feet by 50 feet, given a total of 200 feet of fence. The maximum possible area is 2500 square feet. The domain for this function is [0, 100] and the range is [0, 2500].
Explanation:This problem deals with maximizing an area given a certain length of fencing. It is a part of optimization problems often encountered in calculus. Firstly, we understand that a rectangle will give maximum area for a fixed perimeter. Considering a rectangle, the dimensions would be x and 100-x, where x is one of the sides of the rectangle.
(a) To find the dimension that maximizes the area, we first need to find the derivative A'(x) of A(x), which equals to -2x + 100. Setting A'(x) to zero gives us x = 50. That means the dimensions that will give us maximum area are 50 feet by 50 feet.
(b) Substituting x = 50 back into the original function, we find that the maximum area is A(50)=50(100-50)=50*50=2500 square feet.
(c) For the domain of A in this application, it would be all possible values of x. So, x could be any number from 0 to 100 (both inclusive). The range of A would be from 0 to the maximum value, 2500 square feet.
Learn more about Optimization here:https://brainly.com/question/37742146
#SPJ3
Scores on a certain test are normally distributed with a variance of 88. a researcher wishes to estimate the mean score achieved by all adults on the test. find the sample size needed to assure with 95 percent confidence that the sample mean will not differ from the population mean by more than 3 units.
To solve for the problem, the formula is:
n = [z*s/E]^2
where:
z is the z value for the confidence interval
s is the standard deviation
E is the number of units
So plugging that in our equation, will give us:
= [1.96*9.38/3]^2
= (18.3848/3)^2
= 6.1284^2
= 37.5 or 38
Which of the following is the simplified form of the seventh root of x times the seventh root of x times the seventh root of x ?