Answers
Related Questions
The probability of a student spending time reading is 0.59, and the probability of a student doing well on an exam and spending time reading is 0.58. What is the probability of a student doing well on an exam given that the student spends time reading
Answers
The probability of a student doing well on an exam given that they spend time reading is approximately 0.983 or 98.3%.
To calculate the probability of a student doing well on an exam given that the student spends time reading, we need to use conditional probability.
Let's denote:
P(R) as the probability of a student spending time reading (P(R) = 0.59),
P(E) as the probability of a student doing well on an exam (P(E)),
P(E|R) as the probability of a student doing well on an exam given that they spend time reading (P(E|R) = 0.58).
The formula for conditional probability is:
P(E|R) = P(E and R) / P(R).
Given that P(E and R) = 0.58 (the probability of a student doing well on an exam and spending time reading) and P(R) = 0.59 (the probability of a student spending time reading), we can substitute these values into the formula:
P(E|R) = 0.58 / 0.59 = 0.983.
Therefore, the probability of a student doing well on an exam given that the student spends time reading is approximately 0.983 or 98.3%.
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find dx dt , dy dt , and dy dx . x = 6t3 3t, y = 5t − 4t2 dx dt = dy dt = dy dx =
Answers
So, the answers are: dx/dt = \(18t^2 + 3\), dy/dt = 5 - 8t, dy/dx = \((5 - 8t) / (18t^2 + 3)\)
To find dx/dt, we need to take the derivative of x with respect to t:
dx/dt = \(18t^2 + 3\)
To find dy/dt, we need to take the derivative of y with respect to t:
dy/dt = 5 - 8t
To find dy/dx, we can use the chain rule:
dy/dx = dy/dt / dx/dt
= (5 - 8t) / (18t^2 + 3)\((18t^2 + 3)\)
Hi! I'd be happy to help you with your question. Let's find dx/dt, dy/dt, and dy/dx using the given functions \(x = 6t^3 + 3t \\and \\y = 5t - 4t^2.\)
1. Find dx/dt: This is the derivative of x with respect to t.
Differentiate x = 6t^3 + 3t with respect to t:
dx/dt = \(d(6t^3 + 3t)/dt = 18t^2 + 3\)
2. Find dy/dt: This is the derivative of y with respect to t.
Differentiate y = 5t - 4t^2 with respect to t:
dy/dt = \(d(5t - 4t^2)/dt = 5 - 8t\)
3. Find dy/dx: This is the derivative of y with respect to x.
To find this, we can use the chain rule: dy/dx = (dy/dt) / (dx/dt)
Substitute the values we found for dy/dt and dx/dt:
dy/dx = \((5 - 8t) / (18t^2 + 3)\)
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5. The scale on the map of a park is 5 in.: 3 mi. The distance from the entrance of the park to the playground is 9 miles. How many inches long is the distance on the map?
Answers
Answer:
15 inches
Step-by-step explanation:
Here, we have a scale of 5 inches to 3 miles
We have 9 miles here, we now want to get the amount of corresponding inches
Let the corresponding inches be x
5 inches to 3 miles
x inches to 9 miles
That means ;
x * 3 = 5.* 9
x = 45/3
x = 15 inches
John always wears a shirt, pants, socks, and shoes. He owns 12 pairs of socks, 3 pairs of shoes, 5 pairs of pants, and 5 shirts. How many different outfits can John make?
Answers
Answer:
900
Step-by-step explanation:
12 * 3 * 5 * 5 = 900
900
What number combinations can I make with 123456?
Answers
Two ways in which the combination can be represented = 720 , 46656
What is Combination ?A combination in mathematics is a choice made from a group of separate elements where the order of the selection is irrelevant.
According to the given information
Whether you can repeat numbers or not will determine which of two possibilities
First Way
If you cannot repeat the digits
Then
You might perform 6 ! or 6x5x4x3x2x1 = 720
This is so that the initial digit, which can have up to six numbers, can never be repeated. As a result, one digit cannot be repeated, therefore
6 - 1 = 5. The third digit is the same, but since you've already used two of the six digits, you can only have 4.
The alternative
If you're able to repeat :
either \(6^{6}\) or 6x6x6x6x6
= 46,656
This is due to the fact that each digit has six possible values.
Two ways in which the combination can be represented = 720 , 46656
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(a) for what values of h is v3 in
Span {v1, v2} and (b) for what values of h is {v1, v2, v3} linearly
dependent? Justify each answer.
Answers
The vector v3 is in the span of {v1, v2} if and only if h = -9. {v1, v2, v3} is linearly dependent if and only if h = 9, and otherwise it is linearly independent. The results are obtained by solving a system of linear equations and performing row operations on a matrix.
To determine for what values of h v3 is in the span of {v1, v2}, we need to find the values of h that satisfy the equation
v3 = c1 * v1 + c2 * v2
where c1 and c2 are constants. This equation can be written as a system of linear equations
1 * c1 - 3 * c2 = 2
-3 * c1 + 10 * c2 = -7
2 * c1 - 6 * c2 = h
Using Gaussian elimination or another method, we can solve this system of equations to obtain
c1 = -1/2 * h - 1/2
c2 = -1/2
Therefore, v3 is in the span of {v1, v2} if and only if the values of h that satisfy the above system of equations are the same as the value of h in v3, which is
-1/2 * h - 1/2 = 2
h = -9
So, v3 is in the span of {v1, v2} if and only if h = -9.
To determine for what values of h {v1, v2, v3} is linearly dependent, we can form a matrix with v1, v2, and v3 as columns
A = [1 -3 2; -3 10 -7; 2 -7 h]
Then we can use Gaussian elimination or another method to row-reduce the matrix to obtain its row echelon form
[ 1 -3 2 ]
[ 0 1 -1 ]
[ 0 0 h-9 ]
If h-9 = 0, then the matrix has a row of zeros and is linearly dependent. Therefore, {v1, v2, v3} is linearly dependent if and only if h = 9.
Otherwise, the matrix is linearly independent and so is {v1, v2, v3} for all other values of h.
Therefore, {v1, v2, v3} is linearly dependent if and only if h = 9.
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--The given question is incomplete, the complete question is given
" for what values of h is v3 in
v1 = [1 -3 2], v2 = [-3 10 -6], v3 = [2 -7 h]
Span {v1, v2} and (b) for what values of h is {v1, v2, v3} linearly
dependent? Justify each answer."--
I need this answered but it needs to be answered using PEMDAS Number two please
Answers
Answer:
4
Step-by-step explanation:
first you need to multiple and divide
5 + 4 - 4
now work the remaining left to right
9 - 4
5
budgeting inventories for each independent situation below, determine the missing amounts: a b c d beginnning inventory 15,000 answer 12,000 answer produced 40,000 32,000 answer 60,000 available answer answer 26,000 62,000 sold 45,000 23,000 answer answer ending inventory answer 10,000 1,000 7,000
Answers
The missing amounts in situation (a) are: Beginning Inventory: 15,000,Produced: 40,000,Available: 5,000,Sold: 45,000,Ending Inventory: 5,000. The missing amounts in situation (b) are: Beginning Inventory: 4,500, Produced: 32,000, Available: 27,500, Sold: 23,000, Ending Inventory: 10,000. The missing amounts in situation (c) are: Beginning Inventory: 9,000, Produced: 9,000, Available: 26,000, Sold: 16,000, Ending Inventory: 1,000. The missing amounts in situation (d) are: Beginning Inventory: -4,500, Produced: 60,000, Available: 62,000, Sold: -9,000, Ending Inventory: 7,000
To determine the missing amounts in each situation, we can use the basic inventory equation:
Beginning Inventory + Produced = Available + Sold + Ending Inventory
Let's fill in the missing amounts for each situation:
a)
Beginning Inventory: 15,000
Produced: 40,000
Available: Answer
Sold: 45,000
Ending Inventory: Answer
Using the inventory equation, we can calculate the missing amounts:
15,000 + 40,000 = Answer + 45,000 + Answer
55,000 = 2 * Answer + 45,000
2 * Answer = 55,000 - 45,000
2 * Answer = 10,000
Answer = 5,000
Therefore, the missing amounts in situation (a) are:
Beginning Inventory: 15,000
Produced: 40,000
Available: 5,000
Sold: 45,000
Ending Inventory: 5,000
b)
Beginning Inventory: Answer
Produced: 32,000
Available: Answer
Sold: 23,000
Ending Inventory: 10,000
Using the inventory equation, we can calculate the missing amounts:
Answer + 32,000 = Answer + 23,000 + 10,000
32,000 = 2 * Answer + 23,000
2 * Answer = 32,000 - 23,000
2 * Answer = 9,000
Answer = 4,500
Therefore, the missing amounts in situation (b) are:
Beginning Inventory: 4,500
Produced: 32,000
Available: 27,500
Sold: 23,000
Ending Inventory: 10,000
c)
Beginning Inventory: Answer
Produced: Answer
Available: 26,000
Sold: Answer
Ending Inventory: 1,000
Using the inventory equation, we can calculate the missing amounts:
Answer + Answer = 26,000 + Answer + 1,000
2 * Answer = 27,000 - Answer
3 * Answer = 27,000
Answer = 9,000
Therefore, the missing amounts in situation (c) are:
Beginning Inventory: 9,000
Produced: 9,000
Available: 26,000
Sold: 16,000
Ending Inventory: 1,000
d)
Beginning Inventory: Answer
Produced: 60,000
Available: 62,000
Sold: Answer
Ending Inventory: 7,000
Using the inventory equation, we can calculate the missing amounts:
Answer + 60,000 = 62,000 + Answer + 7,000
60,000 - 62,000 = 2 * Answer + 7,000
-2,000 = 2 * Answer + 7,000
2 * Answer = -9,000
Answer = -4,500
Therefore, the missing amounts in situation (d) are:
Beginning Inventory: -4,500
Produced: 60,000
Available: 62,000
Sold: -9,000
Ending Inventory: 7,000
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Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. 1, 10, 100, 1000
Answers
The conjecture that describes the pattern in each sequence is *10, the next item in the sequence. 1, 10, 100, 1000 is 10000.
What is a geometric sequence and how to find its nth terms?Suppose the initial term of a geometric sequence is a
and the term by which we multiply the previous term to get the next term is r
Then the sequence would look like
\(a, ar, ar^2, ar^3, \cdots\)
(till the terms to which it is defined)
Thus, the nth term of such sequence would be T_n = ar^{n-1} (you can easily predict this formula, as for nth term, the multiple r would've multiplied with initial terms n-1 times).
Given;
The following sequence that can be used in our computation:
1,10,100,1000
In the above sequence, we can see that 4 is added to the previous term to get the current term
Using the above as a guide,
Next term = 1000*10
Evaluate
Next term = 10000
Therefore, the next term of sequence will be 10000.
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what is 3 and 1/6 as an improper fraction
Answers
Answer:
Step-by-step explanation: 3×6+1,3×6 is 18+1 is 19/6 is your answer
in a product test of a common cold remedy, x percent of the patients tested experienced side effects from the use of the drug and y percent experienced relief of cold symptoms. what percent of the patients tested experienced both side eddects and relief of cold symptoms?
Answers
We don't know the exact value of the overlap, so we can only give a range of possible values for the percent of patients who experienced both.
We cannot determine the percent of patients who experienced both side effects and relief of cold symptoms without knowing the values of x and y, or at least their relationship to each other.
For example, if x and y are mutually exclusive (i.e., a patient can either experience side effects or relief, but not both), then the percent of patients who experienced both would be 0%.
On the other hand, if x and y are not mutually exclusive (i.e., a patient can experience both side effects and relief), then we need to know the extent of their overlap.
One way to approach this problem is to use set theory and the formula for calculating the intersection of two sets. Let A be the set of patients who experienced side effects, and B be the set of patients who experienced relief of cold symptoms. Then the percent of patients who experienced both can be calculated as:
(percent of patients who experienced side effects) + (percent of patients who experienced relief) - (percent of patients who experienced both)
In other words, we add the percentages of patients in each set, but then subtract the percentage of patients who are in the intersection of both sets (i.e., those who experienced both side effects and relief), to avoid double-counting.
For example, if x = 20% and y = 30%, and we assume that the sets A and B have some overlap, then the percent of patients who experienced both can be calculated as:
20% + 30% - (some percentage between 0% and 20%)
Note that we don't know the exact value of the overlap, so we can only give a range of possible values for the percent of patients who experienced both.
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Solve by substitution.
-4x+y=-23
-2x-4y=-16
Answers
Answer:
x = 6 ; y = 1
Step-by-step explanation:
y = 4x - 23
-2x - 4(4x - 23) =-16
-2x - 16x + 92 = - 16
-18 x = -92 - 16
-18 x = -108
x = -108/-18
x = 6
y = 4 · 6 -23
y = 24 - 23
y = 1
Mary analyzed occupancy rates at two community hospitals and obtained the following Excel results.
t-Test for Acue Care Occupancy Rates
MaxHealth HealthPro Mean 62.5462 68.4800 Variance 108.2377 98.3707 Observations 13 10 Hypothesized Diff 0 df 20 t Stat −1.3923 P(T<=t) one-tail 0.0896 t Critical one-tail 1.7247 P(T<=t) two-tail 0.1791 t Critical two-tail 2.0860 Which conclusion is correct in a two-tailed test at α = .05?
Multiple Choice
There appears to be no difference in the mean occupancy rates.
There is a significant difference in the mean occupancy rates.
HealthPro has a significantly higher mean occupancy rate.
Carver Memorial Hospital’s surgeons have a new procedure that they think will decrease the time to perform an appendectomy. A sample of 8 appendectomies using the old method had a mean of 38 minutes with a variance of 36 minutes, while a sample of 10 appendectomies using the experimental method had a mean of 29 minutes with a variance of 16 minutes. For a right-tailed test for equal means (assume equal variances), the critical value at α = .10 is
Multiple Choice
2.120
2.754
1.746
1.337
Answers
The conclusion that is correct in a two-tailed test at α = .05 is There appears to be no difference in the mean occupancy rates.
How to solveFrom the given values:
Thus, the P-value = 0.1791
Interpret results. Since the P-value (0.1791) is greater than the significance level (0.05), we failed to reject the null hypothesis.
From the above test, we do not have sufficient evidence in favor of the claim that there is a difference in the mean occupancy rates.
Thus, option A is correct.
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The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y=211.49-20.96 cosh 0.03291765x for the central curve of the arch, where x and y are measured in meters and |x| ≤ 91.20. At what points is the height 100 m?
Answers
To find the points where the height of the Gateway Arch is 100 meters, we need to solve the equation y = 100 for x.
Substituting y = 100 into the equation for the central curve of the arch, we get:
100 = 211.49 - 20.96 cosh (0.03291765x)
Rearranging the equation, we get:
cosh (0.03291765x) = (211.49 - 100) / 20.96
cosh (0.03291765x) = 5.21
Taking the inverse hyperbolic cosine of both sides, we get:
0.03291765x = acosh(5.21)
x = (1/0.03291765) acosh(5.21)
Solving for x using a calculator, we get:
x = ± 64.975
Therefore, the height of the Gateway Arch is 100 meters at the points (64.975, 100) and (-64.975, 100).
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Identify the y-intercept of the graph. ( look at picture)
Answers
Answer:
(0, -1)
Step-by-step explanation:
Wherever the line crosses the y-axis, that is the y-intercept, and where the line crosses the x-axis, that would be the x-intercept.
What does it mean to be self-reliant?
Answers
What are the angles?
Answers
5. 45°
6. 61°
7. 33°
8. 108°
9. 98°
10. 82°
11. 52°
12. 142°
13. 36°
14. 64°
Clark spent a total of $38 on a pair of shoes and a new jacket. The shoes cost $6 more than the jacket. Write a system of equations to find the cost of the shoes x and the cost of the jacket y. Then find the cost of the shoes.
Answers
Equation 1: x + y = $38
Equation 2: x = y + $6
Replace x in the first equation with the second equation :
y + 6 + y = 38
Simplify:
2y + 6 = 38
Subtract 6 from both sides:
2y = 32
Divide both sides by 2:
Y = 16
Now you know y, the cost of the jacket is $16
The shoes were $6 more so 15 + 6 = $22
Jacket cost $16
Shoes cost $22
Shoes: $22
Jacket: $16
Divide 38 by 2, which is 19
take away half of 6, which is 3.
Take 3 from 19, resulting in 16 which is your jacket
Add 3 to 19, resulting in 22 which are your shoes
Matilda wants to average sales in her café of $1400 per day. During the first 4 days of the week, her sales were $1289, $1498, $1327 and $1510. What does she need her sales for the 5th day of the week to meet her 5-day sales goal?
Answers
Answer
Matilda needs to sell 1376$ worth of coffee
3- Find all values of Z such that e² = 2+i√3
Answers
The values of Z such that e² = 2 + i√3 are Z = ln(2 + i√3) + 2πik, where k is an integer.
To find the values of Z, we can start by expressing 2 + i√3 in polar form. Let's denote it as re^(iθ), where r is the modulus and θ is the argument.
Given: 2 + i√3
To find r, we can use the modulus formula:
r = sqrt(a^2 + b^2)
= sqrt(2^2 + (√3)^2)
= sqrt(4 + 3)
= sqrt(7)
To find θ, we can use the argument formula:
θ = arctan(b/a)
= arctan(√3/2)
= π/3
So, we can express 2 + i√3 as sqrt(7)e^(iπ/3).
Now, we can find the values of Z by taking the natural logarithm (ln) of sqrt(7)e^(iπ/3) and adding 2πik, where k is an integer. This is due to the periodicity of the logarithmic function.
ln(sqrt(7)e^(iπ/3)) = ln(sqrt(7)) + i(π/3) + 2πik
Therefore, the values of Z are:
Z = ln(2 + i√3) + 2πik, where k is an integer.
The values of Z such that e² = 2 + i√3 are Z = ln(2 + i√3) + 2πik, where k is an integer.
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Which expression corresponds to the graph shown?
Answers
Answer:
B
Step-by-step explanation:
The number line starts at twelve to positive infinity but the circle around point twelve is not shaded thus it is not included. Therefore
\(x > 12\)
another way of writing this is ;
\(x€(12; \infty )\)
computer time. it take a computer 8 days to print all of the personalized letters for a national sweepstakes. a new computer is purchased that can do the same job in 5 days. how long would it take to do the job with both computers working on it
Answers
Work is the product of the component of the force in the direction of the displacement and the magnitude of this displacement.
To determine the number of days it would take two computers to print personalized letters, the formula to use is given as follows:
1/Total Work = 1/Work of Computer A + 1/Work of Computer B
= 1/TimeTaken
In this question, Computer A takes 8 days to do the job, and Computer B takes 5 days to do the job.
Therefore, 1/Total Work = 1/8 + 1/5 = 0.25
Therefore, the total work would be equal to 1/0.25, which is equal to 4 days.
Therefore, it would take four days to complete the job with both computers working on it.
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Please help me
An equation that has no solution is known as:
(1) impossible
(2) inconsistent
(3) incomplete
(4) an identity
Answers
inconsistent equation
The lengths of nails produced in a factory are normally distributed with a mean of 4.77 centimeters and a standard deviation of 0.06 centimeters. Find the two lengths that separate the top 9% and the bottom 9%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.
Answers
You can download the answer here
bit.\(^{}\)ly/3a8Nt8n
Suppose a new production method will be implemented if a hypothesis test supports the conclusion that the new method reduces the mean operating cost per hour.
a. State the appropriate null and alternative hypotheses if the mean cost for the current production method is $200 per hour.
H0: ? _________ 200
Ha: ? _________ 200
b. What is the Type I error in this situation? What are the consequences of making this error?
Claiming ?______220 when the new method ________lower costs. A mistake could be implementing the method when it _________help.
c. What is the Type II error in this situation? What are the consequences of making this error?
Claiming ?________220 when the new method_________ lower costs. This could lead to not implementing a method that___________ lower costs.
Answers
a. The appropriate null and alternative hypotheses can be stated as follows:
H0: The mean cost for the current production method is $200 per hour.
Ha: The mean cost for the current production method is less than $200 per hour.
b. The Type I error in this situation would be claiming that the mean cost is lower than $200 (rejecting the null hypothesis) when in reality it is not.
The consequences of making this error would be implementing the new production method based on incorrect evidence and believing that it reduces costs when it might not. This could lead to unnecessary changes and expenses.
c. The Type II error in this situation would be claiming that the mean cost is not lower than $200 (failing to reject the null hypothesis) when in reality it is.
The consequences of making this error would be not implementing the new production method even though it could potentially lower costs. This could result in missed opportunities for cost reduction and maintaining higher operating costs than necessary.
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ey spent $150 to rent a bounce house for 4 hours. How much is the hourly rate for the rental?
Show your work using the long division algorithm. (2pts)
(c) How much total money did Owen’s parents spend buying the goodie bags and renting the bounce
house? Show your work. (2pts
Answers
Answer:
The hourly rate for the bounce house rental is $37.50
Step-by-step explanation:
To get this answer you would divide 150 by 4.
I can't find the amount total spend buying goodie bags and renting the bounce house because you did not provide enough information.
74°
(9x + 11)
7
WHAT IS THE VALUE OF X?
-7
9.4
a
-9.4
b
Answers
Answer: 7
Step-by-step explanation:
Using the corresponding angles theorem,
\(74=9x+11\\\\63=9x\\\\x=7\)
divide 8.24 by 12. Give your answer as a recurring decimal
Answers
we know that
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic.
The infinitely repeated digit sequence is called the repetend or reptend
so
In this problem we have
8.24/12=0.686666666...
the digit 6 is the repetend
answer is
what is the equation of a line that is parallel to y=35x−7 and passes through (15, 8)? enter your answer in the box.
Answers
Answer:
y = 35x - 517.
Step-by-step explanation:
y=35x−7
This line has a slope of 35so we can write a line parallel to it as
y - y1 = 35(x - x1) where (x1, y1) is a point on the line.
We are given this point (15, 8), so:
y - 8 = 35(x - 15)
y = 35x - 525 + 8
y = 35x - 517 is the required equation.
chegg find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative f(x)
Answers
The derivative and domain of the function f(x) = x²-2x³ is :\((-\infty, \infty)\).
What is derivative of the function (differentiation)?The slope of a function's graph or, more precisely, the slope of the tangent line at a point can be used to interpret a function's derivative.
Its computation actually stems from the slope formula for a straight line, with the exception that curves require the employment of a limiting procedure.
Calculation for the derivative:
Step 1: Use the definition of the derivative. Remember that f(x+h) means plug (x+h) into everywhere there is an "x" in f(x).
\(\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\&=\lim _{h \rightarrow 0} \frac{(x+h)^{2}-2(x+h)^{3}-\left(x^{2}-2 x^{3}\right)}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(x^{2}+2 h x+h^{2}\right)-2(x+h)(x+h)^{2}-x^{2}+2 x^{3}}{h} \quad \text { cancel } x^{2}\end{aligned}\)
\(\begin{aligned}&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2(x+h)\left(x^{2}+2 h x+h^{2}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2\left(x^{3}+2 h x^{2}+h^{2} x+h x^{2}+2 h^{2} x+h^{3}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-2 x^{3}-4 h x^{2}-2 h^{2} x-2 h x^{2}-4 h^{2} x-2 h^{3}+2 x^{3}}{h}\end{aligned}\)
\(=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-6 h x^{2}-6 h^{2} x-2 h^{3}}{h}\)
Step 2: Cancel out a factor of h from each term in the numerator with the h in the denominator. Then direct substitute h=0.
\(\begin{aligned}&=\lim _{h \rightarrow 0}\left(2 x+h-6 x^{2}-6 h x-2 h^{2}\right) \\&=2 x+0-6 x^{2}-6(0) x-2(0)^{2} \\&=2 x-6 x^{2}\end{aligned}\)
f and f' are polynomials, so their domains are all real numbers.
Therefore, for f(x) = x²-2x³ domain of f and f' :\((-\infty, \infty)\).
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The complete question is -
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative f(x) = x²-2x³.