let f(x) = integral of x^2-3x to -2, et^2 dt, at which value of x is f(x) a minimum
Answers
f''(0) is negative and f''(3) is positive, we can conclude that x = 3 corresponds to the minimum of f(x). at x = 3, f(x) is a minimum.
To find the value of x at which f(x) is a minimum, we need to find the critical points of the function f(x). The critical points occur where the derivative of f(x) is equal to zero or does not exist. Let's calculate the derivative of f(x) and find its critical points.
First, let's find the indefinite integral of the function x² - 3x:
∫(x^2 - 3x) dx = (1/3)x³ - (3/2)x² + C
Next, let's evaluate the definite integral of et² with the limits from -2 to x:
f(x) = ∫[(1/3)t³ - (3/2)t²] from -2 to x
Applying the Fundamental Theorem of Calculus, we can evaluate the definite integral:
f(x) = [(1/3)x³ - (3/2)x²] - [(1/3)(-2)³ - (3/2)(-2)²]
= (1/3)x³ - (3/2)x² - (8/3) + 6
Simplifying further:
f(x) = (1/3)x³ - (3/2)x² + 10/3
To find the critical points, we need to find where the derivative of f(x) is equal to zero:
f'(x) = d/dx [(1/3)x³ - (3/2)x² + 10/3]
= x² - 3x
Setting f'(x) equal to zero:
x² - 3x = 0
Factoring out x:
x(x - 3) = 0
This equation has two solutions:
x = 0 and x = 3
Now, let's analyze these critical points to determine which one corresponds to a minimum.
To do that, we can calculate the second derivative of f(x) and evaluate it at each critical point:
f''(x) = d²/dx² [(1/3)x³ - (3/2)x² + 10/3]
= 2x - 3
Substituting x = 0 into f''(x):
f''(0) = 2(0) - 3 = -3
Substituting x = 3 into f''(x):
f''(3) = 2(3) - 3 = 3
Since f''(0) is negative and f''(3) is positive, we can conclude that x = 3 corresponds to the minimum of f(x).
Therefore, at x = 3, f(x) is a minimum.
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Related Questions
Stacy put a fence around a rectangular flower bed with a length of 18ft and a width of 4.5 ft. Next, he enclosed a square flower bed that had the same area. What number fo feet of fence did he need to enclose the entire square flower bed?
Answers
Answer:
36 feet
Step-by-step explanation:
Let us first find the area of the of the flower bed. It is a rectangle, so:
A = 18 * 4.5 = 81 square feet
The square flower bed has the same area, therefore, the length of the sides of the flower bed can be gotten. The area of a square is:
\(A = L^2\)
\(81 = L^2\\\\L = \sqrt{81}\)
L = 9 feet
Next, we need to find the perimeter of the square flower bed:
P = 4L
=> P = 4 * 9 = 36 feet
He needed 36 feet of fence to enclose the entire square flower bed.
the big box (made up of 5 small boxes) on the ecg paper measures:
Answers
When we measure a ECG waveform the large boxes represents 0.2 seconds.
To determine the measurements of the big box on ECG paper, we need to consider the standard calibration of ECG paper.
In most ECG papers, the standard calibration is as follows:
The horizontal distance (width) of a big box is typically 0.20 seconds.The vertical distance (height) of a big box is typically 0.5 millivolts (mV).
These measurements allow for accurate interpretation of the ECG waveform and analysis of heart activity.
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3(2x - 3) + 2x = 3x + 11
Answers
Answer:
X=4
Step-by-step explanation:
Answer:
6x-9+2x=3x+11, finish the brackets
Collect like terms
6x+2x-3x=11+9
Solve
5x=20
x=20/5
X=4
Step-by-step explanation:
I hope I explained well, please give me brainliest
What is an expression involving division that is equivalent to the expression below?
2/9 x 3/4
Answers
Answer:
2/9 ÷ 4/3
Step-by-step explanation:
2. (15 points) You need to study for your math and physics finals. Your grade on both exams is proportional to the number of hours per day (so capped at 24 total hours) you spend studying that subject times your ability to absorb the material. You estimate that your ability to absorb math material falls from 100% by 4% for every hour you spend studying math and Page 2 of 4 by 3% for every hour you spend studying physics. Similarly, you find your ability to absorb physics material falls by 2% for each hour of studying math and by 5% for each hour of studying physics. (a) Find the optimal number of hours that you should study math and physics to maximize your average grade. Set the problem up and define all the variables and find the answer. (b) If you studied the optimal number of hours you found in part (a), which exam would you be better prepared for? (c) Now assume that your exam scores in both subjects improves by an additional 10% for every hour of the day that you do not study math or physics. Now what is the optimal number of hours spent studying for the two exams? (d) Examine the sensitivity of your answer to the additional 10\% benefit he gets from the time not studying. How would increasing the effect of this benefit affect your average grade?
Answers
A. These equations will give us the optimal values of x and y.
B. (b) To determine which exam you would be better prepared for, we compare the Math_grade and Physics_grade using the optimal values obtained in part (a).
C. We can follow the same steps as in part (a) to set up the equations and find the optimal number of hours spent studying for the two exams.
D. the specific impact would depend on the magnitude of the increase and how it interacts with the other variables and constraints in the problem.
(a) Let's define the variables:
Let x represent the number of hours spent studying math.
Let y represent the number of hours spent studying physics.
We want to maximize the average grade, which is proportional to the product of the hours spent studying and the ability to absorb the material.
The math grade is proportional to x * (1 - 0.04x) * (1 - 0.02y)
The physics grade is proportional to y * (1 - 0.03y) * (1 - 0.05x)
We want to maximize the average grade:
Average grade = (math grade + physics grade) / 2
Now, let's set up the optimization problem:
Maximize: (x * (1 - 0.04x) * (1 - 0.02y) + y * (1 - 0.03y) * (1 - 0.05x)) / 2
Subject to:
0 ≤ x ≤ 24 (maximum hours per day)
0 ≤ y ≤ 24 (maximum hours per day)
To solve this optimization problem, we can use calculus or numerical optimization methods.
(b) To determine which exam you would be better prepared for, calculate the average grade for the optimal number of hours obtained in part (a). Compare the average grade of math and physics to determine which one is higher.
(c) Now, assuming the exam scores improve by an additional 10% for every hour not studying, we need to modify the objective function in the optimization problem. Let's redefine the variables:
Let x represent the number of hours spent studying math.
Let y represent the number of hours spent studying physics.
Let z represent the number of hours not spent studying (24 - x - y).
The math grade is proportional to x * (1 - 0.04x) * (1 - 0.02y)
The physics grade is proportional to y * (1 - 0.03y) * (1 - 0.05x)
The additional benefit is 10% * z.
The new objective function to maximize the average grade is:
(x * (1 - 0.04x) * (1 - 0.02y) + y * (1 - 0.03y) * (1 - 0.05x) + 0.1 * z) / 2
Subject to:
0 ≤ x ≤ 24 (maximum hours per day)
0 ≤ y ≤ 24 (maximum hours per day)
z = 24 - x - y
(d) To examine the sensitivity of the answer to the additional 10% benefit from not studying, you can perform sensitivity analysis by changing the percentage value and observing the effect on the average grade. Increasing the effect of this benefit would likely result in a higher average grade as more emphasis is placed on the additional benefit gained from not studying. However, the exact impact would depend on the specific values used and the relationships between the variables and grades.
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two car rental companies charge a one-time fee and a millage rate for renting a car the graph below compares the cost for renting cars from the two companies based on the graph which of the following statements appears to be true about the costs of renting cars from the two companies
Answers
From the graph we notice that the lines are parallel. This means that the slopes of the lines representing the cost are the same. Since the slope represents the price per mile travel we conclude that:
The mileage at both companies is the same.
Therefore the answer is B
I need help. Please answer if you know the answer.
Thank you.
picture ↓
Answers
Answer:
your answer is 3.2 in
Step-by-step explanation:
A and B are two events. Let P(A) = 0.65, P (B) = 0.17, P(A|B) = 0.65 and P(B|4) = 0.17 Which statement is true?
1. A and B are not independent because P(A|B) + P(A) and P(B|4) + P(B).
2. A and B are not independent because P (A|B) + P(B) and P(B|4) + P(A)
3. A and B are independent because P (A|B) = P(A) and P(BIA) = P(B).
4. A and B are independent because P (A|B) = P(B) and P(B|A) = P(A).
Answers
Answer:
the statement that is true is: A and B are not independent because P(AIB) + P(B) is not equal to P(BIA) + P(A)
Step-by-step explanation:
ur welcome
Identify the angle pair as either corresponding angles, alternate interior angles, same side interior angles.Find the measure of each angle indicated.7)8)+71Angle TypeAngle Measure7
Answers
7) alternate interior angle, Angle measure = 71°
8) alternate interior angle, Angle measure = 90°
Explanation:
Using an illustration:
7) The angle type is alternate interior angle
Alternate interior angle angles are equal.
Hence, the angle measure which is represented with question mark = 71°
8) The angle type is alternate interior angle
Alternate interior angle angles are equal.
The square in red represents 90°
Hence, the angle measure which is represented with question mark is 90°
what is x and y of 2x-4y=10&x+5y=40
Answers
Answer:
\(x = 15, y = 5\)
Step-by-step explanation:
We are given that
\(2x - 4y = 10\\x + 5y = 40\)
When given these types of equations, we can manipulate them in order to cancel out a variable. Cancelling either the x or y works, choose which one is easier.
In this case, we can multiply the second equation by -2 and easily cancel out the x's
\(2x - 4y = 10\\-2x - 10y = -80\)
Now we can add the two equations
\(-4y + (-10y) = 10 + (-80)\\-4y - 10y = -70\\-14y = -70\\y = 5\)
We can substitute y into the second equation and solve for x
\(x + 5(5) = 40\\x + 25 = 40\\ x = 15\)
Check:
2(15) - 4(5)
30 - 20 = 10
it works
A bank advertises a 2/1 arm at 2.75% with a 2/10 cap. what is the maximum interest rate that can be charged during the fifth year? a. 2.75% b. 4.75% 6c. 75% d. 8.75%
Answers
The maximum interest rate that can be charged during the fifth year is 6.75%. The correct option is the third option c. 6.75%
Calculating the maximum interest rate that can be chargedFrom the question, we are to determine the maximum interest rate that can be charged during the fifth year
A 2/1 ARM means that the interest rate is fixed for the first two years and then adjusts annually based on a specified index plus a margin. The "2/10 cap" means that the interest rate can only increase or decrease by a maximum of 2 percentage points each year, and can never be more than 10 percentage points higher than the initial rate.
Since the initial rate is 2.75%, the maximum rate that can be charged during the fifth year is:
2.75% + (2 x 2%) = 6.75%
Hence, the maximum interest rate that can be charged during the fifth year is 6.75%.
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Dalvin took a taxi from his house to the airport. The taxi company charged a pick-up fee of $4.80 plus $2 per mile. The total fare was $30.80, not including the tip. Which tape diagram could be used to represent the context if mm represents the number of miles in the taxi ride?
Answers
Answer:
The equation should be 2.5x+1.8=59.3
Step-by-step explanation:
The number of miles in the taxi ride is 10
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
Given;
Dalvin took a taxi from his house to the house. The taxi company charged a pick-up fee of $4.80;
The fare per mile is $2
The total fare was $30.80, not including the tip;
Let the number of mile is x
Total fare = pick up fee + fee per mile
30.80 = 4.80 + 2x
2x = 26
x = 13
Therefore, the number of miles by algebra Dalvin took a taxi will be 13
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you count 55 cells in the picture. the field of view is 1.85 mm x 1.23 mm. estimate how many cells are in your t75 flask.
Answers
Based on the given information, the estimate for the number of cells in a T75 flask can be calculated by comparing the number of cells in the picture to the field of view area and then scaling it up to the size of the T75 flask.
Given that there are 55 cells in the picture, we can use this information to estimate the density of cells in the field of view. The field of view has dimensions of 1.85 mm x 1.23 mm, which gives an area of 2.7095 square millimeters (\(mm^2\)). To calculate the cell density, we divide the number of cells (55) by the area (2.7095 \(mm^2\)), resulting in an approximate cell density of 20.3 cells per \(mm^2\).
Now, to estimate the number of cells in a T75 flask, we need to know the size of the flask's growth area. A T75 flask typically has a growth area of about 75 \(cm^2\). To convert this to \(mm^2\), we multiply by 100 to get 7500 \(mm^2\).
To estimate the number of cells in the T75 flask, we multiply the cell density (20.3 cells/\(mm^2\)) by the growth area of the flask (7500 \(mm^2\)). This calculation gives us an approximate estimate of 152,250 cells in the T75 flask. It's important to note that this is just an estimate, and actual cell counts may vary depending on various factors such as cell size, confluency, and experimental conditions.
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RS: R(2, -1) and S(6,-5)
Translation: along (3, 4)
Reflection: in x = 2
Answers
Answer:
69r 420s
Step-by-step explanation:
you gotta do that and do this to that and after that you get that
HELP ASAP WILL GIVE FIRST ANSWER BRAINLEIST
Which graph represents the linear equation y equals one fourth times x plus 1 on the coordinate plane?
Answers
Answer:
Below
Step-by-step explanation:
y= 1/4 x + 1
The graph has a y-axis intercept of +1 (y = mx + b form)
last one looks correct....at x = 4 the value is 2 which is correct
The speed of a Garden-Snail is 50 meters per hour and that of the Cheetah is 120 kilome-
ters per hour. Find the ratio of the speeds
Answers
Answer:
50/120 =12:5
Step-by-step explanation:
First we are divide 50/120
answer is 12:5
Answer: 0.05 : 120. OR
50 : 120000 or 1:2400
Step-by-step explanation:
Speed of garden snail
= 50 meters per hour = 0.05 kph
Speed of cheetah
= 120 km per hour or 120000 mph
Ratio of snail : cheetah
= 0 05 : 120
What type of angles are 2 and 7?
A. Supplementary angles
B. Alternate exterior angles
C. Alternate interior angles
D. Vertical angles
Answers
Answer:
B: Alternate Exterior Angles
Can you please help me out with a question
Answers
Formular for total surface area of a cylinder
\(TSAofacylinder=2nr^2\text{ + 2nrh}\)\(\begin{gathered} T\mathrm{}S\mathrm{}A\text{ = 2 }\times\text{ }\pi\text{ }\times7^2\text{ + 2 }\times\pi\times7\times21ft^2^{} \\ \text{ = 307.876 + }923.628 \\ \text{ = 1231.504 ft}^2 \\ =1231.5ft^2\text{ (nearest tenth)} \end{gathered}\)S = 1231.5 sq ft
The number of hours Brittany has been hiking varies directly with the total distance she has traveled, in miles. She can hike 8 miles in 3 hours. How many miles can Brittany hike in 12 hours?
Answers
Answer:
24 miles
Step-by-step explanation:
for every 3 miles she can hike 8 miles so she can hike 24 miles in 12 hours
asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in span {a1, a2, a3}.
Answers
To determine if the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution, we can check whether the vector b is in the span of the vectors {a1, a2, a3}.
In linear algebra, the augmented matrix represents a system of linear equations. The columns a1, a2, and a3 correspond to the coefficients of the variables in the system, while the column b represents the constants on the right-hand side of the equations. To check if the system has a solution, we need to determine if the vector b is a linear combination of the vectors a1, a2, and a3.
If the vector b lies in the span of the vectors {a1, a2, a3}, it means that b can be expressed as a linear combination of a1, a2, and a3. In other words, there exist scalars (coefficients) that can be multiplied with a1, a2, and a3 to obtain the vector b. This indicates that there is a solution to the linear system.
On the other hand, if b is not in the span of {a1, a2, a3}, it implies that there is no linear combination of a1, a2, and a3 that can yield the vector b. In this case, the linear system does not have a solution.
Therefore, determining whether the vector b is in the span of {a1, a2, a3} allows us to determine if the linear system corresponding to the augmented matrix [a1 a2 a3 b] has a solution or not.
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What is the radius of a hemisphere with a volume of 5320\text{ in}^3,5320 in 3 , to the nearest tenth of an inch?
Answers
Answer:
13.6 in
Step-by-step explanation:
volume of a hemisphere = (2/3) x (n) x (r^3)
n = 22/7
r = radius
5320 = 2/3 x r^3 x 22/7
r^3 = 5320 x 3/2 x 7/22
find the cube root of both sides
r = 13.6
A town's population in 2019 was 438,000. The town's current population is 561,000. What is the percent of change?
Answers
Answer:
28.1%
Step-by-step explanation:
A town's population in 2019 was 438,000. The town's current population is 561,000. What is the percent of change?
The formula for Percent change =
Final value - Initial value/Initial value × 100
= 561000 - 438000/438000 × 100
= 123000/438000 × 100
= 28.082191781%
Approximately = 28.1%
Therefore, percent change = 28.1%
Evaluate the discriminant for each equation. Determine the number of real solutions. x²+4 x+5=0 .
Answers
The roots of the equation will be imaginary. Then the number of real solutions will be zero.
What is the discriminant of the quadratic equation?The discriminant represents the types of the root. And the discriminant (D) of the equation ax² + bx + c = 0 is given as
D = b² - 4ac
The type of roots will be given below.
D > 0, Real and Distinct rootsD = 0, Real and Equal rootsD < 0, Imaginary rootsThe equation is given below.
x² + 4x + 5 = 0
The discriminant of the given equation will be
D = 4² - 4 x 1 x 5
D = 16 - 20
D = - 4
D < 0
The roots of the equation will be imaginary. Then the number of real solutions will be zero.
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Can someone please help me with this?
Answers
Answer:
36\(\pi\) square units
Step-by-step explanation:
The formula for the circumference of a circle is C = 2\(\pi\)r.
If the circumference is 12\(\pi\), then that means:
12\(\pi\) = 2\(\pi\)r
After diving both sides by 2\(\pi\), you get 6 = r, meaning our radius is r.
The formula for the area of a circle is A = \(\pi r^{2}\).
As we've now found the radius to be 6, the area is equal to:
A = \(\pi (6)^{2}\)
After squaring 6, we get:
A = \(36\pi\)
Therefore, the area of a circle that has a circumference of \(12\pi\) is \(36\pi\) square units.
————————————————————————————-
Answers
Answer:q
Step-by-step explanation:
34. The distance a spring will stretch varies directly with how much weight is attached to the spring. If a spring stretches 8 inches with 90 pounds attached, how far will it stretch with 55 pounds attached? Round to the nearest tenth an inch
Answers
A cone is 14cm deep and has a base radius of 9/2cm. Calculate to 2dp the volume of the cone that will fill the cone halfway
Answers
The volume of the cone that will fill the cone halfway is \(198.72 cm3\) by the given base radius of 9/2cm and 14cm deep.
The formula for the volume of the cone is
\(V = (1/3)πr^2h\)
where h is the height of the cone, r is its base's radius and is a mathematical constant roughly equal to 3.14159.
We must first determine the height of the cone that is filled halfway in to calculate the volume of the cone that will fill it halfway.
Half of the cone's entire height, or 14 cm divided by 2, or 7 cm, corresponds to the height of the half-filled cone.
The cone's base has a radius of 9/2 cm.
We can calculate the volume of the cone that will fill the cone halfway using the formula for a cone's volume as follows:
\(V = (1/3)πr^2h\)
\(V = (1/3)π(9/2)^2(7) (7)\)
\(V ≈ 198.72 cm^3\) (rounded to 2 decimal places) (rounded to 2 decimal places)
Therefore, \(198.72 cm3\) is the volume of the cone that will fill it halfway.
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Let Y 1
,Y 2
and Y 3
be independent and identically distributed random variables with expectation E[Y i
]=μ and variance V[Y i
]=σ 2
for i=1,2,3. Suppose we want to estimate the expectation μ, and we propose to combine our three observations Y 1
,Y 2
and Y 3
by defining a weighted average: Y
ˉ
ω
=ω 1
Y 1
+ω 2
Y 2
+ω 3
Y 3
where ω 1
,ω 2
and ω 3
are constants between 0 and 1 . 1 (a) Calculate E[ Y
ˉ
ω
]. What condition do the weights ω 1
,ω 2
and ω 3
need to satisfy for Y
ˉ
ω
to be an unbiased estimator for μ ? (b) Calculate V[ Y
ˉ
ω
]. Using the condition you found on (a), find the set of weights that minimize the variance of Y
ˉ
ω
. Justify your answer.
Answers
The weighted average estimator Yˉω is considered to estimate the expectation μ. To be an unbiased estimator, the weights ω1, ω2, and ω3 should satisfy the condition that their sum is equal to 1.
(a) To calculate E[Yˉω], we can take the expectation inside the summation:
E[Yˉω] = E[ω1Y1 + ω2Y2 + ω3Y3]
= ω1E[Y1] + ω2E[Y2] + ω3E[Y3]
= ω1μ + ω2μ + ω3μ
= μ(ω1 + ω2 + ω3)
For Yˉω to be an unbiased estimator, its expectation should be equal to the parameter being estimated, which is μ. Therefore, we have the condition ω1 + ω2 + ω3 = 1.
(b) To calculate V[Yˉω], we need to determine the variance inside the weighted average:
V[Yˉω] = V[ω1Y1 + ω2Y2 + ω3Y3]
= ω1^2V[Y1] + ω2^2V[Y2] + ω3^2V[Y3]
= ω1^2σ^2 + ω2^2σ^2 + ω3^2σ^2
= σ^2(ω1^2 + ω2^2 + ω3^2)
To minimize the variance V[Yˉω], we need to find the weights ω1, ω2, and ω3 that minimize the expression ω1^2 + ω2^2 + ω3^2, while still satisfying the condition ω1 + ω2 + ω3 = 1. One approach to find the minimum variance is by using calculus techniques, such as Lagrange multipliers, to optimize the expression under the constraint. Solving this optimization problem will yield the specific weights that minimize the variance of Yˉω, and the justification lies in the mathematical derivation of the optimal solution.
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What is 88=10y + 30 - 4y?
Answers
Answer:
y = \(\frac{29}{3}\)
Step-by-step explanation:
88 = 10y + 30 - 4y , that is
88 = 6y + 30 ( subtract 30 from both sides )
58 = 6y ( divide both sides by 6 )
\(\frac{58}{6}\) = y , that is
y = \(\frac{29}{3}\)
Sales of a popular toy were about 20 million in 2000 and growing about 5% each year. At this growth rate, the function f(x) = 20(1.05)^x gives the annual number of toys sold in million in the xth year after 2000. Using this model, in about what year will the annual sales surpass 37 million?
Answers
===========================
Work Shown:
Plug in f(x) = 37. Solve for x. Use logarithms.
f(x) = 20(1.05)^x
37 = 20(1.05)^x
20(1.05)^x = 37
1.05^x = 37/20
1.05^x = 1.85
log( 1.05^x ) = log( 1.85 )
x*log( 1.05 ) = log( 1.85 )
x = log( 1.85 )/log( 1.05 )
x = 12.6088044498867
Round up to the nearest whole number to get x = 13.
In the year 2013, sales exceed 37 million.