) Solve the initial value problem: f'(x)= secx tanx +2. given ( 1 ) = √2 + 1 / 2 b) Evaluate: (x - 1)(x + 7 ) ax Q1 dx + cscx cotx dx ay Use substitution method to evaluate Sex cosx dx , 12 Marks)
Answers
a) The solution to the initial value problem is:
f(x) = (1/2)\(sec^2x\) + 2sec(x) + √2 + 1/2 - (1/2)\(sec^2(1)\) - 2sec(1)
b) ∫(x - 1)(x + 7)ax Q1 dx + ∫csc(x)cot(x) dx = (a/4)\((x + 3)^4\) - (5a/2)\((x + 3)^3\) + (26a/2)\((x + 3)^2\) - 48a(x + 3) - ln|sin(x)| + C2
a) To solve the initial value problem:
f'(x) = sec(x)tan(x) + 2
We can integrate both sides with respect to x to obtain:
f(x) = ∫ [sec(x)tan(x) + 2] dx
Using the substitution u = sec(x), du/dx = sec(x)tan(x), we have:
f(x) = ∫ [u + 2] du
f(x) = (1/2)\(u^2\) + 2u + C, where C is an arbitrary constant of integration.
Substituting back u = sec(x), we get:
f(x) = (1/2)\(sec^2x\) + 2sec(x) + C
To find the value of C, we use the given initial condition f(1) = √2 + 1/2:
f(1) = (1/2)\(sec^2(1)\) + 2sec(1) + C = √2 + 1/2
Solving for C, we have:
C = √2 + 1/2 - (1/2)\(sec^2(1)\) - 2sec(1)
Therefore, the solution to the initial value problem is:
f(x) = (1/2)\(sec^2(x)\) + 2sec(x) + √2 + 1/2 - (1/2)\(sec^2(1)\) - 2sec(1)
b) To evaluate:
∫(x - 1)(x + 7)ax Q1 dx + ∫csc(x)cot(x) dx
We can use the substitution method to evaluate the first integral. Let u = x + 3, du/dx = 1. Then, the integral becomes:
∫(u - 4)(u + 4) a(u - 3) Q1 du
Expanding and simplifying, we have:
∫[\(au^3 - 15au^2 + 52au - 48a\)] Q1 du
Integrating each term, we get:
\((a/4)u^4 - (5a/2)u^3 + (26a/2)u^2 - 48au + C1\)
Substituting back u = x + 3, we obtain:
\((a/4)(x + 3)^4 - (5a/2)(x + 3)^3 + (26a/2)(x + 3)^2 - 48a(x + 3) + C1\)
To evaluate the second integral, we can use the substitution u = sin(x), du/dx = cos(x). Then, the integral becomes:
∫csc(x)cot(x) dx = ∫(1/sin(x))(cos(x)/sin(x)) dx = -∫du/u
Using the fact that csc(x) = 1/sin(x), the integral becomes:
ln|sin(x)| + C2
Therefore, the final answer is:
\((a/4)(x + 3)^4 - (5a/2)(x + 3)^3 + (26a/2)(x + 3)^2 - 48a(x + 3) - ln|sin(x)| + C2\)
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Related Questions
anyone know the answer to this?
Valerie makes and sells bracelets in order to make the bracelets, first she has to buy the materials. the table shows the number of bracelets sold and her profit.
Answers
Answer
20 or 30
Step-by-step explanation:
but i pretty sure that it is $20
hope this helped
Suppose that t₀ = 5 and for each n ≥ 0, tₙ₊₁ = 2tₙ +1. Use induction to show that tₙ > 20 for all natural numbers n ≥ 2.
Answers
Using mathematical induction, we can prove that tₙ > 20 for all natural numbers n ≥ 2, starting from the base case t₂ = 11.
To prove that tₙ > 20 for all natural numbers n ≥ 2 using induction, we will follow the steps of the induction proof:
1. Base case: We need to show that t₂ > 20. Using the given recursive formula, we have t₂ = 2t₁ + 1 = 2(5) + 1 = 11. Since 11 > 20, the base case holds.
2. Inductive hypothesis: Assume that for some k ≥ 2, tₖ > 20.
3. Inductive step: We need to show that tₖ₊₁ > 20 using the assumption from the inductive hypothesis. Using the recursive formula, we have tₖ₊₁ = 2tₖ + 1. Since tₖ > 20 (from the inductive hypothesis), we can write tₖ₊₁ = 2tₖ + 1 > 2(20) + 1 = 41.
4. Conclusion: By completing the base case and the inductive step, we have shown that if tₖ > 20 for some k ≥ 2, then tₖ₊₁ > 20. This establishes that tₙ > 20 for all natural numbers n ≥ 2 by mathematical induction.
Therefore, we have proven that tₙ > 20 for all natural numbers n ≥ 2.
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The function in the graph below has a vertical asymptote
x = ____
Answers
Based on the graph provided, the vertical asymptote of the function appears to be x = -2.
What is function?Functions can be represented in different ways, such as algebraically with equations, graphically with curves, or numerically with tables of values. They are used to model relationships between quantities in a wide variety of applications, from physics and engineering to economics and social sciences.
Here,
As x approaches -2 from the left, the function appears to be approaching negative infinity, indicating a vertical asymptote. As x approaches -2 from the right, the function appears to be approaching positive infinity, further supporting the existence of a vertical asymptote at x = -2.
To calculate this mathematically, we can first analyze the function by factoring the numerator and denominator:
f(x) = (x+2)/(x²-4)
The denominator can be factored as the difference of squares:
f(x) = (x+2)/[(x+2)(x-2)]
Simplifying the expression, we can cancel out the common factor of (x+2) in the numerator and denominator:
f(x) = 1/(x-2)
Now, we can see that the denominator has a zero at x = 2, but there is no zero at x = -2. Therefore, there is a vertical asymptote at x = -2.
Thus, the vertical asymptote of the given function is x = -2.
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2. Craig read 3 books each week for 3 weeks. Joe read 4 books a week for 3 weeks. How many more books did Joe read? Choose the answer that has the correct expression and the correct answer to the question
4x3+4x4; Joe read 28 more books than Craig.
0 4x3 - 3x3; Joe read 27 more books than Craig.
3x3-4x 3; Joe read 3 more books than Craig.
0 4x3 - 3x3; Joe read 3 more books than Craig
Answers
Answer:
4*3 - 3*3;
Joe read 3 more books than Craig
Step-by-step explanation:
Given
Craig:
3 books weekly for 3 weeks
Joe:
4 books weekly for 4 weeks
Required
How many more books did Joe read.
First, we need to determine number of books Craig reads
If in a week, Craig reads 3 books
Then in 3 weeks, Craig reads 3 * 3 books
Next, we need to determine number of books Joe reads
If in a week, Joe reads 3 books
Then in 4 weeks, Joe reads 4 * 3 books
Next, we determine the difference;
\(Difference = 4 * 3 - 3 * 3\)
\(Difference = 12 - 9\)
\(Difference = 3\)
Hence, Joe reads 3 more books
Option D answers the question.
Oscar bought a circular cover for his backyard spa with an area of 48 square feet. How can Oscar best determine the radius of the spa cover? A Divide 48 by , then take the square root of the quotient. B Take the square root of 48, then divide the quotient by T. C Multiply 48 byt, then multiply the product by itself. D Take the square root of 48.
Answers
Answer:
it is very hard questions but i give answer
When 1480 is divided into 12 equal parts, the remainder is 4. What is a correct way to write the quotient? (3 points)
Answers
Answer:
Depending on what type of math class you are in, or how your teacher taught it, it could be either 12 R:4 or 12 + 1/370.
Step-by-step explanation:
Remainders can be written in two ways, R: [remainder] or the whole number quotient + [remainder number]/[dividend]. For this problem you would have to simplify 4/1480 to 1/370 because 1480/4 equals 370. So, your final answer would be 12 + 1/370 or 12 R:4.
Write the equation of the line in slope-intercept form that has a slope of 3 and y-intercept of
-2.
Answers
Answer:
y=3x-2
Step-by-step explanation:
y=mx+b where m=slope and b=y-intercept
y=3x-2
There are 44,000 adults living in Oak city in examining attitudes according to the news of research group as random sample of Oak city adults what is your main source of news the results are shown below based on the sample predict the number of adults in Oak city whose main source of the news is television on your answer to the nearest whole number do not run by any intermittent calculations
Answers
The predicted number of adults in Oak City whose main source of news is newspapers or the radio is 85
To predict the number of adults in Oak City whose main source of news is newspapers or the radio, we need to add the number of adults who answered "Newspapers" to the number of adults who answered "Radio".
According to the sample data
Number of adults who answered "Newspapers": 64
Number of adults who answered "Radio": 21
Therefore, the predicted number of adults in Oak City whose main source of news is newspapers or the radio we have to use the addition
64 + 21 = 85
Rounding this answer to the nearest whole number, we get
85 ≈ 85
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The given question is incomplete, the complete question is:
There are 44,000 adults living in Oak City. In examining attitudes toward the news, a research group asked a random sample of Oak City adults "What is your main source of news?" The results are shown below. Main Source of News Newspapers Number of Adults 64 Internet 80 Television 126 Radio 21 Other 40 Based on the sample, predict the number of adults in Oak City whose main source of news is newspapers or the radio. Round your answer to the nearest whole number.
let f(t)f(t) be the number of us billionaires in year tt. in 1985 there were 13 us billionaires, and in 1990 there were 99 us billionaires. assuming the yearly increase remains constant, find a formula predicting the number of us billionaires in year tt.
Answers
The formula predicting the number of US billionaires in any given year (t) is:f(t) = 17.2t - 34,129. We can assume a linear growth model on the based of given information.
The given data states that in the year 1985, there were 13 US billionaires. Whereas in 1990, there were 99 US billionaires. We have to find out the formula that predicts the number of US billionaires in any given year (t).
The yearly increase remains constant, so we can consider the formula for the linear function.f(t) = mt + b
where
t is the year and f(t) is the number of US billionaires in that year (t).
m is the slope of the line and b is the y-intercept.
The slope of the line is given by the formula:m = (y₂ - y₁) / (x₂ - x₁)
Let's plug in the given values to find the slope of the line.m = (99 - 13) / (1990 - 1985)m = 86 / 5m = 17.2 .The y-intercept of the line can be found by substituting the values of t and f(t) from any of the given points into the equation of the line.
Let's use the point (1985, 13).f(t) = mt + b => f(1985) =17.2(1985) + b => f(1985) = 34,142 + b =>b = 13 - 34,142 & b = -34,129.
The formula predicting the number of US billionaires in any given year (t) is:f(t) = 17.2t - 34,129
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Find the exact length of the third side.
Answers
Answer:
5sqrt3
Step-by-step explanation:
Using the pythagorean theorem, we have a^2+4^2=9^2
a^2 + 16 = 81
a^2 = 75.
a = sqrt75.
75 = 25*3. 25 is a perfect square, so sqrt75 = 5sqrt3
Solve question 10 using substitution or elimination. Show all work.
Answers
Printed paper costs 2.50 each
Given:
Ben = $43.50 (12 rolls solid paper and 9 rolls of
printed paper)
Joel = $51.50 (8 rolls solid paper and 15 rolls of
printed paper)
Let x = solid paper
Let y= printed paper
12x + 9y = 43.50
8x + 15y = 51.50
Find X:
8x + 15y = 51.50
8x = 51.50 - 15y
x= (51.50 - 15y)8
Substitute X:
12x + 9y = 43.50
12(51.50 - 15y)/8 + 9y = 43.50
(618 - 180y)/8 + 9y = 43.50
8 (618 - 180y)/8 + 8*9y = 43.50 * 8
618-180y + 72y =348
-108y = 348 - 618
-108y =-270
-108y/-108=-270/-108
Y= 2.50
x = (51.50 - 15y)/8
x = (51.50 - 15(2.5) /8
x= (51.50 - 37.50\8
X = 14/8
X = 1.75
To check: x = 1.75; y = 2.5
12x + 9y = 43.50
12(1.75) + 9(2.5) = 43.50
21 + 22.50 = 43.50
43.50 = 43.50
8x + 15y = 51.50
8(1.75) + 15(2.5) = 51.50
14 + 37.50 = 51.50
51.50 = 51.50
Hope that helps! Have a good day :)
simplify the following expression. 5.3x − 8.14 3.6x 9.8 a. -2.84x − 1.66 b. 8.9x 1.66 c. -2.84x 17.94 d. 8.9x 17.94
Answers
The simplified expression is (-187.584x + 287.6672) / 6.8, which is equivalent to option A: -2.84x - 1.66.
To simplify the expression 5.3x - 8.14 / 3.6x - 9.8, we can first simplify the division by finding a common denominator for the fractions.
The common denominator for 3.6x and 9.8 is 3.6x * 9.8 = 35.28x.
Next, we can rewrite the expression using the common denominator:
5.3x * (35.28x/35.28x) - 8.14 * (35.28x/35.28x) / 3.6x * (35.28x/35.28x) - 9.8 * (35.28x/35.28x)
Simplifying further, we get:
(5.3 * 35.28x^2 - 8.14 * 35.28x) / (3.6 * 35.28x - 9.8 * 35.28x)
Now, we can simplify the numerator:
(187.584x^2 - 287.6672x) / (-6.8x)
Factoring out an x from the numerator, we have:
x(187.584x - 287.6672) / (-6.8x)
Finally, we can cancel out the x terms:
(187.584x - 287.6672) / -6.8
Therefore, the simplified expression is (-187.584x + 287.6672) / 6.8, which is equivalent to option A: -2.84x - 1.66.
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may need to use the appropriate technology to answer this question ergency 911 calls to a small municipality in Idaho come in at the rate of one every five minutes. Anume that the number of 911 colis is a random variohle that can be described by the Produtobusom ) What is the expected number of 911 calls in thour? 12 ) What the probability of the 911 calls in 5 minutes? (Round your answer to four decimal places) X 0 130 What is the probability of no 911 calls in a 5-minute period
Answers
The expected number of 911 calls in an hour is 12 calls. The probability of no 911 calls in a 5-minute period is 0.3679.
Given that emergency 911 calls come in at the rate of one every five minutes to a small municipality in Idaho.
Therefore, the expected number of 911 calls in one hour = 60/5 × 1 = 12 calls. Therefore, the expected number of 911 calls in an hour is 12 calls. Hence, this is the answer to the first question. In the next part of the question, we need to find the probability of 911 calls in 5 minutes and the probability of no 911 calls in a 5-minute period.
To find the probability of 911 calls in 5 minutes, we need to use the Poisson distribution formula which is:
P(X = x) = (e^-λ * λ^x) / x!
Where λ is the expected value of X.
In this question, the value of λ is 1/5 (because one call is coming every 5 minutes).
Therefore,
λ = 1/5
P(X = 0) = (e^-1/5 * (1/5)^0) / 0!
P(X = 0) = e^-1/5
P(X = 0) = 0.8187
Therefore, the probability of no 911 calls in a 5-minute period is 0.3679. Hence, this is the answer to the third question.
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Linear equations
I need help with these
Instructions: choose two linear equations in each box. Write them below. For #3-#8, find the slope and y-intercept of each equation.
Then graph each line on a coordinate plane on the next page. Label each line neatly with its corresponding equation.
Answers
The graph of the equations are added as attachment
How to determine the graphs of the equationsChoose two linear equations
In this case, we make use of the following linear equations
y = 2x + 4 and y = -3x + 5
The slope and y-intercept of each equation
A linear equation is represented as
y = mx + c
Where
Slope = m and y-intercept = c
This means that the slope and y-intercept of each equation are
y = 2x + 4
Slope = 2
y-intercept = 4
y = -3x + 5
Slope = -3
y-intercept = 5
See attachment for the graph of the functions
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suppose that a is orthogonal to b in rn. what is projab? why? give a geometric interpretation in r2 or r3
Answers
Projab is the projection of vector a onto vector b. It is a vector that is perpendicular to b and has the same magnitude as a, and it is the vector that is closest to a in the direction of b.
In R2, the geometric interpretation is that projab is the vector that is obtained by dropping a perpendicular line from the tip of a onto the line containing b.
In R3, the geometric interpretation is that projab is the vector that is obtained by dropping a perpendicular line from the tip of a onto the plane containing b.
The equation for projab is projab = (a.b/|b|^2)b, where a.b is the dot product of a and b, and |b| is the magnitude of b.
The best estimator's error vector, when seen in terms of mean square errors, is orthogonal to all other feasible estimators, according to the orthogonality principle. The orthogonality principle can be expressed in a variety of ways, but it is most frequently used to describe linear estimators.
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Let f(x) = 3x2 + x − 3 and g(x) = x2 − 5x + 1. Find f(x) − g(x).
2x2 − 4x − 2
2x2 − 4x − 4
2x2 + 6x − 2
2x2 + 6x − 4
Answers
Answer:
D) 2x^2 + 6x - 4
Step-by-step explanation:
I took the quiz and got a 100%
Solve 2x + 3y = 11 and 2x - 4y = -24 and hence find the value of 'm' for which y=mx+3.
Answers
Answer:
m=-1
Step-by-step explanation:
2x+3y=11............. (eqn 1)
2x-4y=-24...........(eqn 2)
Subtract eqn 1 from eqn 2
7y=35
y=5
Substitute for y in eqn 1
2x+3(5)=11
2x+15=11
2x=11-15
2x=-4
x=-2
y=mx+3
5=-2m+3
5-3=-2m
2=-2m
m=-1
Answer:Here,
2x+3y=11..........(i)2x-4y= -24......... (ii)Subtracting (ii) from (i),we get
2x+3y=11
- 2x+4y=24
___________
7y =35
y=5
Substituting y = 5 in (i) we get
2x+3×5=11
or,2x +15 =11
or,2x = 11- 15
or,x= -2
Now,
We know,y=mx+c
5=m ×(-2) +3
or,2=m×(-2)
or,2/-2 =m
m= -1
What is the equation of the graphed line written in standard form?
2x + 3y = -6
2x + 3y = 6
y=-2/3x-2
y= 2/3x-2
Answers
Answer:
What is the equation of the graphed line written in standard form?" has four options: 2x + 3y = -6, 2x + 3y = 6, y=-2/3x-2, and y= 2/3x-2. The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. In this case, the first two options, 2x + 3y = -6 and 2x + 3y = 6 are already in standard form. The last two options, y=-2/3x-2 and y= 2/3x-2 are not in standard form. However, without additional information such as a graph or coordinates of points on the line, it is not possible to determine which of these options is the correct equation for the graphed line.
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Determine the volume in milliliters and ounces of 1.5 teaspoons. Note that 1 tablespoon = 3 teaspoons and 1 tablespoon = 0.5 fluid ounces.
Answers
The volume of 1.5 teaspoons is equivalent to 7.39 milliliters or approximately 0.25 fluid ounces.
Given that 1 tablespoon = 3 teaspoons and 1 tablespoon = 0.5 fluid ounces, we can use these conversion factors to convert 1.5 teaspoons to volume in milliliters and ounces.
To convert teaspoons to tablespoons, we can divide the number of teaspoons by 3
1.5 teaspoons ÷ 3 teaspoons/tablespoon = 0.5 tablespoons
To convert tablespoons to fluid ounces, we can multiply the number of tablespoons by 0.5:
0.5 tablespoons x 0.5 fluid ounces/tablespoon = 0.25 fluid ounces
Therefore, 1.5 teaspoons is equivalent to 0.5 tablespoons, which is equivalent to 0.25 fluid ounces. To convert fluid ounces to milliliters, we need to know that 1 fluid ounce is equivalent to approximately 29.57 milliliters.
So, 0.25 fluid ounces x 29.57 milliliters/fluid ounce = 7.39 milliliters
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The concept that a message gives different meanings to different objects is called _____.
a. ​encapsulation
b. ​polymorphism
c. ​linear addressing
d. ​dynamic addressing
Answers
The concept that a message gives different meanings to different objects is called option (b) polymorphism.
Polymorphism is a fundamental concept in object-oriented programming (OOP) that allows different objects to respond to the same message or method invocation in different ways. In other words, it allows objects of different classes to be treated as if they were of the same class, as long as they implement the same method or message.
This can make code more flexible, reusable, and easier to maintain. Polymorphism is achieved through inheritance, interfaces, or overloading methods. For example, a "draw" method could be implemented differently for different shapes, such as circles, rectangles, or triangles.
Therefore, the correct option is (b) polymorphism
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can someone help on my last question plz
Answers
Sure whats the question?
Will mark brainliest! Determine whether the rectangles are similar. If so, write the similarity
ratio and a similarity statement.
Answers
What integer represents a gain of 2 pounds?
Answers
Answer:
+2
Step-by-step explanation:
the toll t charged for driving on a certain stretch of a toll road is $3 except during rush hours (between 7 a.m. and 10 a.m. and between 4 p.m. and 7 p.m.) when the toll is $5.
Answers
The toll charged for driving on a certain stretch of a toll road is $3, except during rush hours when the toll is $5. Here's a step-by-step explanation:
1. The regular toll for driving on the toll road is $3. This means that outside of rush hours, you will need to pay $3 to use the road.
2. Rush hours are between 7 a.m. and 10 a.m. and between 4 p.m. and 7 p.m. During these times, the toll increases to $5. This means that if you drive on the toll road between 7 a.m. and 10 a.m. or between 4 p.m. and 7 p.m., you will need to pay $5 instead of the regular $3.
To summarize, the toll for driving on a certain stretch of a toll road is $3, but during rush hours (between 7 a.m. and 10 a.m. and between 4 p.m. and 7 p.m.), the toll increases to $5.
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what does the activity ratio measure in the value stream
Answers
The activity ratio in the value stream measures the efficiency of the production process by evaluating the ratio of value-added activities to non-value-added activities.
The activity ratio is a performance metric used in value stream mapping, which is a lean management tool for analyzing and improving the flow of materials and information in a production process.
It focuses on distinguishing value-added activities, which directly contribute to meeting customer requirements, from non-value-added activities, which do not add value but consume resources and time.
By calculating the activity ratio, organizations can assess the proportion of time and resources spent on value-added activities versus non-value-added activities.
A higher activity ratio indicates that a greater portion of resources is dedicated to value-added activities, indicating improved efficiency and reduced waste.
Conversely, a lower activity ratio suggests a higher proportion of resources being utilized for non-value-added activities, indicating potential areas for improvement and waste reduction.
The activity ratio serves as a valuable diagnostic tool for organizations to identify process inefficiencies, bottlenecks, and areas for improvement. By analyzing the activity ratio, organizations can streamline their processes, eliminate waste, and optimize resource allocation to enhance overall productivity and customer value.
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Use the distributive property to simplify the expression below:
4x + 2(x+1)
A. 6x + 1
B. 6x + 2
C. 8x + 2x + 2
D. 4x + 2x + 2
Answers
= 6x + 2
So the answer is B
Answer:
B)
Step-by-step explanation:
You would first multiply the variable and constant with the number next to the parenthesis. So you now have 4x+2x+2. Then you add up like terms to get a final answer of 6x+2.
The value of a new car decreases by about 15% in the first year. How much will a car be worth after one year if its initial value was $1,500? If you get stuck, consider using diagrams or a table to organize your work.
Answers
Answer:
$12,750
Step-by-step explanation: when you do the math you get it your welcome my G
2. Solve the following difference equations: (a) \( x_{t+1}=\frac{1}{2} x_{t}+3 \) (b) \( x_{t+1}=-3 x_{t}+4 \)
Answers
(a) ( x_{t+1}=\frac{1}{2} x_{t}+3 ), the solution to this difference equation is x_t = 2^t + 3, The difference equations in this problem are both linear difference equations with constant coefficients.
This can be found by solving the equation recursively. For example, the first few terms of the solution are
t | x_t
--- | ---
0 | 3
1 | 7
2 | 15
3 | 31
The general term of the solution can be found by noting that
x_{t+1} = \frac{1}{2} x_t + 3 = \frac{1}{2} (2^t + 3) + 3 = 2^t + 3
(b) ( x_{t+1}=-3 x_{t}+4 )
The solution to this difference equation is
x_t = 4 \cdot \left( \frac{1}{3} \right)^t + 4
This can be found by solving the equation recursively. For example, the first few terms of the solution are
t | x_t
--- | ---
0 | 4
1 | 5
2 | 2
3 | 1
The general term of the solution can be found by noting that
x_{t+1} = -3 x_t + 4 = -3 \left( 4 \cdot \left( \frac{1}{3} \right)^t + 4 \right) + 4 = 4 \cdot \left( \frac{1}{3} \right)^t + 4
The difference equations in this problem are both linear difference equations with constant coefficients. This means that they can be solved using a technique called back substitution.
Back substitution involves solving the equation recursively, starting with the last term and working backwards to the first term.
In the first problem, the equation can be solved recursively as follows:
x_{t+1} = \frac{1}{2} x_t + 3
x_t = \frac{1}{2} x_{t-1} + 3
x_{t-1} = \frac{1}{2} x_{t-2} + 3
...
x_0 = \frac{1}{2} x_{-1} + 3
The general term of the solution can be found by noting that
x_{t+1} = \frac{1}{2} x_t + 3 = \frac{1}{2} (2^t + 3) + 3 = 2^t + 3
The second problem can be solved recursively as follows:
x_{t+1} = -3 x_t + 4
x_t = -3 x_{t-1} + 4
x_{t-1} = -3 x_{t-2} + 4
...
x_0 = -3 x_{-1} + 4
The general term of the solution can be found by noting that
x_{t+1} = -3 x_t + 4 = -3 \left( 4 \cdot \left( \frac{1}{3} \right)^t + 4 \right) + 4 = 4 \cdot \left( \frac{1}{3} \right)^t + 4
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find the valie of 12 to the 2nd power
Answers
Answer: 144
Step-by-step explanation:
12 x 12 = 144
Answer:
144
Step-by-step explanation:
12 times 12 is 144
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144
it is observed that 50% of e-mail is spam. a software program that filters spam before reaching an in-box has an accuracy of 99% of detecting spam but a 5% chance of tagging non-spam as a spam e-mail. what is the probability that 1 email tagged as spam is not spam?
Answers
The probability that an email tagged as spam is not actually spam is approximately 0.0481 or 4.81%.
To find the probability that an email tagged as spam is not actually spam, we can use conditional probability.
Let's define the following events:
A: Email is tagged as spam
B: Email is actually spam
We want to find P(not B | A), which represents the probability that an email is not spam (not B) given that it is tagged as spam (A).
Using Bayes' theorem, we have:
P(not B | A) = P(A | not B) * P(not B) / P(A)
P(A | not B) represents the probability that an email is tagged as spam (A) given that it is not actually spam (not B). From the problem statement, we know that this is 5% or 0.05.
P(not B) represents the probability that an email is not actually spam. Since 50% of emails are spam, the probability that an email is not spam is 1 - P(B) = 1 - 0.50 = 0.50.
P(A) represents the probability that an email is tagged as spam. This can be calculated by considering two cases: an email is spam and correctly tagged as spam, or an email is not spam and incorrectly tagged as spam. Mathematically, we can write:
P(A) = P(A | B) * P(B) + P(A | not B) * P(not B)
P(A | B) represents the probability that an email is tagged as spam (A) given that it is actually spam (B). From the problem statement, we know that this is 99% or 0.99.
P(B) represents the probability that an email is actually spam. From the problem statement, we know that this is 50% or 0.50.
Plugging in the values, we can calculate P(A):
P(A) = P(A | B) * P(B) + P(A | not B) * P(not B)
= 0.99 * 0.50 + 0.05 * 0.50
= 0.495 + 0.025
= 0.52
Now, we can calculate P(not B | A):
P(not B | A) = P(A | not B) * P(not B) / P(A)
= 0.05 * 0.50 / 0.52
= 0.025 / 0.52
≈ 0.0481
Therefore, the probability that an email tagged as spam is not actually spam is approximately 0.0481 or 4.81%.
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