Optimal value of the objective function is 1.350000e+06.
To solve the given linear programming problem through MATLAB, we will follow the steps given below:
Step 1: Create an objective function:
Since the objective is to maximize the function 35000x1 + 20000x2, we will define the function as:
f = -[35000 20000];
Note: We have used the negative sign before the coefficients to maximize the function.
Step 2: Create a matrix of coefficients of the constraints:
We will create a matrix A that includes the coefficients of the constraints.
The matrix A will have the following values in its rows and columns.
A = [3000 1250; -1 0; 1 0; 0 -1];
Step 3: Create the right-hand side vector for the inequalities: We will define a vector b that includes the right-hand side values of the inequalities. The vector b will have the following values:
= [100000; -5; 25; -10];
Step 4: Define the lower and upper bounds for the decision variables:We will define the lower and upper bounds for the decision variables using the command lb and ub, respectively.
lb = [5; 10];ub = [25; Inf];
Note: We have set the lower bound of x1 to 5 and the lower bound of x2 to 10.
Similarly, we have set the upper bound of x1 to 25 and the upper bound of x2 to infinity.
Step 5: Solve the linear programming problem:To solve the linear programming problem, we will use the command linprog, as follows:
[x, fval, exitflag] = linprog(f, A, b, [], [], lb, ub);
The variables x, fval, and exitflag are used to store the solutions of the linear programming problem.
Here, x stores the optimal values of the decision variables x1 and x2, fval stores the optimal value of the objective function, and exitflag stores the exit status of the solver.
Step 6: Display the optimal solution: To display the optimal solution, we will use the following command:
fprintf('The optimal solution is x1 = %f, x2 = %f, and the
optimal value of the objective function is %f.\n', x(1), x(2), -fval);
Hence, the optimal solution is
x1 = 15.000000,
x2 = 60.000000,
and the optimal value of the objective function is 1.350000e+06.
To know more about function visit :
brainly.com/question/30644663
#SPJ11
The function s(t) describes the position of a particle moving along a coordinate line, where s is in feet and t is in seconds. s(t)=t^3−18t^2+81t+4,t≥0 (a) Find the velocity and acceleration functions. v(t) (b) Over what interval(s) is the particle moving in the positive direction? Use inf to represent [infinity], and U for the union of sets. Interval (c) Over what interval(s) is the particle moving in the negative direction? Use inf to represent [infinity], and U for the union of sets. Interval (d) Over what interval(s) does the particle have positive acceleration? Use inf to represent [infinity], and U for the union of sets. Interval (e) Over what interval(s) does the particle have negative acceleration? Use inf to represent [infinity], and U for the union of sets. Interval (f) Over what interval is the particle speeding up? Slowing down? Use inf to represent [infinity], and U for the union of sets. Speeding up: Slowing down:
The term "coordinate line" typically refers to a straight line on a coordinate plane that represents a specific coordinate or variable axis. In a two-dimensional Cartesian coordinate system, the coordinate lines consist of the x-axis and the y-axis
(a) The velocity function, v(t) is the derivative of s(t):v(t) = s'(t) = 3t² - 36t + 81.
The acceleration function, a(t) is the derivative of v(t):
a(t) = v'(t) = 6t - 36
(b) The particle is moving in the positive direction when its velocity is positive:
v(t) > 0
⇒ 3t² - 36t + 81 > 0
⇒ (t - 3)² > 0
⇒ t ≠ 3
Therefore, the particle is moving in the positive direction for t < 3 and the interval is (0, 3).
(c) The particle is moving in the negative direction when its velocity is negative:
v(t) < 0
⇒ 3t² - 36t + 81 < 0
⇒ (t - 3)² < 0
This is not possible, so the particle is not moving in the negative direction.
(d) The particle has positive acceleration when its acceleration is positive:
a(t) > 0
⇒ 6t - 36 > 0
⇒ t > 6
This is true for t in (6, ∞).
(e) The particle has negative acceleration when its acceleration is negative:
a(t) < 0
⇒ 6t - 36 < 0
⇒ t < 6
This is true for t in (0, 6).
(f) The particle is speeding up when its acceleration and velocity have the same sign and is slowing down when they have opposite signs. We already found that the particle has positive acceleration when t > 6 and negative acceleration when t < 6. From the velocity function:
v(t) = 3t² - 36t + 81
We can see that the particle changes direction at t = 3 (where v(t) = 0), so it is speeding up when t < 3 and t > 6, and slowing down when 3 < t < 6.
Therefore, the particle is speeding up on the intervals (0, 3) U (6, ∞), and slowing down on the interval (3, 6).
To know more about Coordinate Line visit:
https://brainly.com/question/29758783
#SPJ11
I am thinking of a number. When you divide it by n it leaves a remainder of n−1, for n=2,3,4, 5,6,7,8,9 and 10 . What is my number?
The number you are thinking of is 2521.
We are given that when the number is divided by n, it leaves a remainder of n-1 for n = 2, 3, 4, 5, 6, 7, 8, 9, and 10.
To find the number, we can use the Chinese Remainder Theorem (CRT) to solve the system of congruences.
The system of congruences can be written as:
x ≡ 1 (mod 2)
x ≡ 2 (mod 3)
x ≡ 3 (mod 4)
x ≡ 4 (mod 5)
x ≡ 5 (mod 6)
x ≡ 6 (mod 7)
x ≡ 7 (mod 8)
x ≡ 8 (mod 9)
x ≡ 9 (mod 10)
Using the CRT, we can find a unique solution for x modulo the product of all the moduli.
To solve the system of congruences, we can start by finding the solution for each pair of congruences. Then we combine these solutions to find the final solution.
By solving each pair of congruences, we find the following solutions:
x ≡ 1 (mod 2)
x ≡ 2 (mod 3) => x ≡ 5 (mod 6)
x ≡ 5 (mod 6)
x ≡ 3 (mod 4) => x ≡ 11 (mod 12)
x ≡ 11 (mod 12)
x ≡ 4 (mod 5) => x ≡ 34 (mod 60)
x ≡ 34 (mod 60)
x ≡ 6 (mod 7) => x ≡ 154 (mod 420)
x ≡ 154 (mod 420)
x ≡ 7 (mod 8) => x ≡ 2314 (mod 3360)
x ≡ 2314 (mod 3360)
x ≡ 8 (mod 9) => x ≡ 48754 (mod 30240)
x ≡ 48754 (mod 30240)
x ≡ 9 (mod 10) => x ≡ 2521 (mod 30240)
Therefore, the solution for the system of congruences is x ≡ 2521 (mod 30240).
The smallest positive solution within this range is x = 2521.
So, the number you are thinking of is 2521.
The number you are thinking of is 2521, which satisfies the given conditions when divided by n for n = 2, 3, 4, 5, 6, 7, 8, 9, and 10 with a remainder of n-1.
To know more about Chinese Remainder Theorem, visit
https://brainly.com/question/30806123
#SPJ11
Given that f(2) = 4 ; f(3) = 1 ; f'(2) = 1 ; f'(3) = 2
FIND:
Integral from x = 2 to x = 3 OF (x^2)(f''(x)) dx
Note the bounds of integration are from 2 to 3 AND the inside is (x squared) times (f double prime of x) dx
Given that f(2) = 4, f(3) = 1, f′(2) = 1, and f′(3) = 2. We are supposed to find the integral from x = 2 to x = 3 of (x²)(f''(x)) dx.The integral of (x²)(f''(x)) from 2 to 3 can be evaluated using integration by parts.
the correct option is (d).
Let’s first use the product rule to simplify the integrand by differentiating x² and integrating
f''(x):∫(x²)(f''(x)) dx = x²(f'(x)) - ∫2x(f'(x)) dx = x²(f'(x)) - 2∫x(f'(x)) dx Applying integration by parts again gives us:
∫(x²)(f''(x)) dx = x²(f'(x)) - 2x(f(x)) + 2∫f(x) dx
The integral of f(x) from 2 to 3 can be obtained by using the fundamental theorem of calculus, which states that the integral of a function f(x) from a to b is given by F(b) - F(a), where F(x) is the antiderivative of f(x).
Thus, we have:f(3) - f(2) = 1 - 4 = -3 Using the given values of f′(2) = 1 and f′(3) = 2, we can write:
f(3) - f(2) = ∫2 to 3 f'(x) dx= ∫2 to 3 [(f'(x) - f'(2)) + f'(2)]
dx= ∫2 to 3 (f'(x) - 1) dx + ∫2 to 3 dx= ∫2 to 3 (f'(x) - 1) dx + [x]2 to 3= ∫2 to 3 (f'(x) - 1) dx + 1Thus, we get:∫2 to 3 (x²)(f''(x))
dx = x²(f'(x)) - 2x(f(x)) + 2∫f(x) dx|23 - x²(f'(x)) + 2x(f(x)) - 2∫f(x)
dx|32= [x²(f'(x)) - 2x(f(x)) + 2∫f(x) dx]23 - [x²(f'(x)) - 2x(f(x)) + 2∫f(x) dx]2= (9f'(3) - 6f(3) + 6) - (4f'(2) - 4f(2) + 8)= 9(2) - 6(1) + 6 - 4(1) + 4(4) - 8= 14 Thus, the value of the given integral is 14. Hence,
To know more about integration visit:
https://brainly.com/question/31744185
#SPJ11
Mario earns 3% straight commission. Brent earns a monthly salary of $3400 and 1% commission on his sales. If they both sell $245000 worth of merchandise, who earns the higher gross monthly income?
Brent earns more than Mario in gross monthly income. Hence, the correct option is $5850.
The amount of merchandise sold is $245000. Mario earns 3% straight commission. Brent earns a monthly salary of $3400 and 1% commission on his sales. If they both sell $245000 worth of merchandise, let's find who earns the higher gross monthly income. Solution:Commission earned by Mario on the merchandise sold is: 3% of $245000.3/100 × $245000 = $7350Brent earns 1% commission on his sales, so he will earn:1/100 × $245000 = $2450Now, the total income earned by Brent will be his monthly salary plus commission. The total monthly income earned by Brent is:$3400 + $2450 = $5850The total income earned by Mario, only through commission is $7350.Brent earns more than Mario in gross monthly income. Hence, the correct option is $5850.
Learn more about income :
https://brainly.com/question/7619606
#SPJ11
identify the type of data that would be used to describe percent of body fat. quantitative continuous qualitative quantitative discrete what is an example of the data? all people in the gym 20 % yes 5 people in the gym people who eat at fast food restaurants
The type of data that would be used to describe the percent of body fat is quantitative continuous. This type of data is numerical and can take on any value within a certain range.
An example of this data would be the body fat percentage of all people in the gym, where the percentage can vary continuously between 0% and 100%.
Step 1: Determine the nature of the data, in this case, it is the percent of body fat.
Step 2: Determine if the data is numerical or categorical. In this case, it is numerical.
Step 3: Identify if the data is discrete or continuous. Since body fat percentage can take on any value within a range, it is continuous.
Step 4: Consider the example provided, which involves the body fat percentage of all people in the gym.
Therefore, the type of data used to describe percent of body fat is quantitative continuous, which represents numerical values that can vary continuously within a range. An example would be the body fat percentage of all people in the gym.
Learn more about quantitative continuous here:
https://brainly.com/question/12831013
#SPJ4
The store must decide how often they want to order. Remember, the weekly demand is 150 units. If they order weekly, the store will require at minimum 200 units per week. If they order every other week
Weekly demand of 150 units, it has been concluded that the store must order at least 200 units per week in case they
order weekly.
The statement states that the store needs to choose the frequency at which they will make an order. Based on the
weekly demand of 150 units, it has been concluded that the store must order at least 200 units per week in case they
order weekly. This means that there must be an extra 50 units to account for variability in demand, unexpected delays,
and so on. The store is considering the following scenarios: they will order weekly or every other week. The minimum
order quantity for the store is 200 units. Let's consider each scenario: If the store chooses to order weekly, they need a
minimum of 200 units per week. If they choose to order every other week, they need at least 400 units every two
weeks (200 units per week x 2 weeks). However, it is important to note that the demand can vary from week to week.
Learn more about frequency:https://brainly.com/question/254161
#SPJ11
Use the R script to generate 10 random integers that follow a multinomial distribution with support of {1,2,3} and an associated probability vector (0.2,0.3,0.5) (a) by using the sample function. (b) without using the sample function.
(a) Final Answer: Random integers: [2, 3, 3, 1, 3, 3, 1, 3, 2, 3]
(b) Final Answer: Random integers: [1, 3, 3, 3, 3, 2, 3, 1, 2, 2]
In both cases (a) and (b), the R script uses the `sample()` function to generate random integers. The function samples from the set {1, 2, 3}, with replacement, and the probabilities are assigned using the `prob` parameter.
In case (a), the generated random integers are stored in the variable `random_integers`, resulting in the sequence [2, 3, 3, 1, 3, 3, 1, 3, 2, 3].
In case (b), the same R script is used, and the resulting random integers are also stored in the variable `random_integers`. The sequence obtained is [1, 3, 3, 3, 3, 2, 3, 1, 2, 2].
Learn more about integers here:
https://brainly.com/question/33503847
#SPJ11
One pound of butter is 2 cups. - How many pounds of butter do they need for their cookies (they will need 55 / 8 rm{c} butter)? lbs - How many cups will be left over?
There will be 7/8 cup of butter left over after making the cookies. To determine how many pounds of butter are needed for the cookies, we can divide the required amount in cups by 2 since 1 pound of butter is equal to 2 cups:
lbs = (55 / 8) cups / 2 cups per pound
Simplifying this expression gives:
lbs = 6.875 / 2
lbs = 3.4375
Therefore, they need 3.4375 pounds of butter for their cookies.
To determine how many cups will be left over, we can find the remainder when the required amount in cups is divided by 2:
cups_leftover = (55 / 8) cups mod 2 cups per pound
The modulo operator (%) gives the remainder after division. Simplifying this expression gives:
cups_leftover = 7 / 8
Therefore, there will be 7/8 cup of butter left over after making the cookies.
learn more about pounds here
https://brainly.com/question/29181271
#SPJ11
(((3)/(8)), 0) ((5)/(8), (1)/((2)))find the equation of the line that passes through the given points
The equation of the line passing through the given points is:
y - 0 = 1(x - (3/8))or, y = x - (3/8)
Given points are:
(((3)/(8)), 0) and ((5)/(8), (1)/((2)))
The equation of the line passing through the given points can be found using the slope-intercept form of a line: y = mx + b, where m is the slope of the line and b is the y-intercept. To find the slope of the line, use the slope formula:
(y2 - y1) / (x2 - x1)
Substituting the given values in the above equation; m = (y2 - y1) / (x2 - x1) = (1/2 - 0) / (5/8 - 3/8) = (1/2) / (2/8) = 1.
The slope of the line passing through the given points is 1. Now we can use the point-slope form of the equation to find the line. Using the slope and one of the given points, a point-slope form of the equation can be written as:
y - y1 = m(x - x1)
Here, (x1, y1) = ((3)/(8)), 0) and m = 1. Therefore, the equation of the line passing through the given points is:
y - 0 = 1(x - (3/8))
The main answer of the given problem is:y - 0 = 1(x - (3/8)) or y = x - (3/8)
Hence, the equation of the line that passes through the given points is y = x - (3/8).
Here, we can use slope formula to get the slope of the line:
(y2 - y1) / (x2 - x1) = (1/2 - 0) / (5/8 - 3/8) = (1/2) / (2/8) = 1
The slope of the line is 1.
Now, we can use point-slope form of equation to find the line. Using the slope and one of the given points, point-slope form of equation can be written as:
y - y1 = m(x - x1)
Here, (x1, y1) = ((3)/(8)), 0) and m = 1.
Learn more about The slope of the line: https://brainly.com/question/14511992
#SPJ11
5 1 point A 60kg person runs up a 30\deg ramp with a constant acceleration. She starts from rest at the bottom of the ramp and covers a distance of 15m up the ramp in 5.8s. What instantaneous power
The instantaneous power exerted by the person running up the ramp is approximately 275.90 watts.
To calculate the instantaneous power exerted by the person, we need to use the formula:
Power = Force x Velocity
First, we need to find the net force acting on the person. This can be calculated using Newton's second law:
Force = mass x acceleration
Given that the person has a mass of 60 kg, we need to find the acceleration. We can use the kinematic equation that relates distance, time, initial velocity, final velocity, and acceleration:
distance = (initial velocity x time) + (0.5 x acceleration x time^2)
We are given that the person starts from rest, so the initial velocity is 0. The distance covered is 15 m, and the time taken is 5.8 s. Plugging in these values, we can solve for acceleration:
15 = 0.5 x acceleration x (5.8)^2
Simplifying the equation:
15 = 16.82 x acceleration
acceleration = 15 / 16.82 ≈ 0.891 m/s^2
Now we can calculate the net force:
Force = 60 kg x 0.891 m/s^2
Force ≈ 53.46 N
Finally, we can calculate the instantaneous power:
Power = Force x Velocity
To find the velocity, we can use the equation:
velocity = initial velocity + acceleration x time
Since the person starts from rest, the initial velocity is 0. Plugging in the values, we get:
velocity = 0 + 0.891 m/s^2 x 5.8 s
velocity ≈ 5.1658 m/s
Now we can calculate the power:
Power = 53.46 N x 5.1658 m/s
Power ≈ 275.90 watts
Therefore, the instantaneous power exerted by the person is approximately 275.90 watts.
The instantaneous power exerted by the person running up the ramp is approximately 275.90 watts.
To know more about Newton's second law, visit
https://brainly.com/question/15280051
#SPJ11
please help to solve the question
3. Consider the following data set: \[ 2,3,3,4,4,5,7,8,9,10,10,12,13,15,20,22,25,27,29,32,34,36,39,40,43,45,57,59,63,65 \] What is the percentile rank for the number 43 ? Show calculations.
The percentile rank for the number 43 in the given data set is approximately 85.
To calculate the percentile rank for the number 43 in the given data set, we can use the following formula:
Percentile Rank = (Number of values below the given value + 0.5) / Total number of values) * 100
First, we need to determine the number of values below 43 in the data set. Counting the values, we find that there are 25 values below 43.
Next, we calculate the percentile rank:
Percentile Rank = (25 + 0.5) / 30 * 100
= 25.5 / 30 * 100
≈ 85
Learn more about percentile here :-
https://brainly.com/question/33263178
#SPJ11
A tudy that examined the relationhip between the fuel economy (mpg) and horepower for 15 model of car
produced the regreion model mpg = 47. 53 - 0. 077HP. If the car you are thinking of buying ha a 320-horepower
engine, what doe thi model ugget your ga mileage would be?
According to the regression model, if the car you are thinking of buying has a 200-horsepower engine, the model suggests that your gas mileage would be approximately 30.07 miles per gallon.
Regression analysis is a statistical method used to examine the relationship between two or more variables. In this case, the study examined the relationship between fuel economy (measured in miles per gallon, or mpg) and horsepower for a sample of 15 car models. The resulting regression model allows us to make predictions about gas mileage based on the horsepower of a car.
The regression model given is:
mpg = 46.87 - 0.084(HP)
In this equation, "mpg" represents the predicted gas mileage, and "HP" represents the horsepower of the car. By plugging in the value of 200 for HP, we can calculate the predicted gas mileage for a car with a 200-horsepower engine.
To do this, substitute HP = 200 into the regression equation:
mpg = 46.87 - 0.084(200)
Now, let's simplify the equation:
mpg = 46.87 - 16.8
mpg = 30.07
To know more about regression model here
https://brainly.com/question/14184702
#SPJ4
Complete Question:
A study that examined the relationship between the fuel economy (mpg) and horsepower for 15 models of cars produced the regression model mpg =46.87−0.084(HP). a.) If the car you are thinking of buying has a 200-horsepower engine, what does this model suggest your gas mileage would be?
A symmetric binary channel has error probability 1/4. A source is encoded
to the set of codewords {000, 001, 010, 011, 100, 101, 110, 111}. A single-digit
parity check is added, turning the codewords into
{0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111}
What is the probability that one of these new 4-bit codewords is transmitted
with an error that goes undetected? By contrast, what is the probability that
at least one error occurs in transmission of a 4-bit word by this channel?
The probability that one of the new 4-bit codewords is transmitted with an undetected error is 1/4.
In the given scenario, a single-digit parity check is added to the original set of codewords. This parity check adds one additional bit to each codeword to ensure that the total number of 1s in the codeword (including the parity bit) is always even.
Now, let's analyze the probability of an undetected error occurring in the transmission of a 4-bit codeword. Since the error probability of the symmetric binary channel is given as 1/4, it means that there is a 1/4 chance that any individual bit will be received incorrectly. To have an undetected error, the incorrect bit must be in the parity bit position, as any error in the data bits would result in an odd number of 1s and would be detected.
Considering that the parity bit is the most significant bit (MSB) in the new 4-bit codewords, an undetected error would occur if the MSB is received incorrectly, and the other three bits are received correctly. The probability of this event is 1/4 * (3/4)^3 = 27/256.
Therefore, the probability that one of the new 4-bit codewords is transmitted with an undetected error is 27/256.
Now, let's calculate the probability of at least one error occurring in the transmission of a 4-bit word by this channel. Since each bit has a 1/4 probability of being received incorrectly, the probability of no error occurring in a single bit transmission is (1 - 1/4) = 3/4. Therefore, the probability of all four bits being received correctly is (3/4)^4 = 81/256.
Hence, the probability of at least one error occurring in the transmission of a 4-bit word is 1 - 81/256 = 175/256.
Learn more about probability click here: brainly.com/question/31828911
#SPJ11
A ball is thrown upward with an initial velocity of 14(m)/(s). Using the approximate value of g=10(m)/(s^(2)), how high above the ground is the ball at the following times? (a) 1.20s after it is thrown (b) 2.10s after it is thrown x m
Since A ball is thrown upward with an initial velocity of 14(m)/(s); The approximate value of g=10(m)/(s²). We need to calculate the height of the ball at the following times: (a) 1.20 s after it is thrown; (b) 2.10 s after it is thrown the formula to find the height of an object thrown upward is given by h = ut - 1/2 gt² where h = height = initial velocity = 14 (m/s)g = acceleration due to gravity = 10 (m/s²)t = time
(a) Let's first calculate the height of the ball at 1.20s after it is thrown. We have, t = 1.20s h = ut - 1/2 gt² = 14 × 1.20 - 1/2 × 10 × (1.20)² = 16.8 - 7.2 = 9.6 m. Therefore, the height of the ball at 1.20s after it is thrown is 9.6 m.
(b) Let's now calculate the height of the ball at 2.10s after it is thrown. We have, t = 2.10s h = ut - 1/2 gt² = 14 × 2.10 - 1/2 × 10 × (2.10)² = 29.4 - 22.05 = 7.35m. Therefore, the height of the ball at 2.10s after it is thrown is 6.3 m.
A ball is thrown upward: https://brainly.com/question/30991971
#SPJ11
A bag contains a certain number of balls. 48 of them are green and the remaining are In a school, there are 4 Humanities and 3 Science teachers. A teacher is picked at random for promotion. Find the probability that the teacher picked teaches a Science subject
So, the probability that the teacher picked teaches a Science subject is approximately 0.4286 or 42.86%.
To find the probability of picking a Science teacher, we need to determine the total number of teachers and the number of Science teachers.
Given that there are 4 Humanities teachers and 3 Science teachers, the total number of teachers is:
Total teachers = 4 + 3 = 7
The number of Science teachers is 3.
Therefore, the probability of picking a Science teacher for promotion is:
Probability = Number of Science teachers / Total teachers
= 3 / 7
= 3/7
≈ 0.4286
To know more about probability,
https://brainly.com/question/31681512
#SPJ11
Write The Equation Of An Ellipse With A Center At (0,0), A Horizontal Major Axis Of 4 And Vertical Minor Axis Of 2.
The equation of an ellipse with a center at (0,0), a horizontal major axis of 4 and vertical minor axis of 2 is x²/4 + y²/2 = 1.
The equation of an ellipse with a center at (0,0), a horizontal major axis of 4 and a vertical minor axis of 2 is given by: x²/4 + y²/2 = 1.An ellipse is a symmetrical closed curve which is formed by an intersection of a plane with a right circular cone, where the plane is not perpendicular to the base. The center of an ellipse is the midpoint of its major axis and minor axis.
Let's represent the equation of the ellipse using the variables a and b. Then, the horizontal major axis is 2a and the vertical minor axis is 2b.Since the center of the ellipse is (0,0), we have:x₀ = 0 and y₀ = 0Substituting these values into the standard equation of an ellipse,x²/a² + y²/b² = 1,we get the equation:x²/2a² + y²/2b² = 1
Since the horizontal major axis is 4, we have:2a = 4a = 2And since the vertical minor axis is 2, we have:2b = 2b = 1Substituting these values into the equation above, we get:x²/4 + y²/2 = 1Answer: The equation of an ellipse with a center at (0,0), a horizontal major axis of 4 and vertical minor axis of 2 is x²/4 + y²/2 = 1.
To know more about vertical minor axis visit :
https://brainly.com/question/14384186
#SPJ11
dedimal jistes.) (a) Fina the aveage velocity toring eich time centod. (1) [1,2] (in) (1,1 int \operatorname{cim}^{2} (14) \{1,1.011 entere (m) [1,1,00 s) सrys tink
The average velocity during the time intervals [1,2], [1,1.01], [1.01,4], and [1,100] are 0 m/s, 0 m/s, 0.006 m/s, and 0.0003 m/s respectively.
We have given some time intervals with corresponding position values, and we have to find the average velocity in each interval.Here is the given data:Time (s)Position (m)111.0111.0141.0281.041
Average velocity is the displacement per unit time, i.e., (final position - initial position) / (final time - initial time).We need to find the average velocity in each interval:(a) [1,2]Average velocity = (1.011 - 1.011) / (2 - 1) = 0m/s(b) [1,1.01]Average velocity = (1.011 - 1.011) / (1.01 - 1) = 0m/s(c) [1.01,4]
velocity = (1.028 - 1.011) / (4 - 1.01) = 0.006m/s(d) [1,100]Average velocity = (1.041 - 1.011) / (100 - 1) = 0.0003m/s
Therefore, the average velocity during the time intervals [1,2], [1,1.01], [1.01,4], and [1,100] are 0 m/s, 0 m/s, 0.006 m/s, and 0.0003 m/s respectively.
To know more about average velocity visit :
https://brainly.com/question/29125647
#SPJ11
Minimize the following functions to a minimum number of literals in SOP standard form.
(a) (1 Point) F1(a, b, c) = m0 ⋅ m1 (Minterm 0 ANDed with Minterm 1)
(b) (1 Point) F2(a, b, c) = M5 + M1 (Maxterm 5 ORed with Maxterm 2)
(c) (1 Point) F3(a, b, c) = M5 ⋅ m1 (Maxterm 5 ANDed with Minterm 1)
(a) F1(a, b, c) = m0 ⋅ m1 can be minimized to F1(a, b, c) = a' in SOP standard form, reducing it to a single literal. (b) F2(a, b, c) = M5 + M1 can be minimized to F2(a, b, c) = b' + c' in SOP standard form, eliminating redundant variables. (c) F3(a, b, c) = M5 ⋅ m1 can be minimized to F3(a, b, c) = b' + c' in SOP standard form, by removing the common variable 'a'.
(a) To minimize the function F1(a, b, c) = m0 ⋅ m1, we need to find the minimum number of literals in the sum-of-products (SOP) standard form.
First, let's write the minterms explicitly:
m0 = a'bc'
m1 = a'bc
To minimize the function, we can observe that the variables b and c are the same in both minterms. So, we can eliminate them and write the simplified expression as:
F1(a, b, c) = a'
Therefore, the minimum SOP form of F1(a, b, c) is F1(a, b, c) = a'.
(b) To minimize the function F2(a, b, c) = M5 + M1, we need to find the minimum number of literals in the SOP standard form.
First, let's write the maxterms explicitly:
M5 = a' + b' + c'
M1 = a' + b + c
To minimize the function, we can observe that the variables a and c are the same in both maxterms. So, we can eliminate them and write the simplified expression as:
F2(a, b, c) = b' + c'
Therefore, the minimum SOP form of F2(a, b, c) is F2(a, b, c) = b' + c'.
(c) To minimize the function F3(a, b, c) = M5 ⋅ m1, we need to find the minimum number of literals in the SOP standard form.
First, let's write the maxterm and minterm explicitly:
M5 = a' + b' + c'
m1 = a'bc
To minimize the function, we can observe that the variable a is the same in both terms. So, we can eliminate it and write the simplified expression as:
F3(a, b, c) = b' + c'
Therefore, the minimum SOP form of F3(a, b, c) is F3(a, b, c) = b' + c'.
To know more about literal refer here:
https://brainly.com/question/33394013#
#SPJ11
(e) how many ways are there to place a total of m distinguishable balls into n distinguishable urns, with some urns possibly empty or with several balls?
The formula for the number of ways to distribute `m` distinguishable balls into `n` distinguishable urns is: C(m + n - 1, n - 1)
The formula for the number of ways to distribute `m` distinguishable balls into `n` distinguishable urns is:
C(m + n - 1, n - 1)
where C(n, k) represents the binomial coefficient, also known as "n choose k".
In this case, the formula becomes:
C(m + n - 1, n - 1)
This formula accounts for the fact that we can think of placing `m` balls and `n-1` dividers (or "bars") in a line, and the number of ways to arrange them represents the distribution of balls into urns.
The m + n - 1 represents the total number of spaces in the line (balls + dividers), and choosing n-1 spaces to place the dividers separates the line into n sections, corresponding to the urns.
Learn more about Combination here:
https://brainly.com/question/29595163
#SPJ4
44. If an investment company pays 8% compounded quarterly, how much should you deposit now to have $6,000 (A) 3 years from now? (B) 6 years from now? 45. If an investment earns 9% compounded continuously, how much should you deposit now to have $25,000 (A) 36 months from now? (B) 9 years from now? 46. If an investment earns 12% compounded continuously. how much should you deposit now to have $4,800 (A) 48 months from now? (B) 7 years from now? 47. What is the annual percentage yield (APY) for money invested at an annual rate of (A) 3.9% compounded monthly? (B) 2.3% compounded quarterly? 48. What is the annual percentage yield (APY) for money invested at an annual rate of (A) 4.32% compounded monthly? (B) 4.31% compounded daily? 49. What is the annual percentage yield (APY) for money invested at an annual rate of (A) 5.15% compounded continuously? (B) 5.20% compounded semiannually? 50. What is the annual percentage yield (APY) for money invested at an annual rate of (A) 3.05% compounded quarterly? (B) 2.95% compounded continuously? 51. How long will it take $4,000 to grow to $9,000 if it is invested at 7% compounded monthly? 52. How long will it take $5,000 to grow to $7,000 if it is invested at 6% compounded quarterly? 53. How long will it take $6,000 to grow to $8,600 if it is invested at 9.6% compounded continuously?
44. A:
A = P(1 + r/n)^(n*t)
(A) To have $6,000 in 3 years from now:
A = $6,000
r = 8% = 0.08
n = 4 (compounded quarterly)
t = 3 years
$6,000 = P(1 + 0.08/4)^(4*3)
$4,473.10
44. B:
________________________________________________
Using the same formula:
$6,000 = P(1 + 0.08/4)^(4*6)
$3,864.12
45. A:
A = P * e^(r*t)
(A) To have $25,000 in 36 months from now:
A = $25,000
r = 9% = 0.09
t = 36 months / 12 = 3 years
$25,000 = P * e^(0.09*3)
$19,033.56
45. B:
Using the same formula:
$25,000 = P * e^(0.09*9)
$8,826.11
__________________________________________________
46. A:
A = P * e^(r*t)
(A) To have $4,800 in 48 months from now:
A = $4,800
r = 12% = 0.12
t = 48 months / 12 = 4 years
$4,800 = P * e^(0.12*4)
$2,737.42
46. B:
Using the same formula:
$4,800 = P * e^(0.12*7)
$1,914.47
__________________________________________________
47. A:
For an investment at an annual rate of 3.9% compounded monthly:
The periodic interest rate (r) is the annual interest rate (3.9%) divided by the number of compounding periods per year (12 months):
r = 3.9% / 12 = 0.325%
APY = (1 + r)^n - 1
r is the periodic interest rate (0.325% in decimal form)
n is the number of compounding periods per year (12)
APY = (1 + 0.00325)^12 - 1
4.003%
47. B:
The periodic interest rate (r) is the annual interest rate (2.3%) divided by the number of compounding periods per year (4 quarters):
r = 2.3% / 4 = 0.575%
Using the same APY formula:
APY = (1 + 0.00575)^4 - 1
2.329%
__________________________________________________
48. A.
The periodic interest rate (r) is the annual interest rate (4.32%) divided by the number of compounding periods per year (12 months):
r = 4.32% / 12 = 0.36%
Again using APY like above:
APY = (1 + (r/n))^n - 1
APY = (1 + 0.0036)^12 - 1
4.4037%
48. B:
The periodic interest rate (r) is the annual interest rate (4.31%) divided by the number of compounding periods per year (365 days):
r = 4.31% / 365 = 0.0118%
APY = (1 + 0.000118)^365 - 1
4.4061%
_________________________________________________
49. A:
The periodic interest rate (r) is equal to the annual interest rate (5.15%):
r = 5.15%
Using APY yet again:
APY = (1 + 0.0515/1)^1 - 1
5.26%
49. B:
The periodic interest rate (r) is the annual interest rate (5.20%) divided by the number of compounding periods per year (2 semiannual periods):
r = 5.20% / 2 = 2.60%
Again:
APY = (1 + 0.026/2)^2 - 1
5.31%
____________________________________________________
50. A:
AHHHH So many APY questions :(, here we go again...
The periodic interest rate (r) is the annual interest rate (3.05%) divided by the number of compounding periods per year (4 quarterly periods):
r = 3.05% / 4 = 0.7625%
APY = (1 + 0.007625/4)^4 - 1
3.08%
50. B:
The periodic interest rate (r) is equal to the annual interest rate (2.95%):
r = 2.95%
APY = (1 + 0.0295/1)^1 - 1
2.98%
_______________________________________________
51.
We use the formula from while ago...
A = P(1 + r/n)^(nt)
P = $4,000
A = $9,000
r = 7% = 0.07 (annual interest rate)
n = 12 (compounded monthly)
$9,000 = $4,000(1 + 0.07/12)^(12t)
7.49 years
_________________________________________________
52.
Same formula...
A = P(1 + r/n)^(nt)
$7,000 = $5,000(1 + 0.06/4)^(4t)
5.28 years
_____________________________________________
53.
Using the formula:
A = P * e^(rt)
A is the final amount
P is the initial principal (investment)
r is the annual interest rate (expressed as a decimal)
t is the time in years
e is the base of the natural logarithm
P = $6,000
A = $8,600
r = 9.6% = 0.096 (annual interest rate)
$8,600 = $6,000 * e^(0.096t)
4.989 years
_____________________________________
Hope this helps.
Which of the above diagrams correctly portray the demand (D) and marginal revenue (MR) curves of a purely competitive seller?
The diagrams that correctly portray the demand (D) and marginal revenue (MR) curves of a purely competitive seller is the C.
What is the relationship between demand and marginal revenue?The price elasticity of demand, or the responsiveness of quantity demanded to a change in price, is connected to margin revenue. Demand is elastic when marginal revenue is positive and inelastic when marginal revenue is negative.
As the MR curve and the demand curve have the same vertical intercept and the MR curve's horizontal intercept is half that of the demand curve, the MR curve will have a slope that is twice as steep as the demand curve.
Learn more about marginal revenue at:
https://brainly.com/question/13444663
#SPJ4
If f(x) = 4x (sin x+cos x), find
f'(x) =
f'(1) =
Therefore, f'(1) = 8 cos 1.Therefore, f'(x) = (4 + 4x) cos x + (4 - 4x) sin x.
Given that f(x) = 4x (sin x + cos x)
To find: f'(x) = , f'(1)
=f(x)
= 4x (sin x + cos x)
Taking the derivative of f(x) with respect to x, we get;
f'(x) = (4x)' (sin x + cos x) + 4x [sin x + cos x]
'f'(x) = 4(sin x + cos x) + 4x (cos x - sin x)
f'(x) = 4(cos x + sin x) + 4x cos x - 4x sin x
f'(x) = 4 cos x + 4x cos x + 4 sin x - 4x sin x
f'(x) = (4 + 4x) cos x + (4 - 4x) sin x
Therefore, f'(x) = (4 + 4x) cos x + (4 - 4x) sin x.
Using the chain rule, we can find the derivative of f(x) with respect to x as shown below:
f(x) = 4x (sin x + cos x)
f'(x) = 4 (sin x + cos x) + 4x (cos x - sin x)
f'(x) = 4 cos x + 4x cos x + 4 sin x - 4x sin x
The answer is: f'(x) = 4 cos x + 4x cos x + 4 sin x - 4x sin x.
To find f'(1), we substitute x = 1 in f'(x)
f'(1) = 4 cos 1 + 4(1) cos 1 + 4 sin 1 - 4(1) sin 1
f'(1) = 4 cos 1 + 4 cos 1 + 4 sin 1 - 4 sin 1
f'(1) = 8 cos 1 - 0 sin 1
f'(1) = 8 cos 1
Therefore, f'(1) = 8 cos 1.
To know more about sin visit;
brainly.com/question/19213118
#SPJ11
(t/f) if y is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.
If y is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix is a True statement.
In an orthogonal set of vectors, each vector is orthogonal (perpendicular) to all other vectors in the set.
Therefore, the dot product between any two vectors in the set will be zero.
Since the vectors are orthogonal, the weights in the linear combination can be obtained by taking the dot product of the given vector y with each of the orthogonal vectors and dividing by the squared magnitudes of the orthogonal vectors. This allows for a direct computation of the weights without the need for row operations on a matrix.
Learn more about Linear Combination here:
https://brainly.com/question/30888143
#SPJ4
For each of the following statements, find the negation of the statement. (a) For all integers x,x 2
is nonnegative. (b) For all integers a and b, if a
(a) The negation of the statement "For all integers x, x² is nonnegative" is "There exists an integer x such that x² is negative or x is not an integer."
(b) The negation of the statement "For all integers a and b, if a < b then a² < b²" is "There exist integers a and b such that a < b and a² ≥ b²."
Explanation:
(a)The original statement is "For all integers x, x² is nonnegative."This statement can be translated into the symbolic form ∀x ∈ Z, x² ≥ 0.
The negation of this statement is "There exists an integer x such that x² is negative or x is not an integer."
This statement can be translated into the symbolic form ∃x ∈ Z, x² < 0 or x ∉ Z.
(b)The original statement is "For all integers a and b, if a < b then a² < b²."
This statement can be translated into the symbolic form ∀a, b ∈ Z, a < b → a² < b².
The negation of this statement is "There exist integers a and b such that a < b and a² ≥ b²."
This statement can be translated into the symbolic form ∃a, b ∈ Z, a < b ∧ a² ≥ b².
To know more about negation, visit:
https://brainly.com/question/30426958
#SPJ11
Let W= computers with Winamp), with ∣W∣=143, R={ computers with RealPlayer }, with ∣R∣=70, and C={ computers with a CD writer }, with ∣C∣=33. Also, let ∣W∩C∣=20,∣R∩C∣=7, and ∣W∩R∣=28, and let 193 machines have at least one of the three. How many computers have Winamp, RealPlayer, and a CD writer?
According to the given information, there are 2 computers that have Winamp, RealPlayer, and a CD writer among the total of 193 machines with at least one of the three applications.
Let's solve this problem using the principle of inclusion-exclusion. We know that there are a total of 193 machines that have at least one of the three software applications.
We can start by adding the number of computers with Winamp, RealPlayer, and a CD writer. Let's denote this as ∣W∩R∩C∣. However, we need to be careful not to count this group twice, so we subtract the overlapping counts: ∣W∩C∣, ∣R∩C∣, and ∣W∩R∣.
Using the principle of inclusion-exclusion, we have:
∣W∪R∪C∣ = ∣W∣ + ∣R∣ + ∣C∣ - ∣W∩R∣ - ∣W∩C∣ - ∣R∩C∣ + ∣W∩R∩C∣.
Substituting the given values, we have:
193 = 143 + 70 + 33 - 28 - 20 - 7 + ∣W∩R∩C∣.
Simplifying the equation, we find:
∣W∩R∩C∣ = 193 - 143 - 70 - 33 + 28 + 20 + 7.
∣W∩R∩C∣ = 2.
Therefore, there are 2 computers that have Winamp, RealPlayer, and a CD writer among the total of 193 machines with at least one of the three applications.
To learn more about number click here
brainly.com/question/3589540
#SPJ11
a website streams movies and television shows to its subscribers. employees know that the average time a user spends per session on their website is 222 hours. the website changed its design, and they wanted to know if the average session length was longer than 222 hours. they randomly sampled 505050 users and found that their session lengths had a mean of 2.752.752, point, 75 hours and a standard deviation of 1.551.551, point, 55 hours. the employees want to use these sample data to conduct a ttt test on the mean. assume that all conditions for inference have been met. identify the correct test statistic for their significance test.
The appropriate conclusion:
The evidence suggests that the mean session length is longer than 2 hours.
Since the P-value (0.015) is less than the significance level (0.05), we have sufficient evidence to reject the null hypothesis.
The test statistic (t ≈ 2.24) also supports the conclusion that the mean session length is longer than 2 hours.
Thus, the appropriate conclusion at the significance level α = 0.05 is:
The evidence suggests that the mean session length is longer than 2 hours.
Learn more about Hypothesis here:
https://brainly.com/question/31319397
#SPJ4
the question attached here seems it be incomplete, the complete question is:
A website streams movies and television shows to its subscribers. Employees know that the average time a user spends per session on their website is 2 hours. The website changed its design, and they wanted to know if the average session length was longer than 2 hours. They randomly sampled 50 users to test H_{0} / mu = 2 hours versus H_{a} / mu > 2 hours, where μ is the mean session length.
Users in the sample had a mean session length of 2.49 hours and a standard deviation of 1.55 hours. These results produced a test statistic of t \approx 2.24 and a P-value of approximately 0.015,
Assuming the conditions for inference were met, what is an appropriate conclusion at the significance level? alpha = 0.05
Choose 1 answer:
The evidence suggests that the mean session length is shorter than 2 hours.
The evidence suggests that the mean session length is longer than 2 hours.
The evidence suggests that the mean session length is exactly 2 hours.
They cannot conclude the mean session length is longer than 2 hours.
Linear Approximation]
Let f(x,y)=√( 5+2x+3xy^2)
(a) Find the equation of the plane tangent to the graph of z=f(x,y) at (x,y)=(4,1). (b) Give the linear approximation for f(4.1,1.05).
(c) Give the linear approximation for f(3.75,0.5). (d) Use a calculator to determine the exact values for parts (b) and (c). What is the error in each part? Which part had a better approximation, and why?
(a) The equation of the plane tangent to the graph of f(x, y) at (4, 1) is given by
z - f(4, 1) = f x(4, 1)(x - 4) + f y(4, 1)(y - 1)
On solving for z, we get
z = 3 + (x - 4) / 3 + (y - 1) / 2
(b) The linear approximation for f(4.1, 1.05) is given by:
Δz = f x(4, 1)(4.1 - 4) + f y(4, 1)(1.05 - 1)
On substituting the values of f x(4, 1) and f y(4, 1), we get
Δz = 0.565
(c) The linear approximation for f(3.75, 0.5) is given by:
Δz = f x(4, 1)(3.75 - 4) + f y(4, 1)(0.5 - 1)
On substituting the values of f x(4, 1) and f y(4, 1), we get
Δz = -0.265
(d) Using a calculator, we get
f(4.1, 1.05) = 3.565708...f(3.75, 0.5) = 2.66629...
The error in part (b) is given by
Error = |f(4.1, 1.05) - Δz - f(4, 1)|= |3.565708 - 0.565 - 3|≈ 0.0007
The error in part (c) is given by
Error = |f(3.75, 0.5) - Δz - f(4, 1)|= |2.66629 + 0.265 - 3|≈ 0.099
The better approximation is part (b) since the error is smaller than part (c).
Learn more about the plane tangent: https://brainly.com/question/33052311
#SPJ11
Use synthetic division to deteine whether the given number k is a zero of the polynomial function. If it is not, give the value of f(k). See Examples 2 and 3. f(x)=x ^2+2x−8;k=2 f(x)=x ^2+4x−5;k=−5 f(x)=x ^3−3x ^2 +4x−4;k=2
f(x)=x ^3 +2x ^2−x+6;k=−3
f(x)=2x ^3−6x ^2−9x+4;k=1
The k is not a zero of the given polynomial function and the value of k is k=1.
We are required to use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of f(k).
Example 2:
f(x) = x^2 + 2x - 8; k = 2
Taking the synthetic division of f(x) = x^2 + 2x - 8, and substituting k = 2 in the synthetic division:
2 -4 0-8
We get a remainder of 0. Therefore, k = 2 is a zero of the given polynomial function.
Example 3:
f(x) = x^2 + 4x - 5; k = -5
Taking the synthetic division of f(x) = x^2 + 4x - 5, and substituting k = -5 in the synthetic division:
-5 -1 6-5
We get a remainder of 0. Therefore, k = -5 is a zero of the given polynomial function.
Example 4:
f(x) = x^3 - 3x^2 + 4x - 4; k = 2
Taking the synthetic division of f(x) = x^3 - 3x^2 + 4x - 4, and substituting k = 2 in the synthetic division:
2 -3 1 4-6
We get a remainder of -6. Therefore, k = 2 is not a zero of the given polynomial function. f(2) = -6.
Example 5:
f(x) = x^3 + 2x^2 - x + 6; k = -3
Taking the synthetic division of f(x) = x^3 + 2x^2 - x + 6, and substituting k = -3 in the synthetic division:
-3 1 2 -1-3 -3 6-6
We get a remainder of -6. Therefore, k = -3 is not a zero of the given polynomial function. f(-3) = -6.
Example 6: f(x) = 2x^3 - 6x^2 - 9x + 4; k = 1
Taking the synthetic division of f(x) = 2x^3 - 6x^2 - 9x + 4, and substituting k = 1 in the synthetic division:
1 -6 -15 -9-6 -12 3-6
We get a remainder of -6.
Therefore, k = 1 is not a zero of the given polynomial function. f(1) = -6.
To know more about polynomial here:
https://brainly.com/question/4142886
#SPJ11
square room is covered by a number of whole rectangular slabs of sides Calculate the least possible area of the room in square metres (3mks )
The least possible area of the room in square metres is Nlw, where N is the smallest integer that satisfies the equation LW = Nlw.
Let the length, width, and height of the square room be L, W, and H, respectively. Let the length and width of each rectangular slab be l and w, respectively. Then, the number of slabs required to cover the area of the room is given by:
Number of Slabs = (LW)/(lw)
Since we want to find the least possible area of the room, we can minimize LW subject to the constraint that the number of slabs is an integer. To do so, we can use the method of Lagrange multipliers:
We want to minimize LW subject to the constraint f(L,W) = (LW)/(lw) - N = 0, where N is a positive integer.
The Lagrangian function is then:
L(L,W,λ) = LW + λ[(LW)/(lw) - N]
Taking partial derivatives with respect to L, W, and λ and setting them to zero yields:
∂L/∂L = W + λW/l = 0
∂L/∂W = L + λL/w = 0
∂L/∂λ = (LW)/(lw) - N = 0
Solving these equations simultaneously, we get:
L = sqrt(N)l
W = sqrt(N)w
Therefore, the least possible area of the room is:
LW = Nlw
where N is the smallest integer that satisfies this equation.
In other words, the area of the room is a multiple of the area of each slab, and the least possible area of the room is obtained when the room dimensions are integer multiples of the slab dimensions.
Therefore, the least possible area of the room in square metres is Nlw, where N is the smallest integer that satisfies the equation LW = Nlw.
learn more about integer here
https://brainly.com/question/15276410
#SPJ11
Let e>0. For each of the following, find a δ>0 such that ∣f(x)−ℓ∣<ε for all x satisfying 0<|x-a|<δ.
(a.) f(x)=3x+7,a=4,ℓ=19
(b) f(x)==1/x,a=2,ℓ=1/2
(c.) f(x) = x²,ℓ=a²
(d.) f(x) = √∣x∣,a=0,ℓ=0
The value of δ for each of the given functions is:
(a) δ = (ε + 12)/3, for ε > 0
(b) δ
Given information is:
(a.) f(x) = 3x + 7, a = 4, ℓ = 19
(b) f(x) = 1/x, a = 2, ℓ = 1/2
(c) f(x) = x², ℓ = a²
(d) f(x) = √|x|, a = 0, ℓ = 0
In order to find δ > 0, we need to first evaluate the limit value, which is given in each of the questions. Then we need to evaluate the absolute difference between the limit value and the function value, |f(x) - ℓ|. And once that is done, we need to form a delta expression based on this value. Hence, let's solve the questions one by one.
(a) f(x) = 3x + 7, a = 4, ℓ = 19
First, let's evaluate the absolute difference between f(x) and ℓ:
|f(x) - ℓ| = |3x + 7 - 19| = |-12 + 3x| = 3|x - 4| - 12
Now, for |f(x) - ℓ| < ε, 3|x - 4| - 12 < ε
⇒ 3|x - 4| < ε + 12
⇒ |x - 4| < (ε + 12)/3
Therefore, δ = (ε + 12)/3, for ε > 0
(b) f(x) = 1/x, a = 2, ℓ = 1/2
First, let's evaluate the absolute difference between f(x) and ℓ:
|f(x) - ℓ| = |1/x - 1/2| = |(2 - x)/(2x)|
Now, for |f(x) - ℓ| < ε, |(2 - x)/(2x)| < ε
⇒ |2 - x| < 2ε|x|
Now, we know that |x - 2| < δ, therefore,
δ = min{2ε, 1}, for ε > 0
(c) f(x) = x², ℓ = a²
First, let's evaluate the absolute difference between f(x) and ℓ:
|f(x) - ℓ| = |x² - a²| = |x - a| * |x + a|
Now, for |f(x) - ℓ| < ε, |x - a| * |x + a| < ε
⇒ |x - a| < ε/(|x + a|)
Now, we know that |x - a| < δ, therefore,
δ = min{ε/(|a| + 1), 1}, for ε > 0
(d) f(x) = √|x|, a = 0, ℓ = 0
First, let's evaluate the absolute difference between f(x) and ℓ:
|f(x) - ℓ| = |√|x| - 0| = √|x|
Now, for |f(x) - ℓ| < ε, √|x| < ε
⇒ |x| < ε²
Now, we know that |x - 0| < δ, therefore,
δ = ε², for ε > 0
Learn more about value here :-
#SPJ11