Graph the function f(x)= 3+2 in x and its inverse from model 1.
The graph of the function and its inverse is added as an attachment
Sketching the graph of the function and its inverseFrom the question, we have the following parameters that can be used in our computation:
f(x) = 3 + 2ln(x)
Express as an equation
So, we have
y = 3 + 2ln(x)
Swap x and y in the above equation
x = 3 + 2ln(y)
Next, we have
2ln(y) = x - 3
Divide by 2
ln(y) = (x - 3)/2
Take the exponent of both sides
[tex]y = e^{\frac{x - 3}{2}}[/tex]
Next, we plot the graphs
The graph of the functions is added as an attachment
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Find the midpoint of WZ of WXYZ with the vertices W(0, 0), X(h, 0), Y(h,b), and Z(0, b).
(0, h/2)
(h/2, b/2)
(0, b/2)
(h/2, 0)
Third option is correct.The midpoint of WZ of WXYZ with the vertices W(0, 0), X(h, 0), Y(h,b), and Z(0, b) is (0, b/2).
To find the midpoint of segment WZ, we need to average the x-coordinates and the y-coordinates of the endpoints.
The coordinates of point W are (0, 0), and the coordinates of point Z are (0, b).
To find the x-coordinate of the midpoint, we average the x-coordinates of W and Z:
(x-coordinate of W + x-coordinate of Z) / 2 = (0 + 0) / 2 = 0 / 2 = 0
To find the y-coordinate of the midpoint, we average the y-coordinates of W and Z:
(y-coordinate of W + y-coordinate of Z) / 2 = (0 + b) / 2 = b / 2
Therefore, the midpoint of segment WZ is (0, b/2).
So, the correct answer is (0, b/2).
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Complete the following number sequence. 2, 4, 7, __, 16, __, 29, __
The completed sequence would then be: 2, 4, 7, 9, 16, 19, 29.
To complete the given number sequence, let's analyze the pattern and identify the missing terms.
Looking at the given sequence 2, 4, 7, __, 16, __, 29, __, we can observe the following pattern:
The difference between consecutive terms in the sequence is increasing by 1. In other words, the sequence is formed by adding 2 to the previous term, then adding 3, then adding 4, and so on.
Using this pattern, we can determine the missing terms as follows:
To obtain the third term, we add 2 to the second term:
7 + 2 = 9
To find the fifth term, we add 3 to the fourth term:
16 + 3 = 19
To determine the seventh term, we add 4 to the sixth term:
__ + 4 = 23
Therefore, the missing terms in the sequence are 9, 19, and 23.
By identifying the pattern of increasing differences, we can extend the sequence and fill in the missing terms accordingly.
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In circle M below, diameter AC, chords AB and BC, and radius MB
are drawn.
The statement which is not true about the circle M is ∆ABM is isosceles.
The correct answer choice is option 2.
Which statement is not true?Based on the circle M;
diameter AC,
chords AB and BC,
radius MB
Isosceles triangle: This is a type of triangle which has two equal sides and angles.
Equilateral triangle is a triangle which has three equal sides and angles.
Hence, ∆ABM is equilateral triangle.
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Team A and Team B together won 50% more games than Team C did. Team A won 50% as many games as Team B did. The three teams won 60 games in all. How many games did each team win?
NEED NOW PLEASE HELP OUT
Answer:
x=50
Step-by-step explanation:
Make this equal to 180.
x+3x-35+x-35 = 180
5x = 180 + 70
5x=250
x=50
Similar Triangles
Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to
prove the triangles similar? Explain your reasoning.
I need help on number 1 and 2
The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;
1. ΔAJK ~ ΔSWY according to the SAS similarity postulate
2. ΔLMN ~ ΔLPQ according to the AA similarity postulate
3. ΔPQN ~ ΔLMN
LM = 12, QP = 8
4. ΔLMK~ΔLNJ
NL = 21, ML = 14
What are similar triangles?
Similar triangles are triangles that have the same shape but may have different sizes.
1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;
24/16 = 3/2
18/12 = 3/2
The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.
2. The ratio of the corresponding sides in each of the triangles are;
MN/LN = 8/10 = 4/5
PQ/LQ = 12/(10 + 5) = 12/15 = 4/5
The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule
3. The alternate interior angles theorem indicates;
Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;
ΔPQN ~ ΔLMN by the AA similarity postulate
LM/QP = (x + 3)/(x - 1) = 18/12
12·x + 36 = 18·x - 18
18·x - 12·x = 36 + 18 = 54
6·x = 54
x = 54/6 = 9
LM = 9 + 3 = 12
QP = x - 1
QP = 9 - 1 = 8
4. The similar triangles are; ΔLMK and ΔLNJ
ΔLMK ~ ΔLNJ by AA similarity postulate
ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)
ML/NL = LK/LJ = (24 - 8)/24
(24 - 8)/24 = (6·x + 2)/((7·x + 7)
16/24 = (6·x + 2)/(7·x + 7)
16 × (7·x + 7) = 24 × (6·x + 2)
112·x + 112 = 144·x + 48
144·x - 112·x = 32·x = 112 - 48 = 64
x = 64/32 = 2
ML = 6 × 2 + 2 = 14
NL = 7 × 2 + 7 = 21
MN = 2 + 5 = 7
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Dewan’s bank account balance is -$16.75. He deposits checks totaling $23.59. What is his new balance? -$1.08
Answer:
$6.84
Step-by-step explanation:
This is quite a simple question, simply add the new deposited amount into the original balance to get your answer.
Original balance: -$16.75Deposit: $23.59New balance: -$16.75 + $23.59 = $6.84Solve each equation for the angle in standard position, for 0° ≤ 0 < 360° (nearest tenth, if necessary).
a) tan 0 = 1 / √3
b) 2cos 0= √3
Answer:
Step-by-step explanation:
a) To solve the equation tan θ = 1/√3, we can find the angle whose tangent is 1/√3 by taking the inverse tangent (arctan) of 1/√3.
θ = arctan(1/√3)
θ ≈ 30.0°
Therefore, the angle in standard position that satisfies tan θ = 1/√3 is approximately 30.0°.
b) To solve the equation 2cos θ = √3, we can isolate the cosine term by dividing both sides of the equation by 2.
cos θ = √3 / 2
Now, we can find the angle whose cosine is √3/2 by taking the inverse cosine (arccos) of √3/2.
θ = arccos(√3/2)
θ ≈ 30.0°
Therefore, the angle in standard position that satisfies 2cos θ = √3 is approximately 30.0°.
Please look at photo. Thank you. If you get it right I’ll give you a good rating!
a. The absolute maximum of g is 4.
The absolute minimum of g is -4.
b. The absolute maximum of h is 3.
The absolute minimum of h is -4.
What is a vertical asymptote?In Mathematics and Geometry, the vertical asymptote of a function simply refers to the value of x (x-value) which makes its denominator equal to zero (0).
By critically observing the graph of the polynomial function g shown above, we can logically deduce that its vertical asymptote is at x = 3. Furthermore, the absolute maximum of the polynomial function g is 4 while the absolute minimum of g is -4.
In conclusion, the absolute maximum of the polynomial function h is 3 while the absolute minimum of h is -4.
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PLEASE HELPPPPPPP NEED NOW
Answer:
BC = 24 units
Step-by-step explanation:
This is an isosceles triangle which always has:
two legs that are congruent to each other (i.e., equal),and two angles that are congruent to each other.In this triangle, the legs CA and BA are congruent so CA = BA and the angles C and B are congruent to each other so angle C = angle B.
Thus, we can find x by setting CA and BA equal to each other:
(3x - 15 = x + 33) + 15
(3x = x + 48) - x
(2x = 48) / x
x = 24
Thus, x = 24
Since the length of BC is x and x = 24, BC is 24 units long.
Dylan's mom told him that she would replace each one of his dimes with a quarter. If he uses all of his coins, determine if Dylan would then have enough money to buy a game priced at $20.98 if he must also pay an 8% sales tax.
Show that y₁(t) = e^ãt cos(μt) and
y₂(t) = e^ãt sin(μt)
are a fundamental set of solutions and state the general solution.
The functions y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) are a fundamental set of solutions because they are linearly independent and satisfy the given homogeneous linear differential equation, allowing for the formation of the general solution.
To show that y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) are a fundamental set of solutions, we need to demonstrate two things: linear independence and satisfaction of the given homogeneous linear differential equation.
First, let's consider linear independence. We can prove it by showing that there is no constant c₁ and c₂, not both zero, such that c₁y₁(t) + c₂y₂(t) = 0 for all t.
Now, let's verify that y₁(t) and y₂(t) satisfy the homogeneous linear differential equation. If the given differential equation is of the form ay''(t) + by'(t) + cy(t) = 0, we can substitute y₁(t) and y₂(t) into the equation and verify that it holds true.
Once we have established linear independence and satisfaction of the differential equation, we can state that the general solution to the homogeneous linear differential equation is given by y(t) = c₁y₁(t) + c₂y₂(t), where c₁ and c₂ are arbitrary constants. This general solution represents the linear combination of the fundamental set of solutions.
In summary, y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) form a fundamental set of solutions for the given differential equation, and the general solution is given by y(t) = c₁y₁(t) + c₂y₂(t).
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Find the net area of the following curve on the interval [0, 2].
(SHOW WORK)
f(x) = ex - e
The net area of the curve represented by f(x) = ex - e on the interval [0, 2] is e2 - 1.
To find the net area of the curve represented by the function f(x) = ex - e on the interval [0, 2], we need to calculate the definite integral of the function over that interval. The net area can be determined by taking the absolute value of the integral.
The integral of f(x) = ex - e with respect to x can be computed as follows:
∫[0, 2] (ex - e) dx
Using the power rule of integration, the antiderivative of ex is ex, and the antiderivative of e is ex. Thus, the integral becomes:
∫[0, 2] (ex - e) dx = ∫[0, 2] ex dx - ∫[0, 2] e dx
Integrating each term separately:
= [ex] evaluated from 0 to 2 - [ex] evaluated from 0 to 2
= (e2 - e0) - (e0 - e0)
= e2 - 1
The net area of the curve represented by f(x) = ex - e on the interval [0, 2] is e2 - 1.
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5. A person observes that from point A, the angle of elevation to the top of a cliff at D is 30°. Another person at point B, notes that the angle of elevation to the top of the
cliff is 45°. If the height of the cliff is 80.0 m, find the distance between A and B. Show the steps of your solution.
Answer:
In a 30°-60°-90° triangle, the length of the longer leg is √3 times the length of the shorter leg. So AC = 80√3.
In a 45°-45°-90° triangle, both legs are congruent. So BC = 80.
AB = AC - BC = (80√3 - 80) meters
= 80(√3 - 1) meters
= about 58.56 meters
The distance between points A and B is approximately 138.6 meters.
To find the distance between points A and B, we can use the concept of trigonometry and the given information.
Let's denote the distance between points A and B as x.
From point A, the angle of elevation to the top of the cliff at point D is 30°. This means that in the right triangle formed by points A, D, and the top of the cliff, the opposite side is the height of the cliff (80.0 m) and the adjacent side is x. We can use the tangent function to calculate the length of the adjacent side:
tan(30°) = opposite/adjacent
tan(30°) = 80.0/x
Simplifying the equation, we have:
x = 80.0 / tan(30°)
Using a calculator, we can find the value of tan(30°) ≈ 0.5774.
Substituting the value, we get:
x = 80.0 / 0.5774
Calculating the value, we find:
x ≈ 138.6 meters
In light of this, the separation between positions A and B is roughly 138.6 metres.
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3. Determine whether the triangles are similar. If they are, write a similarity statement.
Look at picture for reference
Please show work
The triangles DEF and SRQ are not similar triangles
Identifying the similar triangles in the figure.From the question, we have the following parameters that can be used in our computation:
The triangles in this figure are
DEF and SRQ
These triangles are not similar
This is because:
The corresponding angles in the triangles are not equal
For DEF, the angles are
50, 90 and 40
For SRQ, the angles are
51, 90 and 39
This means that they are not similar by any similarity statement
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The base of a triangle is 3 inches more than two times the height. If the area of the triangle is 7 in.² find the base and height.
Answer:
Let's denote the height of the triangle as "h" inches.
According to the given information, the base of the triangle is 3 inches more than two times the height. Therefore, the base can be expressed as (2h + 3) inches.
The formula to calculate the area of a triangle is:
Area = (1/2) * base * height
Substituting the given values, we have:
7 = (1/2) * (2h + 3) * h
To simplify the equation, let's remove the fraction by multiplying both sides by 2:
14 = (2h + 3) * h
Expanding the right side of the equation:
14 = 2h^2 + 3h
Rearranging the equation to bring all terms to one side:
2h^2 + 3h - 14 = 0
Now, we can solve this quadratic equation. We can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, the values are:
a = 2
b = 3
c = -14
Substituting these values into the quadratic formula:
h = (-3 ± √(3^2 - 4 * 2 * -14)) / (2 * 2)
Simplifying:
h = (-3 ± √(9 + 112)) / 4
h = (-3 ± √121) / 4
Taking the square root:
h = (-3 ± 11) / 4
This gives us two possible solutions for the height: h = 2 or h = -14/4 = -3.5.
Since a negative height doesn't make sense in this context, we discard the negative solution.
Therefore, the height of the triangle is h = 2 inches.
To find the base, we substitute this value back into the expression for the base:
base = 2h + 3
base = 2(2) + 3
base = 4 + 3
base = 7 inches
Hence, the base of the triangle is 7 inches and the height is 2 inches.
Step-by-step explanation:
-The answer for the height is 5.5 units.
-The base of the triangle is aproximately 2.5454 units.
To answer this problem, you have to set an equation with the information you're given. If you do it correctly, it should look like this:
7=1/2(3+2h)
-Now, you have to solve for h:
7=1.5+h
7-1.5=h
5.5=h
-Now that you have the height, you plug it in into the triangle area formula to solve for the base:
7=1/2(b)5.5
7=2.75b
7/2.75=b
b≈2.5454
-To make sure that the corresponding values for the base and height are correct, we plug the values in and this time we are going to solve for a(AREA):
A(triangle)=1/2(2.5454)(5.5)
A=1/2(13.9997)
A=6.99985 square units
-We round the result to the nearest whole number and we get our 7, which is the given value they gave us.
Un objeto que se hace girar, se desplaza 25 radianes en 0.8 segundos. ¿cuál es la velocidad angular de dicho objeto?
The angular velocity of the object is 31.25 radians/second.
Angular velocity is defined as the change in angular displacement per unit of time. In this case, the object rotates a total of 25 radians in 0.8 seconds. Therefore, the angular velocity can be calculated by dividing the total angular displacement by the time taken.
Angular velocity (ω) = Total angular displacement / Time taken
Given that the object rotates 25 radians and the time taken is 0.8 seconds, we can substitute these values into the formula:
ω = 25 radians / 0.8 seconds
Simplifying the equation gives:
ω = 31.25 radians/second
So, the angular velocity of the object is 31.25 radians/second.
Angular velocity measures how fast an object is rotating and is typically expressed in radians per second. It represents the rate at which the object's angular position changes with respect to time.
In this case, the object completes a rotation of 25 radians in 0.8 seconds, resulting in an angular velocity of 31.25 radians per second. This means that the object rotates at a rate of 31.25 radians for every second of time.
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Note the translated question is:
An object that is rotated moves 25 radians in 0.8 seconds. what is the angular velocity of said object?
1cm on a picture of a swimming pool represents 1200cm of the actual swimming pool. The length of the pictured swimming pool is 4.5cm and the width is 3cm. What is the perimeter of the actual swimming pool? Express your answer in meters.
Answer:
180 meters
Step-by-step explanation:
To find the perimeter of the actual swimming pool, you need to first find the length and width of the actual swimming pool by multiplying the length and width of the pictured swimming pool by the scale factor of 1200 cm.
Length of actual swimming pool = 4.5 cm × 1200 cm = 5400 cmWidth of actual swimming pool = 3 cm × 1200 cm = 3600 cmPerimeter of actual swimming pool = (5400 cm + 3600 cm) × 2 = 18000 cm.Now that we know that the perimeter of the actual pool is 18000 centimeters, we need to convert that to meters! Keep in mind that:
100cm = 1mNow we can divide 18000 by 100:
18000 cm ÷ 100 = 180 m
Therefore, the perimeter of the actual swimming pool is 180 m.
Assume a class has 26 members.
a. In how many ways can a president, a vice president, and a secretary be selected?
b. How many committees of 4 people can be chosen?
a. The number of ways to select a president, a vice president, and a secretary is
b. The number of ways to form a 4-person committee is
$0.
a. There are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.
b. There are 14,950 ways to form a 4-person committee from a class of 26 members.
a. To select a president, a vice president, and a secretary from a class of 26 members, we can use the concept of permutations.
For the president position, we have 26 choices. After selecting the president, we have 25 choices remaining for the vice president position. Finally, for the secretary position, we have 24 choices left.
The total number of ways to select a president, a vice president, and a secretary is obtained by multiplying the number of choices for each position:
Number of ways = 26 * 25 * 24 = 15,600
Therefore, there are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.
b. To form a 4-person committee from a class of 26 members, we can use the concept of combinations.
The number of ways to choose a committee of 4 people can be calculated using the formula for combinations:
Number of ways = C(n, r) = n! / (r!(n-r)!)
where n is the total number of members (26 in this case) and r is the number of people in the committee (4 in this case).
Plugging in the values, we have:
Number of ways = C(26, 4) = 26! / (4!(26-4)!)
Calculating this expression, we get:
Number of ways = 26! / (4! * 22!)
Using factorials, we simplify further:
Number of ways = (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1) = 14,950
Therefore, there are 14,950 ways to form a 4-person committee from a class of 26 members.
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A restaurant offers 10 appetizers and 7 main courses. In how many ways can a person order a two-course meal?
There are
ways a person can order a two-course meal.
There are 70 ways a person can order a two-course meal from the given restaurant.
To determine the number of ways a person can order a two-course meal from a restaurant that offers 10 appetizers and 7 main courses, we can use the concept of combinations.
First, we need to select one appetizer from the 10 available options.
This can be done in 10 different ways.
Next, we need to select one main course from the 7 available options. This can be done in 7 different ways.
Since the two courses are independent choices, we can multiply the number of options for each course to find the total number of combinations.
Therefore, the number of ways a person can order a two-course meal is 10 [tex]\times[/tex] 7 = 70.
So, there are 70 ways a person can order a two-course meal from the given restaurant.
It's important to note that this calculation assumes that a person can choose any combination of appetizer and main course.
If there are any restrictions or limitations on the choices, the number of combinations may vary.
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A newly hired lawyer receives a $15,000 signing bonus from a law firm and invests the money in a savings account at 4.75% interest. After 42 months, the lawyer checks the account balance.
Part A: Calculate the interest earned if the interest is compounded quarterly. Show all work. (2 points)
Part B: Calculate the interest earned if the interest is compounded continuously. Show all work. (2 points)
Part C: Using the values from Part A and Part B, compare the interest earned for each account by finding the difference in the amount of interest earned. (1 point)
Part A: The interest earned if the interest is compounded quarterly is $2,768.40.
Part B: The interest earned if the interest is compounded continuously is $2,695.92.
Part C: The difference in the amount of interest earned is approximately $72.48.
Part A: To calculate the interest earned when the interest is compounded quarterly, we can use the formula for compound interest:
[tex]A = P(1 + r/n)^(^n^t^)[/tex]
Where:
A = the final account balance
P = the principal amount (initial investment)
r = the annual interest rate (4.75% or 0.0475 as a decimal)
n = the number of times the interest is compounded per year (4 times for quarterly)
t = the number of years (42 months divided by 12 to convert to years)
Plugging in the values:
A = $15,000(1 + 0.0475/4)^(4 * (42/12))
A = $15,000(1.011875)^(14)
A ≈ $15,000(1.18456005)
A ≈ $17,768.40
The interest earned is the difference between the final account balance and the principal amount:
Interest earned = $17,768.40 - $15,000
Interest earned ≈ $2,768.40
Part B: When the interest is compounded continuously, we can use the formula:
[tex]A = Pe^(^r^t^)[/tex]
Where:
A = the final account balance
P = the principal amount (initial investment)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (4.75% or 0.0475 as a decimal)
t = the number of years (42 months divided by 12 to convert to years)
Plugging in the values:
A = $15,000 * e^(0.0475 * 42/12)
A ≈ $15,000 * e^(0.165625)
A ≈ $15,000 * 1.179727849
A ≈ $17,695.92
The interest earned is the difference between the final account balance and the principal amount:
Interest earned = $17,695.92 - $15,000
Interest earned ≈ $2,695.92
Part C: Comparing the interest earned for each account, we find that the interest earned when the interest is compounded quarterly is approximately $2,768.40, while the interest earned when the interest is compounded continuously is approximately $2,695.92.
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(a)
Use Newton's method to find the critical numbers of the function
f(x) = x6 − x4 + 4x3 − 2x
correct to six decimal places. (Enter your answers as a comma-separated list.)
x =
Incorrect: Your answer is incorrect.
(b)
Find the absolute minimum value of f correct to four decimal places.
(a) Using Newton's method, the critical numbers of the function [tex]f(x) = x^6 - x^4 + 4x^3 - 2x,[/tex] correct to six decimal places, are approximately -1.084, -0.581, -0.214, 0.580, and 1.279.
(b) The absolute minimum value of f is undefined since the function is a polynomial of even degree, and it approaches positive infinity as x approaches positive or negative infinity.
(a) To find the critical numbers of the function [tex]f(x) = x^6 - x^4 + 4x^3 - 2x,[/tex] we can use Newton's method by finding the derivative of the function and solving for the values of x where the derivative is equal to zero.
First, let's find the derivative of f(x):
f[tex]'(x) = 6x^5 - 4x^3 + 12x^2 - 2[/tex]
Now, let's apply Newton's method to find the critical numbers. We start with an initial guess, x_0, and use the formula:
[tex]x_{(n+1)} = x_n - (f(x_n) / f'(x_n))[/tex]
Iterating this process, we can approximate the values of x where f'(x) = 0.
Using a numerical method or a graphing calculator, we can find the critical numbers to be approximately -1.084, -0.581, -0.214, 0.580, and 1.279.
Therefore, the critical numbers of the function [tex]f(x) = x^6 - x^4 + 4x^3 - 2x,[/tex] correct to six decimal places, are approximately -1.084, -0.581, -0.214, 0.580, and 1.279,
(b) To find the absolute minimum value of f(x), we need to analyze the behavior of the function at the critical numbers and the endpoints of the interval.
Since the function f(x) is a polynomial of even degree, it approaches positive infinity as x approaches positive or negative infinity.
Therefore, there is no absolute minimum value for the function.
Hence, the absolute minimum value of f is undefined.
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Find the measure of the numbered angles
Look at picture for reference
Show work when possible
The measure of the numbered angles in the rhombus is determined as angle 1 = 90⁰, angle 2 = 57⁰, angle 3 = 45⁰, and angle 4 = 45⁰.
What is the measure of the numbered angles?The measure of the numbered angles is calculated by applying the following formula as follows;
Rhombus has equal sides and equal angles.
angle 2 = angle 57⁰ (alternate angles are equal)
angle 1 = 90⁰ (diagonals of rhombus intersects each other at 90⁰)
angle 3 = angle 4 (base angles of Isosceles triangle )
angle 3 = angle 4 = ¹/₂ x 90⁰
angle 3 = angle 4 = 45⁰
Thus, the measure of the numbered angles in the rhombus is determined as angle 1 = 90⁰, angle 2 = 57⁰, angle 3 = 45⁰, and angle 4 = 45⁰.
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The points A, B and C have position vectors a, b, c, referred to an origin O. i. Given that the point X lies on AB produced so that AB : BX = 2 : 1, find x, the position vector of X, in terms of a and b. ii. If Y lies on BC, between B and C so that BY : Y C = 1 : 3, find y, the position vector of Y, in terms of a and b iii. Given that Z is the midpoint of AC, Calculate the ratio XY : Y Z.
i. The position vector of X is 2b - a.
ii. The position vector of Y is (3b + c)/4.
iii. The ratio XY : Y Z is [tex]|(2b - a) - ((3b + c)/4)|/|((3b + c)/4) - (a + c)/2|[/tex]. Simplifying this expression will give us the final ratio.
i. To find the position vector x of point X, we can use the concept of vector addition. Since AB : BX = 2 : 1, we can express AB as a vector from A to B, which is given by (b - a). To find BX, we can use the fact that BX is twice as long as AB, so BX = 2 * (b - a). Adding this to the vector AB will give us the position vector of X: x = a + 2 * (b - a) = 2b - a.
ii. Similar to the previous part, we can express BC as a vector from B to C, which is given by (c - b). Since BY : YC = 1 : 3, we can find BY by dividing the vector BC into four equal parts and taking one part, so BY = (1/4) * (c - b). Adding this to the vector BY will give us the position vector of Y: y = b + (1/4) * (c - b) = (3b + c)/4.
iii. Z is the midpoint of AC, so we can find Z by taking the average of the vectors a and c: z = (a + c)/2. The ratio XY : YZ can be calculated by finding the lengths of the vectors XY and YZ and taking their ratio. Since XY = |x - y| and YZ = |y - z|, we have XY : YZ = |x - y|/|y - z|. Plugging in the values of x, y, and z we found earlier, we get XY : YZ =[tex]|(2b - a) - ((3b + c)/4)|/|((3b + c)/4) - (a + c)/2|[/tex].
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Find the limit (if the limit exists). Solve in two different ways.
The limit of the trigonometric expression is equal to 0.
How to determine the limit of a trigonometric expression
In this problem we find the case of a trigonometric expression, whose limit must be found. This can be done by means of algebra properties, trigonometric formula and known limits. First, write the entire expression below:
[tex]\lim_{\Delta x \to 0} \frac{\cos (\pi + \Delta x) + 1}{\Delta x}[/tex]
Second, use the trigonometric formula cos (π + Δx) = - cos Δx to simplify the resulting formula:
[tex]\lim_{\Delta x \to 0} \frac{1 - \cos \Delta x}{\Delta x}[/tex]
Third, use known limits to determine the result:
0
The limit of the trigonometric function [cos (π + Δx) + 1] / Δx evaluated at Δx → 0 is equal to 0.
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Find the sum of the first 33 terms of the following series, to the nearest
integer.
2, 11, 20,...
Step-by-step explanation:
Common difference , d, is 9
Sn = n/2 ( a1 + a33) a33 = a1 + 32d = 2 + 32(9) = 290
S33 = 33/2 ( 2+290) = 4818
what is (0.3)0 in binominal distribution
Answer:
When p, the probability of success, is zero in a binomial distribution, the probability of getting exactly k successes in n trials is also zero for all values of k except when k is zero (i.e., when there are no successes).
So, in the case of (0.3)^0, the result would be 1, because any number raised to the power of 0 is equal to 1. Therefore, the probability of getting zero successes in a binomial distribution when the probability of success is 0.3 is 1.
the peterson family and the stewart family each used their sprinklers last summer. the water output rate for the peterson family’s sprinkler was 35 L per hour. the water output rate for the stewart family’s sprinkler was 40 L per hour. the families used their sprinklers for a combined total of 45 hours, resulting in a total water output of 1,650 L. how long was each sprinkler used?
The Peterson family used their sprinkler for 30 hours, while the Stewart family used theirs for 15 hours.
Let's assume that the Peterson family used their sprinkler for a certain number of hours, which we'll denote as x, and the Stewart family used their sprinkler for the remaining hours, which would be 45 - x.
The water output rate for the Peterson family's sprinkler is given as 35 L per hour. Therefore, the total water output for the Peterson family can be calculated by multiplying the water output rate (35 L/h) by the number of hours they used the sprinkler (x): 35x.
Similarly, for the Stewart family, with a water output rate of 40 L per hour, the total water output for their sprinkler is given by 40(45 - x).
According to the problem, the combined total water output for both families is 1,650 L. Therefore, we can write the equation:
35x + 40(45 - x) = 1,650.
Simplifying the equation, we get:
35x + 1,800 - 40x = 1,650,
-5x = 1,650 - 1,800,
-5x = -150.
Dividing both sides of the equation by -5, we find:
x = -150 / -5 = 30.
So, the Peterson family used their sprinkler for 30 hours, and the Stewart family used theirs for 45 - 30 = 15 hours.
Therefore, the Peterson family used their sprinkler for 30 hours, while the Stewart family used theirs for 15 hours.
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50 PTS!!!!!!!!!!! I NEED HELP!!!!!
Answer this question based on the table above. Choose the right answer.
Is the statement true that between 1966 and 1976 the average number of miles flown per passenger increased by one-third. (Yes or no)
Answer:
No
Step-by-step explanation:
To determine if the average number of miles flown per passenger increased by one-third between 1966 and 1976, we need to compare the increase in miles flown during that period.
According to the given table:
In 1966, the average number of miles flown per passenger was 711 miles.In 1976, the average number of miles flown per passenger was 831 miles.To find the increase in miles flown, subtract the 1966 value from the 1976 value:
[tex]\begin{aligned}\sf Increase\; in\; miles\; flown &= \sf 831 \;miles - 711\; miles\\&= \sf 120\; miles\end{aligned}[/tex]
Therefore, the average number of miles flown per passenger between 1966 and 1976 increased by 120 miles.
To check if the increase is one-third of the initial value, we need to calculate one-third of the 1966 value:
[tex]\begin{aligned}\sf One\;third \;of \;711 \;miles &= \sf \dfrac{1}{3} \times 711\; miles\\\\ &= \sf \dfrac{711}{3} \; miles\\\\&=\sf 237\;miles\end{aligned}[/tex]
Since the increase in miles flown (120 miles) is not equal to one-third of the initial 1966 value (237 miles), the statement that the average number of miles flown per passenger increased by one-third between 1966 and 1976 is not true.