Answer:
True
Step-by-step explanation:
The wording of the question is a little tricky, but here's what I think.
If a function f(x) is not defined at a point c, then the function has a discontinuity at that point. In order for a function to be continuous on an interval, it must be defined and have no abrupt changes or jumps within that interval. Since f(x) is not defined at c, it violates the condition of continuity, and therefore f(x) cannot be continuous on any interval that includes c.
The given statement "If f(x) is not defined at c, then f(x) cannot be continuous on any interval." is false because it does not automatically mean that f(x) cannot be continuous on any interval.
Continuity of a function depends on the behavior of the function around the point of interest, rather than just the absence of a definition at a single point. A function can still be continuous on an interval except at the specific point where it is not defined.
For example, consider the function f(x) = 1/x. This function is not defined at x = 0, but it is continuous on any interval that does not include x = 0. This is because f(x) approaches positive or negative infinity as x approaches 0 from the left or right side, respectively, indicating that there is no abrupt jump or discontinuity.
In general, the continuity of a function is determined by its behavior around a point, including its limit as x approaches that point. The absence of a definition at a single point does not automatically imply that the function cannot be continuous on any interval.
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From a point 340 meters from the base of the Hoover Dam, the angle of elevation to
the top of the dam is 33°. Find the height of the dam to the nearest meter.
Answer:
221 m
Step-by-step explanation:
Let h = height of dam
tan 33 = h/340
340 tan 33 = h
h = 221 m
The height of the dam to the nearest meter will be 221 m.
What is height?The vertical distance between the object's top and bottom is defined as height. It is measured in centimeters, inches, meters, and other units.
From the trigonometry;
[tex]\rm tan \theta = \frac{P}{B} \\\\ \rm tan 33^0 = \frac{h}{340} \\\\ h=221 \ m[/tex]
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488, 460, 520, 544, 535
What is the range of the data?
Answer:
84
Step-by-step explanation:
To find the range, find the difference between the largest value and the smallest value.
544 - 460 = 84
Find the derivative of the given function. y = - 4xln(x + 12) 3х 3x A. + 3ln (x + 12) x+12 B. - 3ln (x + 12) x+12 4x 4x C. - 4ln (x + 12) D. - x+12 + 4ln (x + 12) x+12
Using chain rule the derivative for the function y = -4xln(x + 12) is -4ln(x + 12) - 4x / (x + 12).
To find the derivative of the given function y = -4xln(x + 12), we can use the product rule and the chain rule.
The product rule states that for two functions u(x) and v(x), the derivative of their product is given by:
(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)
In this case, u(x) = -4x and v(x) = ln(x + 12). Let's calculate their derivatives:
u'(x) = -4
v'(x) = 1 / (x + 12)
Applying the product rule, we have:
(d/dx)(-4xln(x + 12)) = (-4)(ln(x + 12)) + (-4x)(1 / (x + 12))
Simplifying further:
= -4ln(x + 12) - 4x / (x + 12)
Therefore, the derivative of y = -4xln(x + 12) is -4ln(x + 12) - 4x / (x + 12).
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The question is -
Find the derivative of the given function.
y = -4xln(x + 12)
A. 3x / x + 12 + 3ln(x+12)
B. 3x / x + 12 - 3ln(x+12)
C. - (3x / x + 12) - 3ln(x+12)
D. - (3x / x + 12) + 4ln(x+12)
find the number of Primitives to 250 find the reminders when zo is divided by 11 find the reminders when ah! divided by 37
The number of primes up to 250 is 54.
To find the number of primes up to 250, we need to check each number up to 250 to determine whether it is prime or not. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.
We can use a simple algorithm to determine whether a number is prime or not. We start by checking if the number is divisible by 2. If it is divisible by 2, then it is not prime unless it is 2 itself. If the number is not divisible by 2, we check if it is divisible by any odd numbers starting from 3 up to the square root of the number.
Applying this algorithm to each number up to 250, we can count the number of primes. By doing so, we find that there are 54 prime numbers up to 250.
Therefore, the main answer is that there are 54 primes up to 250.
Note: The explanation provided assumes that by "Primitives," you meant prime numbers.
The second part of the question.
Remainder when "zo" is divided by 11:
To find the remainder when "zo" is divided by 11, we need to assign numerical values to the letters and then perform the division.
In this case, let's assign the values as follows:
z = 26
o = 15
Now, we calculate the value of "zo":
zo = 26 * 10 + 15 = 265
To find the remainder when 265 is divided by 11, we perform the division:
265 ÷ 11 = 24 remainder 1
Therefore, the remainder when "zo" is divided by 11 is 1.
Remainder when "ah!" is divided by 37:
To find the remainder when "ah!" is divided by 37, we assign numerical values to the letters and then perform the division.
In this case, let's assign the values as follows:
a = 1
h = 8
Now, we calculate the value of "ah!":
ah! = 1 * 10 + 8 = 18
To find the remainder when 18 is divided by 37, we perform the division:
18 ÷ 37 = 0 remainder 18
Therefore, the remainder when "ah!" is divided by 37 is 18.
Note: The explanation assumes that the letters in "zo" and "ah!" represent their corresponding positions in the English alphabet (e.g., a = 1, b = 2, etc.).
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If the shaded strip diagram represents 100% then which strip diagram represents 150%
Answer:
Your answer should be C.
Step-by-step explanation:
In order to make 150% you need a whole which represents the 100%. Which leaves B out of the question. A Is not close to half of a bar, which you can also eliminate. C and D are somewhat the same but it has to have the same amount of boxes shaded and unshaded, in this case D is 4 shaded and 2 unshaded which is wrong. So, your answer is C because there is 3 shaded and unshaded boxes in the model, hope this helps!
The diagram that represents 150% should be C.
What is the percentage?A percentage is a minimum number or ratio that is measured by a fraction of 100.
In order to make 150% we need a whole which represents the 100%. Which leaves B out of the question.
Option A Is not close to half of a bar, which we can also eliminate.
Option C and D are somewhat the same but it has to have the same amount of boxes shaded and unshaded,
In this case D is 4 shaded and 2 unshaded which is wrong.
Therefore, the answer is C because there is 3 shaded and unshaded boxes in the model.
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The last one the one at the bottom
Answer:
Step-by-step explanation
Her bank account decreased by 3 times.
Please let me know if this helps you!
300 mm
40 cm
50 cm
2 dm
Slice of a cake
Answer:
The correct answer in each case is:
Surface area = 60 [tex]cm^{2}[/tex]Surface area = 6000 [tex]mm^{2}[/tex]Surface area = 0.006 [tex]m^{2}[/tex] Volume = 1200 [tex]cm^{3}[/tex]Step-by-step explanation:
First, to calculate the surface area or the volume you must have all the measures in the same units, for the exercise we're gonna use centimeters and after we can replace the units in the answer if we need, then:
300 mm = 30 cm 40 cm 50 cm 2 dm = 20 cmNow, to obtain the surface area of the triangle we're gonna use the next formula:
Surface area = (base * height) / 2And we replace the values in centimeters:
Surface area = (30 cm * 40 cm) / 2Surface area = (120 [tex]cm^{2}[/tex]) / 2Surface area = 60 [tex]cm^{2}[/tex]To obtain this same value now in square milimeters, you must know:
1 [tex]cm^{2}[/tex] = 100 [tex]mm^{2}[/tex]Now, you must multiply:
60 [tex]cm^{2}[/tex] * 100 = 6000 [tex]mm^{2}[/tex]60 [tex]cm^{2}[/tex] = 6000 [tex]mm^{2}[/tex]To obtain this value now in [tex]m^{2}[/tex], you must know:
1 [tex]m^{2}[/tex] = 10000 [tex]cm^{2}[/tex]You must divide:
60 [tex]cm^{2}[/tex] / 10000 = 0.006 60 [tex]cm^{2}[/tex] = 0.006 [tex]m^{2}[/tex]By last, to obtain the volume of the piece of cake, you can use the next formula:
Volume of the piece of cake: surface area * depthAnd we replace the surface area in [tex]cm^{2}[/tex] because the answer must be in [tex]cm^{3}[/tex]:
Volume of the piece of cake: 60 [tex]cm^{2}[/tex] * 20 cmVolume of the piece of cake: 1200 [tex]cm^{3}[/tex]Rework problem 29 from section 2.3 of your text, involving the selection of officers in an advisory board. Assume that you have a total of 13 people on the board: 3 out-of-state seniors, 4 in-state seniors, 1 out-of-state non-senior, and 5 in-state non-seniors. University rules require that at least one in-state student and at least one senior hold one of the three offices. Note that if individuals change offices, then a different selection exists. In how many ways can the officers be chosen while still conforming to University rules?
There are 80 ways to choose the officers while conforming to University rules.
To determine the number of ways the officers can be chosen while conforming to University rules, we need to consider the different possibilities based on the required conditions.
First, let's consider the positions that must be filled by in-state students and seniors. Since there are 4 in-state seniors and 5 in-state non-seniors, we can select the in-state senior for one position in 4 ways and the in-state non-senior for the other position in 5 ways.
Next, let's consider the remaining position. This can be filled by any of the remaining individuals, which includes 3 out-of-state seniors and 1 out-of-state non-senior. Therefore, there are 4 options for filling the remaining position.
To determine the total number of ways the officers can be chosen, we multiply the number of options for each position: 4 (in-state senior) × 5 (in-state non-senior) × 4 (remaining position) = 80.
Hence, there are 80 ways to choose the officers while conforming to University rules.
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Someone help what is the answer
4h + 14 > 38 =
19 points
Answer:
its in there
Step-by-step explanation:
Answer: Solve the Inequality for h
Solve for h
Graph
Convert to Interval Notation
Evaluate
Write in y=mx+b Form
Find the Exact Value
Solve the Absolute Value Inequality for h
Convert to Set Notation
Plot
Write in Slope-Intercept Form
Solve the Rational Equation for h
Simplify
Add
Solve by Factoring
Describe the Transformation
Evaluate the Summation
Find the Absolute Max and Min over the Interval
Find the Product
Convert from Degrees to Radians
Convert to a Decimal
Subtract
Find the Slope
Find the Domain and Range
Find the Derivative - d/dh
Solve Using the Square Root Property
Find the Direction Angle of the Vector
Multiply
Find the Function Rule
Convert to a Simplified Fraction
Find the Area Between the Curves
Solve the Function Operation
Solve the System of Inequalities
Solve for h in Degrees
Find Where Increasing/Decreasing
Find the Maximum/Minimum Value
Write with Rational (Fractional) Exponents
Evaluate the Limit
Find the Slope and y-intercept
Solve by Isolating the Absolute Value for h
Find All Complex Solutions
Factor over the Complex Numbers
Find the Intersection
Find the Center and Radius
Convert to Radical Form
Find the Integral
Evaluate Using Summation Formulas
Evaluate Using Scientific Notation
Simplify/Condense
Determine if Linear
Graph Using a Table of Values
Reduce
Rationalize the Denominator
Find the Union
Find the Critical Points
Solve for h in Radians
Find the x and y Intercepts
Find the Inverse
Find the Perpendicular Line
Convert from Interval to Inequality
Solve by Completing the Square
Simplify the Matrix
Maximize the Equation given the Constraints
Convert to Regular Notation
Determine if the Relation is a Function
Write in Standard Form
Convert to Trigonometric Form
Split Using Partial Fraction Decomposition
Find the Zeros by Completing the Square
Find the Prime Factorization
Find the Local Maxima and Minima
Find the Quotient
Multiply the Matrices
Factor
BRAINIEST TO WHOEVER RIGHT PLZ HELP
Answer:
a) 2.7 sec
b) 2.6 sec
c) 30.3 ft, 1.3 sec
Step-by-step explanation:
graphed the equation and determined answers from the curve
this this this!!!!! can u answer
Answer:
A= 30
B= 24.3
Step-by-step explanation:
The amount Troy charges to mow a lawn is proportional to the time it takes him to mow the lawn. Troy charges $30 to mow a lawn that took him 1.5 hours to mow.
Which equation models the amount in dollars, , Troy charges when it takes him h hours to mow a lawn?
A word processing system selling firm records track of the number of customers who call on any one day and the number of orders placed on any one day. Let X, denote the number of calls, Xz denote the number of orders placed, and p( x1,x2) the joint probability function for (X1,X2); records indicate that P(0,0)= 0.06 P (2,0)= 0.20 P(1,0)= 0.14 P(2,1)= 0.30 P(1,1)= 0.10 P(2,2) = 0.20 For any given day, the probability of say, two calls and one order is 0.30. Find the correlation coefficient and interpret your results.
The correlation coefficient is:r = -0.332 / (0.683 × 0.566) = -0.994
Correlation Coefficient The correlation coefficient, denoted by r, is a statistical measure of the strength of the relationship between two quantitative variables.
It is always between -1 and 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation at all.
In this problem, we have the joint probability function for X1 and X2 as follows:
P(0,0) = 0.06P(2,0) = 0.20P(1,0) = 0.14P(2,1) = 0.30P(1,1) = 0.10P(2,2) = 0.20
The probability of two calls and one order is given as 0.30. Thus:P(2,1) = 0.30
Now, let us find the marginal probabilities of X1 and X2.P(X1 = 0) = P(0, 0) + P(0, 1) + P(0, 2) = 0.06 + 0 + 0 = 0.06P(X1 = 1) = P(1, 0) + P(1, 1) + P(1, 2) = 0.14 + 0.10 + 0 = 0.24P(X1 = 2) = P(2, 0) + P(2, 1) + P(2, 2) = 0.20 + 0.30 + 0.20 = 0.70Similarly,P(X2 = 0) = P(0, 0) + P(1, 0) + P(2, 0) = 0.06 + 0.14 + 0.20 = 0.40P(X2 = 1) = P(0, 1) + P(1, 1) + P(2, 1) = 0 + 0.10 + 0.30 = 0.40P(X2 = 2) = P(0, 2) + P(1, 2) + P(2, 2) = 0 + 0 + 0.20 = 0.20
The expected values of X1 and X2 are:
E(X1) = 0 × 0.06 + 1 × 0.24 + 2 × 0.70 = 1.64
E(X2) = 0 × 0.40 + 1 × 0.40 + 2 × 0.20 = 0.80
Let us now find the expected value of X1X2:
E(X1X2) = 0 × 0.06 + 1 × 0.00 + 2 × 0.30 + 0 × 0.14 + 1 × 0.10 + 2 × 0.20 = 1.14
Thus, the covariance of X1 and X2 is:Cov(X1, X2) = E(X1X2) - E(X1)E(X2) = 1.14 - 1.64(0.80) = -0.332
Finally, the correlation coefficient is given as
r = Cov(X1, X2) / (σ(X1)σ(X2))
Where σ(X1) is the standard deviation of X1 and σ(X2) is the standard deviation of X2.
Let us find the standard deviations of X1 and X2:
Variance of X1:Var(X1) = E(X1^2) - [E(X1)]^2 = 0^2(0.06) + 1^2(0.24) + 2^2(0.70) - (1.64)^2 = 0.4676
Standard deviation of X1:σ(X1) = √Var(X1) = √0.4676 = 0.683
Variance of X2:Var(X2) = E(X2^2) - [E(X2)]^2 = 0^2(0.40) + 1^2(0.40) + 2^2(0.20) - (0.80)^2 = 0.3200
Standard deviation of X2:σ(X2) = √Var(X2) = √0.3200 = 0.566
Thus, the correlation coefficient is:r = -0.332 / (0.683 × 0.566) = -0.994
Therefore, the correlation coefficient is very close to -1.
This means that there is a very strong negative correlation between the number of calls and the number of orders placed. This can be interpreted as follows: As the number of calls increases, the number of orders placed decreases, and vice versa.
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can anyone help with this pls
Answer:
9
Step-by-step explanation:
Answer:
9 square meter
Step-by-step explanation:
1/2xbasexheight
=1/2x6x3
=9
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Three integers have a mean of 10, a median of 12 and a range of 8.
Find the three integers.
Answer:
The answers are
x=5
y=12
z=13
Step-by-step explanation:
let the numbers be x,y,z
[tex] \frac{x + y + z}{3} = 10[/tex]
[tex]y = 12[/tex]
[tex]z - x = 8[/tex]
z=8+x
x+12+x+8/3=10
2x+20/3=10
2x+20=30
2x=30-20
2x=10
divide both sides by 2
2x/2=10/2
x=5
z=8+5
z=13
(200 x 3) + (50 x 3)
200x3=600
50x3=150
600+150=750
750
if both expressions have the same value after substituting two different values and simplifying, then they are . When p = 2, the first expression is and the second expression is 16. When p = 8, the first expression is 40 and the second expression is . The expressions are .
Answer:
If both expressions have the same value after substituting and simplifying two different values for the variable, then they are
✔ equivalent
.
!Step-by-step explanation:
hop3 it helped
Find the value of x. Round the length to the nearest tenth.
ANSWER: A)7.2 ft
Answer:
7.2
Step-by-step explanation:
Answer:
I dont think you realize you posted the answer...
Step-by-step explanation:
7.2
Solve this ODE with the given initial conditions. y" + 4y' + 4y = 68(t-π) with y(0) = 0 & y'(0) = 0
The specific solution to the given ODE with the initial conditions is:
y(t) = (8.5π - 8.5t) [tex]e^{(-2t)[/tex] + 8.5(t - π)
To solve the given ordinary differential equation (ODE) with the initial conditions, we can use the method of undetermined coefficients.
The characteristic equation for the homogeneous part of the ODE is:
r² + 4r + 4 = 0
Solving this quadratic equation, we find a repeated root:
(r + 2)² = 0
r + 2 = 0
r = -2
Since we have a repeated root, the general solution to the homogeneous part is:
[tex]y_{h(t)[/tex]= (C₁ + C₂t) [tex]e^{(-2t)[/tex]
Next, we need to find a particular solution to the non-homogeneous part of the ODE. We assume a particular solution in the form:
[tex]y_{p(t)[/tex] = A(t - π)
Taking the derivatives:
[tex]y'_{p(t)[/tex] = A
[tex]y''_{p(t)[/tex] = 0
Substituting these derivatives into the ODE:
0 + 4A + 4A(t - π) = 68(t - π)
Simplifying:
8A(t - π) = 68(t - π)
8A = 68
A = 8.5
Therefore, the particular solution is:
[tex]y_{p(t)[/tex] = 8.5(t - π)
The general solution to the ODE is the sum of the homogeneous and particular solutions:
[tex]y(t) = y_{h(t)} + y_{p(t)[/tex]
= (C₁ + C₂t) [tex]e^{(-2t)[/tex] + 8.5(t - π)
To find the values of C₁ and C₂, we apply the initial conditions:
y(0) = 0
0 = (C₁ + C₂(0)) [tex]e^{(-2(0))[/tex] + 8.5(0 - π)
0 = C₁ - 8.5π
C₁ = 8.5π
y'(0) = 0
0 = C₂ [tex]e^{(-2(0))[/tex] + 8.5
0 = C₂ + 8.5
C₂ = -8.5
Therefore, the specific solution to the given ODE with the initial conditions is:
y(t) = (8.5π - 8.5t) + 8.5(t - π)
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An English teacher reviewed 2/3 of an essay in 1/4 of an hour. At this rate, how many essays can she review in 1 hour?
Answer: 2 2/3
Step-by-step explanation:
Answer:
2 2/3
Step-by-step explanation:
Find an angle in each quadrant with a common reference angle with 258°, from 0°≤θ<360°
The reference angles are in the first, second, third, and fourth is 78°, 102°, 258°, and 282° respectively.
What is the reference angle?the angle between the terminal side of the angle and the x-axis.
For the first quadrant:
The reference angle = 258 - 180 = 78°
For the second quadrant:
= 180 - 78 = 102°
For the third quadrant:
= 78 + 180
= 258°
For the fourth quadrant:
= 360- 78
= 282°
Thus, the reference angles are in the first, second, third, and fourth is 78°, 102°, 258°, and 282° respectively.
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Which is the correct comparison?
4.25 hours = 260 minutes
260 minutes > 4.25 hours
4.25 hours > 260 minutes
260 minutes < 4.25 hours
Answer:
260 minutes > 4.25 hours
Step-by-step explanation:
1h=60 min.
0.25 h=¼h=¼*60min=15min.
4.25h=4h+0.25h=60*4+15min=240+15=255min
260>255
Answer:
60+60+60+60=240
240/4=4,25
=4,25
How many 1/4 cup serving are in a 6 cup container
Answer:
24
Step-by-step explanation:
6 ÷ 1/4 = 6 × 4 = 24
Answer:
24
Step-by-step explanation:
0.00871 written in scientific notation
Answer:
[tex]8.71*10^{-3}[/tex]
Step-by-step explanation:
Moving the decimal point 3 spots to the right, we get 8.71, which shows that 0.00871 is equal to 8.71*10^-3 in scientific notation
Answer:
Step-by-step explanation:
0.00871 written in scientific notation is 8.71 x 10^(-3).
hope it helps!
Mhanifa can you please help? Look at the picture attached. I will mark brainliest!
Answer:
see explanation
Step-by-step explanation:
Since the marked angles are congruent, let them be x
(6)
The sum of the interior angles of a quadrilateral = 360°
Sum the angles and equate to 360
x + x + 120 + 90 = 360
2x + 210 = 360 ( subtract 210 from both sides )
2x = 150 ( divide both sides by 2 )
x = 75
Then ∠ X = ∠ Y = 75°
---------------------------------------------------------------------
(7)
The sum of the interior angles of a hexagon = 720°
Sum the angles and equate to 720
x + x + 108 + 103 + 149 + 90 = 720
2x + 450 = 720 ( subtract 450 from both sides )
2x = 270 ( divide both sides by 2 )
x = 135
Then ∠ X = ∠ Y = 135°
Answer:
6) x = 75°, y = 75°7) x = 135°, y = 135°Step-by-step explanation:
Sum of the interior angles of a regular polygon:
S(n) = 180°(n - 2), where n- number of sides Exercise 6Quadrilateral has sum of angles:
S(4) = 180°(4 - 2) = 360°Sum the given angles and consider x = y as marked congruent:
2x + 120° + 90° = 360° 2x + 210° = 360° 2x = 360° - 210° 2x = 150°x = 75° and y = x = 75° Exercise 7Hexagon has sum of angles:
S(6) = 180°(6 - 2) = 720°Sum the given angles and consider x = y as marked congruent:
2x + 108° + 103° + 149° + 90° = 720° 2x + 450° = 720° 2x = 720° - 450° 2x = 270°x = 135° and y = x = 135°Choose all that apply!!!
Answer:
56 degree and 68 degree
plz mark me as brainliest
Answer:
62°
Step-by-step explanation:
Since the triangle is isosceles then the 2 base angles are congruent.
Given there is one angle measuring 56° then it must be the vertex angle.
Thus the 2 congruent base angles are
[tex]\frac{180-56}{2}[/tex] = [tex]\frac{124}{2}[/tex] = 62°
Cari is searching online for airline tickets. Two weeks ago, the cost to fly from Boston to Hartiord was
$225. Now the cost is $335. What is the percent increase? What would be the percent increase the
ainline charges an additional $50 baggage fee with the new ticket price?
The percent increase of the airline ticket is %
Answer:
The percent of the air ticket went up 148% plus the baggage fee would be 168%. The new total is 420$
Step-by-step explanation:
Marilyn has 24 hair ribbons. 9 of her ribbons are red. What percent of her hair
ribbons are red?
A. 30%
B. 37.5%
C. 25%
D. 40%
Answer:
B. 37.5
Step-by-step explanation:
100/24 = 4.1666666667
4.1666666667*9 = 37.5
Given the disk of the radius r = 1, i.e., = {(x₁, x₂) € R² | x² + x² <1} find the smallest and largest values that the function f(x₁, x₂) = x₁ + x₂ achieves on the set D. a) Formulate the problem as an optimization problem and write down the optimality conditions. b) Find the point(s) in which the function f achieves maximum and minimum on the set D What is the largest and smallest value of f ? Comments: Make sure that you properly justify that you find a minimizer and maximizer. c) Denote the smallest value fin. What is the relative change of fin expressed in percents if the radius of the disk decreases and it is given as D {(1,₂) € R²|x²+x≤0.99}
The smallest value of the function f(x₁, x₂) = x₁ + x₂ on the disk D with a radius of 1 is -√2, and the largest value is √2. The relative change in the smallest value, expressed in percent, can be calculated if the radius of the disk decreases to 0.99.
a) The problem can be formulated as an optimization problem with constraints. We want to find the smallest and largest values that the function f(x₁, x₂) = x₁ + x₂ achieves on the set D, which is defined as the disk with radius r = 1, i.e., D = {(x₁, x₂) ∈ ℝ² | x₁² + x₂² < 1}.
To find the smallest value, we can minimize the function f subject to the constraint that (x₁, x₂) is within the disk D. Mathematically, this can be written as:
Minimize: f(x₁, x₂) = x₁ + x₂
Subject to: x₁² + x₂² < 1
To find the largest value, we can maximize the function f subject to the same constraint. Mathematically, this can be written as:
Maximize: f(x₁, x₂) = x₁ + x₂
Subject to: x₁² + x₂² < 1
b) To find the points at which the function f achieves the maximum and minimum on the set D, we can analyze the problem. The function f(x₁, x₂) = x₁ + x₂ represents a plane with a slope of 1.
Considering the constraint x₁² + x₂² < 1, we observe that it represents a circle with radius 1 centered at the origin.
Since the function f represents a plane with a slope of 1, the maximum and minimum values occur at the points on the boundary of the disk D where the plane is tangent to the disk. In other words, the maximum and minimum values occur at the points where the plane f(x₁, x₂) = x₁ + x₂ is perpendicular to the boundary of the disk.
Considering the disk D: x₁² + x₂² < 1, we can see that the boundary of the disk is x₁² + x₂² = 1 (the equation of a circle).
At the boundary, the gradient of the function f(x₁, x₂) = x₁ + x₂ is parallel to the normal vector of the boundary circle. The gradient of f is (∂f/∂x₁, ∂f/∂x₂) = (1, 1), which represents the direction of steepest ascent of the function.
Thus, at the points where the plane f(x₁, x₂) = x₁ + x₂ is tangent to the boundary circle, the gradient of f is parallel to the normal vector of the circle. Therefore, the gradient of f at these points is proportional to the vector pointing from the origin to the tangent point.
To find the tangent points, we can use the fact that the tangent line to a circle is perpendicular to the radius at the point of tangency. The radius of the circle D is the vector from the origin to any point (x₁, x₂) on the boundary, which is (x₁, x₂).
So, the tangent points occur when the gradient vector (1, 1) is proportional to the radius vector (x₁, x₂), which means:
1/1 = x₁/1 = x₂/1
Simplifying, we get:
x₁ = x₂
Substituting this back into the equation of the boundary circle, we have:
x₁² + x₂² = 1
x₁² + x₁² = 1
2x₁² = 1
x₁² = 1/2
Taking the positive square root, we get:
x₁ = √(1/2)
Since x₁ = x₂, the corresponding values are:
x₂ = √(1/2)
Thus, the points where the function f achieves the maximum and minimum on the set D are (x₁, x₂) = (√(1/2), √(1/2)) and (x₁, x₂) = (-√(1/2), -√(1/2)).
Plugging these values into the function f(x₁, x₂) = x₁ + x₂, we get:
f(√(1/2), √(1/2)) = √(1/2) + √(1/2) = 2√(1/2) = √2
f(-√(1/2), -√(1/2)) = -√(1/2) - √(1/2) = -2√(1/2) = -√2
Therefore, the largest value of f is √2, and the smallest value of f is -√2.
c) Denoting the smallest value as fin = -√2, we can find the relative change in fin expressed in percent if the radius of the disk decreases to D = {(x₁, x₂) ∈ ℝ² | x₁² + x₂² ≤ 0.99}.
To calculate the relative change, we can use the formula:
Relative Change = (New Value - Old Value) / Old Value * 100
The new value of fin, denoted as fin', can be found by minimizing the function f subject to the constraint x₁² + x₂² ≤ 0.99.
Solving the minimization problem, we find the new smallest value fin' on the set D with a radius of 0.99.
Comparing fin' to fin, we can calculate the relative change:
Relative Change = (fin' - fin) / fin * 100
By solving the new minimization problem, you can find the new smallest value fin' and calculate the relative change using the formula provided.
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The scatter plot shows the years of experience and the amount charged per hour by each of 24 dog sitters in Ohio. Also shown is the line of best fit for the data. Fill in the blanks below. y 22 20 18 X 16 X ***** 14 - Xx X X X Amount charged 'in dollars 12 ** 10 X per hour *** Х 8 X 6 4 2 X 0 2 3 4 5 6 7 8 9 10 11 12 13 Years of experience 0 2 3 4 5 6 7 8 9 10 i 12 13 Years of experience Х $ ?. (a) For these 24 dog sitters, as experience increases, the amount charged tends to (Choose one) (b) For these 24 dog sitters, there is (Choose one) V correlation between experience and amount charged. (C) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately (Choose one)
a) For these 24 dog sitters, as experience increases, the amount charged tends to increase.
b) For these 24 dog sitters, there is a positive correlation between experience and amount charged.
c) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately $16.5 per hour.
(a) For these 24 dog sitters, as experience increases, the amount charged tends to increase. This means that the amount charged per hour increases with an increase in years of experience for dog sitters.
(b) For these 24 dog sitters, there is a positive correlation between experience and amount charged. The points on the scatter plot show a generally upward trend, and the line of best fit is also sloping upward.
(c) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately $16.5 per hour. This can be determined by locating the point on the X-axis corresponding to 5 years of experience, and then drawing a vertical line to the line of best fit. From there, we can draw a horizontal line to the Y-axis to find the predicted amount charged per hour, which is about $16.5.
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