Find the parabola of the form y=ax2+b which best fits the points (−1,0), (5,5), (6,10) by minimizing the sum of squares, s, given by
Answers
That is,
{x} = [-1, 5, 6]
{y} = [0, 5, 10]
Let the curve of the best least squares fit be
y = ax² + b
The sum of the squares of errors between the data and the curve is
[tex]E = \sum_{i=1}^{3} (ax_{i}^{2}+b-y_{i})^{2}[/tex]
To minimize the error requires that
[tex] \frac{\partial E}{\partial a} =0\\ 2\sum x_{i}^{2} (ax_{i}^{} + b - y_{i} ) = 0[/tex]
Also,
[tex] \frac{\partial E}{\partial b} =0 \\ 2 \sum (ax_{i}^{2} + b - y_{i} ) = 0[/tex]
Therefore the normal equations are
(x₁⁴+x₂⁴+x₃⁴)a + (x₁²+x₂²+x₃²)b = x₁²y₁+x₂²y₂+x₃²y₃ (1)
(x₁²+x₂²+x₃²)a + 3b = y₁+y₂+y₃ (2)
From the given data, obtain
1922a + 62b = 485
62a + 3b = 15
The solution of these equations yields
a = 0.2732
b = -0.6452
The required curve is
y = 0.2732x² - 0.6452
A graph of the curve and the given data is shown below.
Answer: y = 0.2732x² - 0.6452
The optimal least squares fit for the given data is y = 0.2732x² - 0.6452. This curve minimizes the sum of squared errors, providing an accurate representation of the data.
The provided data points are (-1, 0), (5, 5), and (6, 10), denoted as {x} = [-1, 5, 6] and {y} = [0, 5, 10]. The goal is to find the least squares fit for the curve y = ax² + b. To minimize the sum of squared errors between the data and the curve, the normal equations are derived.
The normal equations are given by:
(x₁⁴ + x₂⁴ + x₃⁴)a + (x₁² + x₂² + x₃²)b = x₁²y₁ + x₂²y₂ + x₃²y₃ (Equation 1)
(x₁² + x₂² + x₃²)a + 3b = y₁ + y₂ + y₃ (Equation 2)
By substituting the given data, the equations become:
1922a + 62b = 485
62a + 3b = 15
Solving these equations yields the values: a = 0.2732 and b = -0.6452. Therefore, the curve of the best least squares fit is y = 0.2732x² - 0.6452.
In conclusion, the curve that minimizes the sum of squared errors for the provided data points is y = 0.2732x² - 0.6452. The graph of this curve, along with the given data, visually represents the accuracy of the least squares fit.
To learn more about least squares
https://brainly.com/question/30634235
#SPJ6
Related Questions
Write the word sentence as an equation. Then solve. A number x multiplied by 2/5 is 3/20 .
Answers
To solve, multiply both sides by 5/2. The answer is x=3/8.
The given word sentence translates into the algebraic equation 2/5 * x = 3/20. Solving this equation returns x = 3/8 as the solution.
Explanation:The word sentence 'A number x multiplied by 2/5 is 3/20' is translated into an algebraic equation as: 2/5 * x = 3/20. To solve for x, you could do cross multiplication or simply divide both sides of the equation by 2/5, which gives: x = (3/20) / (2/5). Simplifying this equation leads to: x = 3/20 * 5/2 = 15/40 = 3/8. Therefore, the solution to the equation is x = 3/8.
Learn more about Solving Algebraic Equations here:https://brainly.com/question/33839116
#SPJ2
Given: Angle 2 and Angle 3 are a linear pair. Angle 3 and Angle 4 are a linear pair. Prove Angle 2 is congruent to Angle 4.
1. Angle 2 and Angle 3 are a linear pair. Angle 3 and Angle 4 are a linear pair. Reason: Given
2. Angle 2 and Angle 3 are supplementary. Angle 3 and Angle 4 are Supplementary.
Reason: ?
3. Angle 2 is congruent to Angle 4.
Reason: ?
Answers
Angle 2 and Angle 4 are congruent because they form a linear pair and their sum is 180 degrees.
Explanation:Given that Angle 2 and Angle 3 are a linear pair and Angle 3 and Angle 4 are a linear pair, we can prove that Angle 2 is congruent to Angle 4.
Since Angle 2 and Angle 3 are a linear pair, they form a straight line and therefore their angles add up to 180 degrees. So, Angle 2 + Angle 3 = 180 degrees. (Supplementary)
Similarly, Angle 3 and Angle 4 are a linear pair, so Angle 3 + Angle 4 = 180 degrees. (Supplementary)
By transitive property, if Angle 2 + Angle 3 = 180 degrees and Angle 3 + Angle 4 = 180 degrees, then Angle 2 + Angle 3 + Angle 4 = 180 degrees.
Since the sum of Angle 2, Angle 3, and Angle 4 is 180 degrees, we can subtract Angle 3 from both sides of the equation to get Angle 2 + Angle 4 = 180 degrees.
Therefore, Angle 2 + Angle 4 = 180 degrees, and since the sum of the angles in a straight line is 180 degrees, we can conclude that Angle 2 is congruent to Angle 4.Learn more about Congruent Angles here:
https://brainly.com/question/37051513
#SPJ2
The reciprocal of a number plus the reciprocal of three times the number equals 1/3. find the number.
Answers
3+1=x
x=4
Find the value of xx that makes m ∥ n.
Answers
If the lines are parallel, they are supplementary.
The sum of the measures of supplementary angles is 180.
150 + 3x - 15 = 180
135 + 3x = 180
3x = 45
x = 15
5. A gym membership costs $30 to join and $12 each month. Write and use an algebraic expression to find the cost of the gym membership for 6 months.
Answers
or
y=12x+30
x is the variable for the number of months.
Y is the total cost of the membership.
What is the sum of the first five terms of the geometric series 3 − 6 + 12 − . . . ?
Hint: r ≠ 1, where a1 is the first term and r is the common ratio.
Hint: image attached
OPTIONS:
A) S5 = −136
B) S5 = −33
C) S5 = 12
D) S5 = 33
Answers
Choose the correct slope of the line that passes through the points (−4, 8) and (−3, −6). (1 point)
7
−14
0
14
Answers
answer
−14
Answer:
-14
Step-by-step explanation:
Find f(-3) if f(x) = x^2.
-9
-6
6
9
It's not -6!!
Answers
= -3 * -3
= 9
Answer:
f(-3) = (-3)²
= -3 * -3
= 9
Step-by-step explanation:
Math problem...
375
x 6
------
Answers
The mathematics question asks to multiply the numbers 375 and 6. By following the multiplication process, we find the answer to be 2250, which is a basic arithmetic operation in middle school mathematics.
The student has presented a multiplication problem, which is a basic arithmetic operation in mathematics. The problem can be solved by multiplying the two numbers given, 375 and 6. The calculation goes as follows:
Multiply the digit in the one's place of the bottom number (6) by each digit of the top number (375), starting from the one's place of the top number.
Write down the result of each multiplication and carry over any tens if necessary.
After you complete the multiplication process, add any carried-over numbers to the resultant product.
The completed multiplication should give you the final answer to the problem.
Example Calculation:
375
x 6
------
2250
Here, the answer to the multiplication problem is 2250.
In right triangle ABC, B = 51° and AB = 14.
BC = _____
8.8
10.8
11.3
Answers
Given: the triangle is a right triangle, find a.
(The side opposite angle A.)
Answer:
a = 8.8104 ≈ 8.8
Step-by-step explanation:
∠A = ?, ∠B = 51°, ∠C = 90°
∠A = 180° - ( ∠B + ∠C )
∠A = 180° - ( 51° + 90° ) = 180° - 39°
∠A = 39°
[tex]\frac{a}{sin A}=\frac{b}{sin B} =\frac{c}{sin C}\\\\\frac{a}{sin39degrees}=\frac{14}{sin90degrees}\\\\\frac{14}{sin90degrees}*sin39degrees\\\\= 14 * 0.629\\= 8.8104[/tex]
So side a or BC = 8.8
The earnings of two employees are given below: Employee A: 6% commission on all sales Employee B: 4% commission on the first $80,000 and 8% on anything over $80,000 How much more does a straight commission employee make than the graduated commission employee for sales of $100,000?
Answers
B: (80000 x 0.04) + (20000 x 0.08) =4800
A makes $1200 more than b
Hope this helps
Employee A makes $1,200 more than Employee B for sales of $100,000.
The earnings of Employee A are:
= Commission rate x Sales
= 6% x 100,000
= $6,000
The earnings of employee B are:
= Commission on first $80,000 + Commission on anything above $80,000
= (4% x 80,000) + (8% x (100,000 - 80,000)
= 3,200 + (8% x 20,000)
= $4,800
The difference is:
= Employee A commission - Employee B commission
= 6,000 - 4,800
= $1,200
In conclusion, Employee A makes $1,200 more than Employee B for sales of $100,000.
Find out more at https://brainly.com/question/24691073.
Roy realized that some other conditions or constraints apply. Write an inequality for each constraint described below.
Answers
The longer base of isosceles trapezoid is 12 cm. the side is 5 cm. the acute angle is 37º. find the shorter base.
Answers
The shorter base of the isosceles trapezoid is approximately 6 cm.
Explanation:To find the shorter base of the isosceles trapezoid, we need to use the properties of the trapezoid. Firstly, let's draw the isosceles trapezoid and label the given information.
The longer base is 12 cm, the side is 5 cm, and the acute angle is 37º. We can see that the trapezoid has two congruent triangles, and we can find the height of the triangle using trigonometry.
Step 1: Find the height of the triangle using the sine of the acute angle.
sine(37º) = opposite / hypotenuse
sin(37º) = h / 5
h = 5 * sin(37º) ≈ 3 cm
Step 2: Use the height to find the shorter base of the trapezoid.
The shorter base can be found by subtracting twice the height from the longer base.
shorter base = longer base - 2 * height
shorter base = 12 cm - 2 * 3 cm
shorter base = 12 cm - 6 cm
shorter base ≈ 6 cm
Therefore, the shorter base of the isosceles trapezoid is approximately 6 cm.
if the discriminant of an equation is positive, which of the following is true of the equation?
A. it has one complex solution
B. it has two real solutions
C. it has one real solution
D. it has two complex solutions
Answers
It can have one solution only if D=0 and complex solutions only if D<0.
D means discriminant
Answer: B it has two real solutions
hope this helps :)
Which expression is equal to (x+2)(−3x2+3x+1)?
−3x2−3x+2
−3x2+5x+3
−3x3−3x2+7x+2
−3x3+9x2−5x+2
Answers
-3x^3 + 3x^2 + x
Multiply everything in the second parenthesis by 2.
-6x^2 + 6x + 2
Combine these two equations together.
-3x^3 + 3x^2 + x - 6x^2 + 6x + 2
Combine like terms.
-3x^3 - 3x^2 + 7x + 2
Hope this helps!
Option C is correct. The required product of the functions is -3x³ - 3x² + 7x + 2
Given the functions (x+2) and (-3x² + 3x + 1)
Taking the product of the functions:
= (x+2)(-3x² + 3x + 1)
Expand the bracket
= x(-3x²) + x(3x) + x(1) + 2(-3x²) + 2(3x) + 2(1)
= -3x³ + 3x² + x - 6x² + 6x + 2
Collect the like terms;
= -3x³ + 3x² - 6x²+ x + 6x + 2
= -3x³ - 3x² + 7x + 2
Hence the required product of the functions is -3x³ - 3x² + 7x + 2
Learn more here: https://brainly.com/question/4854699
Which expression is equal to (−12−2i)+(2+2i)?
10−4i
−10−4i
−10
10
Answers
= -12 - 2i + 2 + 2i
= -12 + 2 - 2i + 2i
= -10
Answer: -10
Answer:
-10!
Step-by-step explanation:
How to solve on a road map of pennsylvania, the disatance from philadelphia to washington,
d.c. is 6.8 centimeters. the map scale is 2 cm;40 mi. what is the actual distance between the cities?
Answers
the pattern below follows the rule "starting with a value of 3, every consecutive row has a value that is 2 less than twice the value of the previous row" what is the fifth value? a. 18 b. 12 c. 34 d. 66
Answers
Answer: a. 18
Step-by-step explanation:
Given phrase : "starting with a value of 3, every consecutive row has a value that is 2 less than twice the value of the previous row"
The recursive formula for the given phrase :-
[tex]a_n=2a_{n-1}-2[/tex]
The first term : [tex]a_1=3[/tex]
The second term : [tex]a_2=2(3)-2=4[/tex]
The third term : [tex]a_3=2(4)-2=6[/tex]
The fourth term : [tex]a_4=2(6)-2=10[/tex]
The fifth term : [tex]a_5=2(10)-2=18[/tex]
Hence, the fifth value of the pattern = 18
Answer:18
Step-by-step explanation:
Suppose we build a sequence of numbers using the method of adding the previous two numbers to build the next one. this time, however, suppose our first two numbers are 2 and 1. generate the first 15 terms. this sequence is called the lucas sequence and is written as l1, l2, l3 , …. compute the quotients of consecutive terms of the lucas sequence as we did with the fibonacci numbers. what number do these quotients approach? what role do the initial values play in determining what number the quotients approach? try two other first terms and generate a sequence. what do the quotients approach?
Answers
quotients are 0.5 ,2,1.333,1.571,1.636,1.611, 1.621,1.617, 1.6184,1.6179,1.6181,1.6180,1.61801, 1.61804
The ratios are approaching the Golden Number.
If we try 2 other first 2 terms say 3 and 2:
we get 3,2,5,7,12,19,31,50,81,131,212,343, 555, 898
quotients are 0.67,2.5,1.4,1.714,1.5833,1.631 , 1.612, 1.62,1.617, 1.6183
,1.6181,1.61801 Again we are approaching the golden number
The sequence is given below:
What is sequence?A sequence is an enumerated collection of objects in which repetitions are allowed and order matters.
Given:
The table showing the sequence of the first 15 terms is:
Sequence Term Quotient
1 2 2
2 1 2 (2 / 1)
3 3 (2 + 1) 0.333 (1 / 3)
4 4 (1 + 3) 0.750 (3 / 4)
5 7 (3 + 4) 0.571 (4 / 7)
6 11 (4 + 7) 0.633 (7 / 11)
7 18 (7 + 11) 0.611 (11 / 18)
8 29 (11 + 18) 0.621 (18 / 29)
9 47 (18 + 29) 0.617 (29 / 47)
10 76 (29 + 47) 0.618 (47 / 76)
11 123 (47 + 76) 0.618 (76 / 123)
12 199 (76 + 123) 0.618 (123 / 199)
13 322 (123 + 199) 0.618 (199 / 322)
14 521 (199 + 322) 0.618 (322 / 521)
15 843 (322 + 521) 0.618 (521 / 843)
The quotient converges around the value of 0.618, as shown in the above table.
The preceding sequence is a Lucas adaptation of the so-called Fibonacci sequence, where the initial two numbers serve as seeds and result in the quotient known as the "Golden Ratio," which is created by adding the previous two consecutive terms in each term.
Learn more about sequence here:
https://brainly.com/question/28583639
#SPJ2
Ken and roberta each sold fifty tickets for the school play. the cost of a student ticket was three fourths the cost of an adult ticket. roberta sold ten more student tickets that adult tickets and collected a total of $212.50.
a. how many tickets did roberta sell?
b. what was the cost of a student ticket?
c. if ken collected $200 in all, how many student tickets did he sell?
Answers
Roberta sold 30 tickets in total, with 10 adult tickets and 20 student tickets. The cost of a student ticket was $3. Ken collected $200 in total and sold 20 student tickets.
Explanation:To solve this problem, we can set up two equations based on the given information. Let's use variables to represent the unknown quantities.
Let's say the cost of an adult ticket is A dollars. Then the cost of a student ticket would be 3/4 of A dollars, or (3/4)A dollars.
Based on the information given, we can set up the following equations:
Robera sold 10 more student tickets than adult tickets, so let's say she sold X adult tickets and X + 10 student tickets.
The total collected by Roberta was $212.50, so we have the equation:
(A * X) + ((3/4)A * (X + 10)) = $212.50
Ken sold 50 tickets in total, so he sold 50 - (X + 10) student tickets. The total collected by Ken was $200, so we have the equation:
((3/4)A * (50 - (X + 10))) + (A * X) = $200
Now we can solve these equations to find the values of X and A.
Plugging in A = 4, we get X = 10. Therefore, Roberta sold 10 adult tickets and 20 student tickets. The cost of a student ticket was (3/4) * 4 = $3.
In conclusion, Roberta sold 30 tickets in total, with 10 adult tickets and 20 student tickets. The cost of a student ticket was $3. Ken collected $200 in total and sold 20 student tickets.
One positive number exceeds another by 4, and their product is 60. Find the numbers.
Answers
1. Solve the system by substitution- 2x+y=-11, 3x-4y=11.
Answers are:
C
B
B
C
A
Answers
Thank you so much T-T
Management theories of leadership focus on which of the following?
A. Situations <--- My Answer
B. The leader's style
C. Rewards and punishments
D. Relationships
2. In which conflict style do people often have a firm idea of what they want to see happen and they will fight to make that happen?
A. Collaborative
B. Competitive
C. Compromising
D. Accomodating
3. With a de-emphasizing approach to conflict, the focus is put on which of the following?
A. Areas of agreement
B. Differences in personality
C. Points of contention
D. All of the above
4. Behavioral theories of leadership argue that leaders are born, not made.
True
False
Answers
2)Competitive
3)Areas of agreement
4)True
Hope this helps!
HELP PLEASE!
What is the value of f(−1) when f(x)=2x+2 ?
Enter your answer in the box.
f(−1)= ?
Answers
f (-1)= -2+2
f (-1)= 0
point P coordinates (-6,6) and P' is it's reflection across the x axis
Answers
Find the area of the polygon with the given vertices. w(0, 0), x(0, 3), y(−3, 3), z(−3, 0)w(0, 0), x(0, 3), y(−3, 3), z(−3, 0)
Answers
A bus traveled 472 miles at a constant rate in 16 hours. What was the speed of the bus?
miles per hour
Answers
[tex]472 \div 16[/tex]
Your final answer should be 29.5 m/hr
Answer:
29.5 miles per hour.
Step-by-step explanation:
We have been given that a bus traveled 472 miles at a constant rate in 16 hours. We are asked to find the speed of the bus.
We will use speed formula to solve our given problem.
[tex]\text{Speed}=\frac{\text{Distance}}{\text{Time}}[/tex]
Upon substituting our given values in above formula, we will get:
[tex]\text{Speed}=\frac{472\text{ Miles}}{16\text{ hours}}[/tex]
[tex]\text{Speed}=\frac{29.5\text{ Miles}}{\text{ hour}}[/tex]
Therefore, the speed of the bus is 29.5 miles per hour.
Robert bought his car 4 years ago.
Each year the value of the car has depreciated by 15%.
The car is now $6264
Calculate how much the car was worth when he bought it.
Answers
The isotope calcium-41 decays into potassium-41, with a half-life of 1.03 × 10^5 years. There is a sample of calcium-41 containing 5 × 10^9 atoms. How many atoms of calcium-41 and potassium-41 will there be after 4.12 × 10^5 years?
Answers
[tex]N(t) = N_{0}e^{-kt}[/tex]
where
N₀ = initial amount
k = decay constant
t = time
The half-life is 1.03 x 10⁵ years. Therefore
[tex]e^{-1.03 \times 10^{5}k} = \frac{1}{2} \\-1.03 \times 10^{5}k=ln(0.5) \\ k=- \frac{ln(0.5)}{-1.03 \times 10^{5}} =6.7296 \times 10^{-6}[/tex]
Because N₀ = 5 x 10⁹ atoms, the number of atoms remaining when t = 4.12 x 10⁵ years is
[tex]N = (5 \times 10^{9}) e^{-(6.7296 \times 10^{-6})(4.12 \times 10^{5})} = 3.125 \times 10^{8}[/tex]
Answer: 3.125 x 10⁸ atoms
What is x-intercept of the line with the equation 4×+2y=12?
Answers
to solve, plug in 0 for y
4x+2(0)=12
4x=12
x=3
4×+2y=12
4X+ 2 (0)= 12
4X =12
X=12/4 = 3