1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Hatshy [7]
4 years ago
8

Fit a quadratic function of the form ????(????)=c0+c1????+c2????2f(t)=c0+c1t+c2t2 to the data points (0,10)(0,10), (1,6)(1,6), (

2,26)(2,26), (3,30)(3,30), using least squares.
Mathematics
1 answer:
nikitadnepr [17]4 years ago
5 0

Answer:

Therefore the quadratic function is

f(x=\frac{19}{2}+\frac{17}{2}t -\frac12 t^2

Step-by-step explanation:

We want find a quadratic function

f(t)=c_0+c_1t+c_2t^2

such that, f(0)=10, f(1)=16, f(2)=26, f(3)=30

These conditions gives the following system of linear equations

c_0+0.c_1+0.c_2=10

c_0+1.c_1+1.c_2=16

c_0+2.c_1+4.c_2=26

c_0+3.c_1+9.c_2=30

Let

A=\left[\begin{array}{ccc}1&0&0\\1&1&1\\1&2&4\\1&3&9\end{array}\right],  \vec x= \left[\begin{array}{c}c_0\\c_1\\c_2\end{array}\right]    and   \vec b=\left[\begin{array}{ccc}10\\16\\26\\30\end{array}\right]

The unique least square solution is A\vec x=\vec b

\vec x= (A^TA)^{-1}A^T \vec b

\left[\begin{array}{c}c_0\\c_1\\c_2\end{array}\right]=\left(\left[\begin{array}{cccc}1&1&1&1\\0&1&2&3\\0&1&4&9\end{array}\right] \left[\begin{array}{ccc}1&0&0\\1&1&1\\1&2&4\\1&3&9\end{array}\right]\right)^{-1}\left[\begin{array}{cccc}1&1&1&1\\0&1&2&3\\0&1&4&9\end{array}\right]\left[\begin{array}{ccc}10\\16\\26\\30\end{array}\right]

\left[\begin{array}{c}c_0\\c_1\\c_2\end{array}\right]=\left(\left[\begin{array}{ccc}4&6&14\\6&14&36\\14&36&98\end{array}\right] \right)^{-1}\left[\begin{array}{cccc}1&1&1&1\\0&1&2&3\\0&1&4&9\end{array}\right] \left[\begin{array}{ccc}10\\16\\26\\30\end{array}\right]

\left[\begin{array}{c}c_0\\c_1\\c_2\end{array}\right]=\left[\begin{array}{ccc}\frac{19}{20}&-\frac{21}{20}&\frac14 \\ \\-\frac{21}{20}&\frac{49}{20}&-\frac{3}{4}\\ \\ \frac14&-\frac34& \frac14 \end{array}\right] \left[\begin{array}{cccc}1&1&1&1\\0&1&2&3\\0&1&4&9\end{array}\right] \left[\begin{array}{ccc}10\\16\\26\\30\end{array}\right]

\left[\begin{array}{c}c_0\\c_1\\c_2\end{array}\right]= \left[\begin{array}{c}\frac{19}{2}\\ \\\frac{17}{2}\\ \\-\frac12\end{array}\right]

Therefore c_0 = \frac{19}{2},  c_1=\frac{17}{2} and c_2=-\frac12.

Therefore the quadratic function is

f(x=\frac{19}{2}+\frac{17}{2}t -\frac12 t^2

You might be interested in
in the following ordinary annuity interest is compounded with each payment and the payment is made at the end of the compounding
stealth61 [152]

Answer: $59313.58

Step-by-step explanation:

We know that formula we use to find the accumulated amount of the annuity ( ordinary annuity interest is compounded ) is given by :-

FV=A(\frac{(1+\frac{r}{m})^{mt})-1}{\frac{r}{m}}), where A is the annuity payment deposit, r is annual interest rate , t is time in years and n is number of periods.

Given : Annuity payment deposit :A= $4500

rate of interest :r= 6%=0.06

No. of periods : m= 1   [∵ its annual]

Time : t= 10 years

Now we get,

FV=(4500)(\frac{(1+\frac{0.06}{1})^{1\times10})-1}{\frac{0.06}{1}})\\\\\Rightarrow\ FV=(4500)(\frac{(1.06)^{10})-1}{0.06})\\\\\Rightarrow\ FV=(4500)(\frac{0.79084769654}{0.06})\\\\\Rightarrow\ FV=(4500)(13.1807949423)\\\\\Rightarrow\ FV=59313.5772407\approx59313.58 \ \

∴ the accumulated amount of the annuity= $59313.58

4 0
4 years ago
Could anyone help me with this? I will mark you as brainliest
Marrrta [24]

Answer:

the right answer is number 7

4 0
3 years ago
Read 2 more answers
Hey lads could you please check out my channel on you.tube please leave a like and subscribe it would help a lot much love stay
nika2105 [10]

Answer:

ok bud, better be good content then ill sub

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
PLS HELP PLS HELP PLS HELP PLS HELP PLS HELP I BEG U PLS BOTH QUESTIONS PLS
STatiana [176]

Answer:14 is obtuse

15 is A

Step-by-step explanation:

For 14 the only remaining angle is over 90

For 15 the first triangle has all same side and angle measures

second triangle has 2 equal side measures

and third has none

6 0
3 years ago
Read 2 more answers
A basketball player attempts 15 baskets in a game, He makes 9 of the attempted basket. which ratio describe the number of basket
Taya2010 [7]

Answer:

3:5

Step-by-step explanation:

We are looking for the ratio of baskets made to baskets attempted

made: attempted

9:15

We can divide each number by 3

9/3: 15/3

3:5

The ratio of baskets made: attempted  is 3:5

3 0
3 years ago
Other questions:
  • What is the slope of the line shown below?
    14·2 answers
  • GIVEN BRAINLIEST 5 STARS + THANKS
    12·1 answer
  • PLEASE HURRY my question is in the picture​
    8·2 answers
  • Help ne with this math
    15·1 answer
  • Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth.
    12·1 answer
  • 24 1/2 simplified what is the answer?
    5·2 answers
  • The graphs of the two lines 4x=3y+23 and 4y+3x= -19 A. do not intersect. B. are identical C. intersect at (-2,5) D.are perpendic
    9·1 answer
  • The length of a rectangle is the width minus 8 units. The area of the rectangle is 9 units. What is the length, in units, of the
    9·1 answer
  • Helppppp me plzzzzzzzzzzz
    9·1 answer
  • Why are prefixes used in naming covalent compounds
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!