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
saul85 [17]
3 years ago
8

1. Approximate the given quantity using a Taylor polynomial with n3.

Mathematics
1 answer:
Jet001 [13]3 years ago
4 0

Answer:

See the explanation for the answer.

Step-by-step explanation:

Given function:

f(x) = x^{1/4}

The n-th order Taylor polynomial for function f with its center at a is:

p_{n}(x) = f(a) + f'(a) (x-a)+\frac{f''(a)}{2!} (x-a)^{2} +...+\frac{f^{(n)}a}{n!} (x-a)^{n}

As n = 3  So,

p_{3}(x) = f(a) + f'(a) (x-a)+\frac{f''(a)}{2!} (x-a)^{2} +...+\frac{f^{(3)}a}{3!} (x-a)^{3}

p_{3}(x) = f(a) + f'(a) (x-a)+\frac{f''(a)}{2!} (x-a)^{2} +...+\frac{f^{(3)}a}{6} (x-a)^{3}

p_{3}(x) = a^{1/4} + \frac{1}{4a^{ 3/4} }  (x-a)+ (\frac{1}{2})(-\frac{3}{16a^{7/4} } ) (x-a)^{2} +  (\frac{1}{6})(\frac{21}{64a^{11/4} } ) (x-a)^{3}

p_{3}(x) = 81^{1/4} + \frac{1}{4(81)^{ 3/4} }  (x-81)+ (\frac{1}{2})(-\frac{3}{16(81)^{7/4} } ) (x-81)^{2} +  (\frac{1}{6})(\frac{21}{64(81)^{11/4} } ) (x-81)^{3}

p_{3} (x) = 3 + 0.0092592593 (x - 81) + 1/2 ( - 0.000085733882) (x - 81)² + 1/6  

                                                                                  (0.0000018522752) (x-81)³

p_{3} (x)  =  0.0092592593 x - 0.000042866941 (x - 81)² + 0.00000030871254

                                                                                                       (x-81)³ + 2.25

Hence approximation at given quantity i.e.

x = 94

Putting x = 94

p_{3} (94)  =  0.0092592593 (94) - 0.000042866941 (94 - 81)² +          

                                                                 0.00000030871254 (94-81)³ + 2.25

         = 0.87037 03742 - 0.000042866941 (13)² + 0.00000030871254(13)³ +    

                                                                                                                       2.25

         = 0.87037 03742 - 0.000042866941 (169) +  

                                                                      0.00000030871254(2197) + 2.25

         = 0.87037 03742 - 0.007244513029 + 0.0006782414503 + 2.25

p_{3} (94)  = 3.113804102621

Compute the absolute error in the approximation assuming the exact value is given by a calculator.

Compute \sqrt[4]{94} as 94^{1/4} using calculator

Exact value:

E_{a}(94) = 3.113737258478

Compute absolute error:

Err = | 3.113804102621 - 3.113737258478 |

Err (94)  = 0.000066844143

If you round off the values then you get error as:

|3.11380 - 3.113737| = 0.000063

Err (94)  = 0.000063

If you round off the values up to 4 decimal places then you get error as:

|3.1138 - 3.1137| = 0.0001

Err (94)  = 0.0001

You might be interested in
Help asap ☺️☺️☺️☺️☺️☺️☺️
never [62]
Hey There!

80+70 is 150

180
-150
is 30 so 30 must be your answer

Hope This Helps!!!
3 0
3 years ago
Read 2 more answers
In a certain town there were 403 robberies last year.this year the number of robberies has gone down 35% how many robberies were
8_murik_8 [283]
262 robberies this year
7 0
3 years ago
Read 2 more answers
Select the statement that best justifies the conclusion based on the given information. B is between A and C Conclusion: A, B an
ivanzaharov [21]
It is c is the collinear
5 0
3 years ago
Read 2 more answers
Solve this question<br>What is 511+85​
labwork [276]

Answer:596

Step-by-step explanation:

511+85​(5 is by itself),(8+1=9)(1+5=6)

511

+85

596

Hope I helped :D

7 0
3 years ago
Need help with this geometric question ASAP, Thank you
USPshnik [31]
Because LP squared plus PM squared must equal LM therefore LP cannot be less.
3 0
2 years ago
Other questions:
  • The lengths of the sides of the base of a right square pyramid are 3 feet, and its slant height is 8 feet. If the sides of the b
    13·1 answer
  • What is the measure of x, given that figure ABCD is a rhombus.
    5·1 answer
  • Simplify and express in exponential form: 8^-1 * 3^3/ 2^-4 * 9^-2
    15·2 answers
  • If master chief has three and a half cookies, and the arbiter has five and three quarters of a cookie, how many cookies do they
    11·1 answer
  • Any natural number and "0".<br> what vocabulary is it ?
    13·1 answer
  • Apple trees cost $ 30 each, and cherry trees cost $ 40 each. Rohan has $ 600 to spend on fruit trees.
    10·1 answer
  • Helppppppppppp plzzzzzzzzzzzzzzzzzz
    10·2 answers
  • Expression 9(2j + 7 + 5j).
    9·2 answers
  • <img src="https://tex.z-dn.net/?f=2x%2B4%2836-x%29" id="TexFormula1" title="2x+4(36-x)" alt="2x+4(36-x)" align="absmiddle" class
    12·2 answers
  • Find x. Leave answers in simplified radical form. (Step by step)
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!