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
tester [92]
3 years ago
5

use Taylor's Theorem with integral remainder and the mean-value theorem for integrals to deduce Taylor's Theorem with lagrange r

emainder
Mathematics
1 answer:
Vadim26 [7]3 years ago
5 0

Answer:

As consequence of the Taylor theorem with integral remainder we have that

f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \int^a_x f^{(n+1)}(t)\frac{(x-t)^n}{n!}dt

If we ask that f has continuous (n+1)th derivative we can apply the mean value theorem for integrals. Then, there exists c between a and x such that

\int^a_x f^{(n+1)}(t)\frac{(x-t)^k}{n!}dt = \frac{f^{(n+1)}(c)}{n!} \int^a_x (x-t)^n d t = \frac{f^{(n+1)}(c)}{n!} \frac{(x-t)^{n+1}}{n+1}\Big|_a^x

Hence,

\int^a_x f^{(n+1)}(t)\frac{(x-t)^k}{n!}d t = \frac{f^{(n+1)}(c)}{n!} \frac{(x-t)^{(n+1)}}{n+1} = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} .

Thus,

\int^a_x f^{(n+1)}(t)\frac{(x-t)^k}{n!}d t = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}

and the Taylor theorem with Lagrange remainder is

f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}.

Step-by-step explanation:

You might be interested in
Which are factors of x2 – 4x – 5?
Kitty [74]

Answer:

(x-5)(x+1)

Step-by-step explanation:

Since the highest power of x is 2, u can assume the equation to be (x +/- A) (x +/- B), whereby A and B are unknowns.

Since the coefficient of X^2 is 1, and the number in the equation is - 5, the only possible values for A and B to get - 5 is either - 5 and 1 or 1 and - 5.

Since coefficient of x is - 4 and number in equation is - 5, imagine inserting those numbers in the equation and cross multiplying back. You would realise that the answer should be as listed above.

7 0
3 years ago
The old time train museum is raising the price of tickets for train rides.Tickets used to cost $4.80 each. The new ticket price
SVETLANKA909090 [29]
Only add $4.80 + $1.25
5 0
3 years ago
Read 2 more answers
Please help me asap i will give brainiest and all my points
Alinara [238K]

Answer:

a. Not Divisible

b. Divisible

c. Divisible

d. Divisible

e. Divisible

Step-by-step explanation:

To find the divisibility we need to follow the PEMDAS rule and check if the total value can be divided by the selected numbers without returning a remainder.

Let's begin with a:

16^{5}+215 by 33

1,048,576 + 215

1,048,791

We now then divide the total amount with 33.

\dfrac{1,048,576}{33}

= 31,781.5454

We can see that the value has a decimal point indicating that the total value divided by 33 will return a remainder. So it is NOT Divisible by 33.

Now let's continue on with the others.

51^{7}-51^{6} by 25

897,410,677,851 - 17,596,287,801

879,814,390,050

We now then divide the total amount with 25.

\dfrac{879,814,390,050}{25}

= 35,192,575,602

The total value did NOT return a decimal point, therefore making the equation true. 51^{7}-51^{6} IS divisible by 25.

Next on we have:

5^{5}-5^{4} +5^{3} by 7

3,125 - 625 + 125

As we know in the PEMDAS rule, that addition comes first. In this situation we have to read the equation from left to right and solve which ever comes first.

2,500 + 125

2,625

We now then divide the total amount with 7.

\dfrac{2,625}{7}

= 375

The total value did NOT return a decimal point, therefore making the equation true. 5^{5}-5^{4} +5^{3} IS divisible by 7.

7^6+7^5-7^4 by 11

117,649+16,807-2,401

134,456-2,401

132,055

We now then divide the total amount with 11.

\dfrac{132,055}{11}

= 12,005

The total value did NOT return a decimal point, therefore making the equation true. 7^6+7^5-7^4 IS divisible by 11.

Last but not the least:

5^{23}-5^{21} by 24

11,920,928,955,078,125-476,837,158,203,125

11,444,091,796,875,000

We now then divide the total amount with 24.

\dfrac{11,444,091,796,875,000}{24}

= 476,837,158,203,125

The total value did NOT return a decimal point, therefore making the equation true. 5^{23}-5^{21} IS divisible by 24.

5 0
3 years ago
I need help asap I have multiplied and added but they don't match the answers.
Alecsey [184]

Answer:

c

Step-by-step explanation:

12.50*6=75

45.25*3=135.75

135.75+75=210.75

3 0
3 years ago
Read 2 more answers
Answer now please ‍♀️
Shtirlitz [24]

Answer:

8/15

Step-by-step explanation:

In a right triangle, where B is an acute angle, the tan B is equal to the measure of the side opposite B divided by the measure of the shorter side at B

tanB = 16/30 = 8/15

3 0
3 years ago
Other questions:
  • Circle D is shown with the measures of the minor arcs.
    8·2 answers
  • A cold front comes through and the temperature changes overnight. What does the integer -14 represent in this context? What does
    8·1 answer
  • You have 22 coins, dimes and nickels. If the number of dimes and nickels were reversed, you would have $0.40 less than you actua
    13·1 answer
  • A tool box is a rectangular solid with sides of 19 in., 12 in., and 9 in. What is its volume?
    13·2 answers
  • 1)a-19=3 2) y/7=10 3)5g=100 solve please
    10·1 answer
  • Help? smzksoxncodmxksms
    7·2 answers
  • A sum of money is divided among Khairul, Michael and Ethan in the ratio
    8·1 answer
  • For the following right triangle, find the side length x. Round your answer to the nearest hundredth. X, 18, and 10
    12·1 answer
  • 5.4x + 20.6 = 14.4<br>10.8x + 18.4 = 17​
    9·1 answer
  • Gannon has swim practice 4 days a week. Practices are the same length each day except Monday, when practice is 2 hours long. If
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!