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
Aleksandr-060686 [28]
2 years ago
6

Use the definition of Taylor series to find the Taylor series, centered at c, for the function. f(x) = sin x, c = 3π/4

Mathematics
1 answer:
anyanavicka [17]2 years ago
3 0

Answer:

\sin(x) = \sum\limit^{\infty}_{n = 0} \frac{1}{\sqrt 2}\frac{(-1)^{n(n+1)/2}}{n!}(x - \frac{3\pi}{4})^n

Step-by-step explanation:

Given

f(x) = \sin x\\

c = \frac{3\pi}{4}

Required

Find the Taylor series

The Taylor series of a function is defines as:

f(x) = f(c) + f'(c)(x -c) + \frac{f"(c)}{2!}(x-c)^2 + \frac{f"'(c)}{3!}(x-c)^3 + ........ + \frac{f*n(c)}{n!}(x-c)^n

We have:

c = \frac{3\pi}{4}

f(x) = \sin x\\

f(c) = \sin(c)

f(c) = \sin(\frac{3\pi}{4})

This gives:

f(c) = \frac{1}{\sqrt 2}

We have:

f(c) = \sin(\frac{3\pi}{4})

Differentiate

f'(c) = \cos(\frac{3\pi}{4})

This gives:

f'(c) = -\frac{1}{\sqrt 2}

We have:

f'(c) = \cos(\frac{3\pi}{4})

Differentiate

f"(c) = -\sin(\frac{3\pi}{4})

This gives:

f"(c) = -\frac{1}{\sqrt 2}

We have:

f"(c) = -\sin(\frac{3\pi}{4})

Differentiate

f"'(c) = -\cos(\frac{3\pi}{4})

This gives:

f"'(c) = - * -\frac{1}{\sqrt 2}

f"'(c) = \frac{1}{\sqrt 2}

So, we have:

f(c) = \frac{1}{\sqrt 2}

f'(c) = -\frac{1}{\sqrt 2}

f"(c) = -\frac{1}{\sqrt 2}

f"'(c) = \frac{1}{\sqrt 2}

f(x) = f(c) + f'(c)(x -c) + \frac{f"(c)}{2!}(x-c)^2 + \frac{f"'(c)}{3!}(x-c)^3 + ........ + \frac{f*n(c)}{n!}(x-c)^n

becomes

f(x) = \frac{1}{\sqrt 2} - \frac{1}{\sqrt 2}(x - \frac{3\pi}{4}) -\frac{1/\sqrt 2}{2!}(x - \frac{3\pi}{4})^2 +\frac{1/\sqrt 2}{3!}(x - \frac{3\pi}{4})^3 + ... +\frac{f^n(c)}{n!}(x - \frac{3\pi}{4})^n

Rewrite as:

f(x) = \frac{1}{\sqrt 2} + \frac{(-1)}{\sqrt 2}(x - \frac{3\pi}{4}) +\frac{(-1)/\sqrt 2}{2!}(x - \frac{3\pi}{4})^2 +\frac{(-1)^2/\sqrt 2}{3!}(x - \frac{3\pi}{4})^3 + ... +\frac{f^n(c)}{n!}(x - \frac{3\pi}{4})^n

Generally, the expression becomes

f(x) = \sum\limit^{\infty}_{n = 0} \frac{1}{\sqrt 2}\frac{(-1)^{n(n+1)/2}}{n!}(x - \frac{3\pi}{4})^n

Hence:

\sin(x) = \sum\limit^{\infty}_{n = 0} \frac{1}{\sqrt 2}\frac{(-1)^{n(n+1)/2}}{n!}(x - \frac{3\pi}{4})^n

You might be interested in
A restaurant chef made 2 7/10 jars of pasta sauce. Each serving of pasta requires 9/10 of a jar of sauce. How many servings of p
Kisachek [45]

The chef will be able to prepare 3 servings of pasta because if you take one away from a whole number you have 9/10ths so you do that process again you have 2 servings but, you have 2 tenths left over so you add that to the 7 tenths.

Hope this helps :)

6 0
3 years ago
Packages of mixed-colored socks contain 8 pairs of socks. In each package, there are 5 pairs of White socks. How many pairs of w
BabaBlast [244]

125 5 diced by 40 = 125


6 0
3 years ago
Mallory needs to go to the airport. It takes her 120 minutes to get there by car. This
sergeinik [125]

Answer:

120(3)=x

120*3=360

x=360 min to go to the Airport by train.

Step-by-step explanation:

6 0
3 years ago
What is the dependent variable in y=x+12
erma4kov [3.2K]

The independent variable is x

y is dependent on the value of x

8 0
1 year ago
What is the y-intercept of the equation 2x + 3y=12?<br> -2/3<br> 12<br> 6<br> 4
pantera1 [17]

Answer:

4

Step-by-step explanation:

3 0
3 years ago
Other questions:
  • a segment has and endpoint at (-6, 8). The midpoint is at (-6, 2). What are the coordinates of the other endpoint?
    12·1 answer
  • What is the y-intercept of the graph that is shown below?
    10·1 answer
  • A soup can in the shape of a right circular cylinder is to be made from two materials. The material for the side of the can cost
    11·1 answer
  • I need the answers quick
    5·1 answer
  • I need help ASAP please
    6·1 answer
  • If the volume of a sphere is 38808 cubes then find its surface area
    12·1 answer
  • Which ratio is equvalent to 7:3
    6·1 answer
  • How do if figure out if g(x) and f(x) are equivilant
    7·1 answer
  • Find the value of x.<br> (3x - 4)<br> X<br> (5x - 2)
    6·1 answer
  • Which equation has no solution?
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!