Find the 3rd order for the taylor polynomial in powers of x for the function f(x)=sinh2x and find the remainder as a function of x and c.
P3(x)=?
R3(x)=?
What's the kth term expression?
To find the 3rd order Taylor polynomial for the function f(x) = sinh(2x) and the remainder as a function of x and c, we can follow these steps:
Step 1: Compute the derivatives of f(x) up to the 3rd order.
The derivatives of f(x) = sinh(2x) are:
f'(x) = 2cosh(2x)
f''(x) = 4sinh(2x)
f'''(x) = 8cosh(2x)
Step 2: Evaluate the derivatives at x = c.
Evaluate f'(x), f''(x), and f'''(x) at x = c:
f'(c) = 2cosh(2c)
f''(c) = 4sinh(2c)
f'''(c) = 8cosh(2c)
Step 3: Write the general formula for the Taylor polynomial up to the 3rd order.
P3(x) = f(c) + f'(c)(x - c) + (f''(c)/2!)(x - c)^2 + (f'''(c)/3!)(x - c)^3
Step 4: Substitute the values into the Taylor polynomial formula.
P3(x) = sinh(2c) + 2cosh(2c)(x - c) + (4sinh(2c)/2!)(x - c)^2 + (8cosh(2c)/3!)(x - c)^3
So, the 3rd order Taylor polynomial for f(x) = sinh(2x) is P3(x) = sinh(2c) + 2cosh(2c)(x - c) + (2sinh(2c))(x - c)^2 + (8cosh(2c)/3!)(x - c)^3.
Now, let's find the remainder R3(x):
Step 5: Write the general formula for the remainder in Taylor series.
R3(x) = (f^(4)(z)/4!)(x - c)^4, where z is a value between x and c.
Step 6: Find the 4th derivative of f(x) and substitute it into the remainder formula.
f^(4)(x) = 16sinh(2x)
R3(x) = (16sinh(2z)/4!)(x - c)^4, where z is a value between x and c.
So, the remainder R3(x) is equal to (16sinh(2z)/4!)(x - c)^4, where z is a value between x and c.
To find the kth term expression, we need to expand the Taylor polynomial up to the kth term. In this case, since we are finding the 3rd order Taylor polynomial, the kth term expression will be:
k = 0: f(c) = sinh(2c)
k = 1: f'(c)(x - c)
k = 2: (f''(c)/2!)(x - c)^2
k = 3: (f'''(c)/3!)(x - c)^3