# How do you find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a f(x) = cos(x), a= pi/4?

Remembering that any function

We can compute the required derivatives (in this case three) and find a general formula for our infinite sum.

Calculations:

Now that we have our derivatives at point

To check our answer, we can always graph both equations:

Graph of

graph{cos x [-7.9, 7.895, -3.95, 3.95]}

Graph of

graph{sqrt(2)/(2) - (sqrt(2)/(2)(x-pi/4))/(1!) - (sqrt(2)/(2)(x-pi/4)^(2))/(2!)+ (sqrt(2)/(2)(x-pi/4)^(3))/(3!) [-7.9, 7.895, -3.95, 3.95]}

Overlapping both graphs gives you the following:

If we increase the number of polynomial terms in our series we automatically make it a better approximation to our function.

By signing up, you agree to our Terms of Service and Privacy Policy

The Taylor polynomials of orders 0, 1, 2, and 3 generated by ( f ) at ( a ) where ( f(x) = \cos(x) ) and ( a = \frac{\pi}{4} ) are as follows:

Order 0: [ P_0(x) = f(a) = f\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} ]

Order 1: [ P_1(x) = f(a) + f'(a)(x-a) = \cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right)(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) ]

Order 2: [ P_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 = \cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right)(x - \frac{\pi}{4}) - \cos\left(\frac{\pi}{4}\right)\left(x - \frac{\pi}{4}\right)^2 ]

Order 3: [ P_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 ] [ = \cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right)(x - \frac{\pi}{4}) - \cos\left(\frac{\pi}{4}\right)\left(x - \frac{\pi}{4}\right)^2 - \sin\left(\frac{\pi}{4}\right)\left(x - \frac{\pi}{4}\right)^3 ]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the Maclaurin series for #f(x)= e^(-1/(x^2)) # centered at 0?
- How do you find the Maclaurin Series for # (sin(x/11))^2#?
- How do you find the maclaurin series expansion of #e^(7x)ln((1-x)/3)#?
- How do you find the taylor series series for # (4-x)^(1/2)#?
- What is the radius of convergence of the series #sum_(n=0)^oo(n*(x+2)^n)/3^(n+1)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7