1. ## Quadratic Polynomail with chain rule

Let f(x)=sin(x)+cos(2x). Find a quadratic polynomial p(x) so that p(0)=f(0), p'(0)=f'(0) and p''(0)=f''(0).

2. Hello Velvet Love

Originally Posted by Velvet Love
Let f(x)=sin(x)+cos(2x). Find a quadratic polynomial p(x) so that p(0)=f(0), p'(0)=f'(0) and p''(0)=f''(0).
First, find the derivatives of f(x)

It is f'(x) = cos(x)-2sin(2x) and

f''(x) = -sin(x) -4cos(2x)

p(x) = ax^2+bx+c

In this excercise we have to find a,b,c and want to work with derivatives, so let's find p'(x) and p''(x) first

p'(x) = 2ax+b

p''(x) = 2a

( It should be p(0)=f(0), p'(0)=f'(0) and p''(0)=f''(0) )

I suggest we start with the second derivative, because p''(x) has only one unknown in it - the 'a'. p(x) contains a,b and c.

So

p''(0) = f''(0)

2a = -sin(0) -4cos(2*0)

2a = 0 -4cos(0)

2a = -4*1

=> a = -2.

So p'(x) = 2ax+b = 2*(-2)a+b

Now consider

p'(0) = f'(0) and solve for b.

then p(0) = f(0) and solve for c.

Done

Do you have some question on that?

Yours
Rapha