1. Numerical Methods Taylor Polynomial

Hey all, this one has me stumped,

Q. The nth Taylor Polynomial for a function f at xo is sometimes referred to as the polynomial of a degree at most n that "best" approximates f near xo.

Find the quadratic polynomial that best approximates a function f near xo = 1 if the tangent line at xo = 1 has equation y = 4x - 1, and if f''(1) = 6.

I'd appreciate anyone taking a look at this

2. Originally Posted by Locke333
Hey all, this one has me stumped,

Q. The nth Taylor Polynomial for a function f at xo is sometimes referred to as the polynomial of a degree at most n that "best" approximates f near xo.

Find the quadratic polynomial that best approximates a function f near xo = 1 if the tangent line at xo = 1 has equation y = 4x - 1, and if f''(1) = 6.

I'd appreciate anyone taking a look at this
Observe that from this information, we can also conclude that $f\!\left(1\right)=3$ and $f^{\prime}\!\left(1\right)=4$.

Thus, we have the taylor polynomial $f\!\left(x\right)\approx f\!\left(x_0\right)+f^{\prime}\!\left(x_0\right)\l eft(x-x_0\right)+\tfrac{1}{2}f^{\prime\prime}\!\left(x_0 \right)\left(x-x_0\right)^2$ $=3+4(x-1)+3(x-1)^2$

I leave it for you to simplify that expression.

Can you take it from here?

3. the 2d Taylor polynomial is:

f(x0) + f ' (x0)(x-x0) + [f "(x0)/2](x-x0)^2

You already have the first 2 terms from the tangentline we have

-1 + 4x +[ f " (1)/2](x-1)^2

= -1 + 4x + 3(x-1)^2

= 3x^2 - 2x + 2

4. Thanks to the both of you, your posts were a great help