A quadratice function, P(x)=Ax^2+Bx+C is called a quadratic approximation of the function f?

Question

for values near a, provided that the following 3 conditions are satisfied.
f(a)=p(a)
f'(a)=p"(a)
f"(a)=p"(a)
A.) For values of x near a=0, find a quadratic polynomial, p(x)=Ax^2+Bx+C, that is a quadratic approximation of the function f(x)=e^-x^2
B.) Use the function p to approximate the value of f at x=1/2 and x=2

Answer 1

For any function f and point a, you can compute the Taylor series of f around a:

T(f,a,x) = f(a) + (1/1!)f'(a)x + (1/2!)f''(a)x^2 + (1/3!)f'''(a)x^3 + ...

If f is well behaved, this sum gives the value of f(a+x)

So, taking only the first three terms, you have a quadratic Q(x) that approximates f(a+x) and has the properties:

Q(0) = f(a)
Q'(0) = f'(a)
Q''(0) = f''(a)

Now let P(x) = Q(x-a), then P is a quadratic and has all the needed properties.