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Help me solve this problem!!!

Let p(x) = 3^x. A general quadratic polynomial can be written as q(x) = ax^2+ bx + C.
Find the quadratic polynomial q(x) such that:
q(0)=p(0)
q’(0)=p'(0)
q'(0)=p"(0).
q(X)
=

Respuesta :

sqdancefan

Answer:

  q(x) ≈ 0.60347x^2 +1.09861x +1

Step-by-step explanation:

The derivative of an exponential function is ...

  d(a^x)/dx = ln(a)·a^x

Then the second derivative is ...

  d²(a^x)/dx² = ln(a)²·a^x

Here, you have a=3, so ...

  q(0) = 3^0 = 1

  q'(0) = ln(3)·3^0 = ln(3) ≈ 1.09861

  q''(0) = ln(3)²·3^0 = ln(3)² ≈ 1.20695

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The derivatives of p(x) are ...

  p'(x) = 2ax +b   ⇒   p'(0) = b = q'(0)

  p''(x) = 2a   ⇒   a = q''(0)/2

So, ...

  q(x) = 1.20695/2x^2 +1.09861x +1

  q(x) ≈ 0.60347x^2 +1.09861x +1