101k views
1 vote
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)
=

User Mork
by
8.4k points

1 Answer

8 votes

Answer:

q(x) ≈ 0.60347x^2 +1.09861x +1

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

__

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

User Mihir Palkhiwala
by
8.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories