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Question

Let . A general quadratic polynomial can be written as .

Find the quadratic polynomial such that:

 ,

 ,

 .

Plot the graphs of and together near . Describe how the graphs are related.

Answer 1

Function p is given, so wed can compute the values p(0), p'(0), p"(0).

q(0) = c

p(0) = 1

So, the value c is computed : c = 1

Derivative of q: q'(x) = 2ax + b

q'(0) = b

What is derivative of p?

p'(x) = ln 3 * 3^x

So, p'(0) = q'(0)

b = ln 3

Similarly, we get the value of a, the first coefficient.

Answer 2

p(0)=1 => c =1

q(x)=2ax +b so that q'(0)=b

p'(x)=ln 3 * 3^x and p'(0)= ln 3

therefore b=ln 3

similarly a will be found to be ln²3

A graph will show that the 2 functions are close near 0.

What you have actually found is the Taylor polynomial of degree 2 for the function 3^x.