Find the quadratic polynomial \( g ( x ) = a x ^ {...

Question

$Find the quadratic polynomial \( g ( x ) = a x ^ {...$

Find the quadratic polynomial $ g ( x ) = a x ^ { 2 } + b x + c $ which best fits the function $ f ( x ) = 3 ^ { x } $ at $ x = 0 $ , in the sense that $ g ( 0 ) = f ( 0 ) $ , and $ g ^ { \prime } ( 0 ) = f ^ { \prime } ( 0 ) $ , and $ g ^ { \prime \prime } ( 0 ) = f ^ { \prime \prime } ( 0 ) $ . $ g ( x ) = $ ___________.(Using a computer or calculator, sketch graphs of $ f $ and $ g $ on the same axes. What do you notice?)

Answer

$g(0)= c= f(0)= 1\\g'(x)= 2ax+ b,f'(x)= 3^x\cdot\ln3\\f'(0)= \ln3= g'(0)= b\\g''(x)= 2a,f''(x)= 3^x\cdot(\ln3)^2\\g''(0)= 2a= f''(0)= (\ln3)^2\Rightarrow a= \frac{(\ln3)^2}{2}\\g(x)= \frac{(\ln3)^2}{2}x^2+ \ln3x+ 1$

Solution

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