Solve the problem.Consider the functions f(x) = x2 and g(x) = x3 and their linearizations at the origin. Over some interval the approximation error for g(x) is less than the approximation error for f(x) for all x within the interval. Derive a reasonable approximation for the value of ?. Show your work. (Hint, the absolute value of the second derivative of each function gives a measure of how quickly the slopes of the function and its linear approximation are deviating from one another.)

Question

What will be an ideal response?

Answer 1

? ? . The slopes of g(x) and its linear approximation are deviating from each other more slowly than the slopes of the corresponding curves for f(x) as long as Since  and  then  or   Thus,