Need math help to consider a function

Dec 7th, 2015
DreamIt
Category:
Engineering
Price: $5 USD

Question description

Consider the function g(x) = 12/ 3+x2(a)  Let L(x) be the local linearization of g(x) at x = 3. Find a formula for L(x). Sketch graphs of the functions L(x) and g(x) on the same set of axes.(b)  Using L(x), approximate g(3.2). Similarly, approximate g(2.8). Are your estimates overestimates or underestimates? Recall that for a function f(x) we define “the quadratic approximation”1 of f(x) at x = a to be the degree 2 polynomial Q(x) such that all of the following hold: • Q(x) and f(x) have the same function value at x = a.• Q(x) and f(x) have the same first derivative at x = a.• Q(x) and f(x) have the same second derivative at x = a.  A formula for Q(x) is given byQ(x) = f(a) + f′(a)(x − a) +( f′′(a)/2)(x − a)^2   (c) With g(x) as above, let P(x) be the quadratic approximation of g(x) at x = 3. Find a formula for P(x). Sketch graphs of the functions g(x),L(x), and P(x) on the same set of axes.(d)  Using P(x), approximate g(3.2) and g(2.8). Are your estimates overestimates or underestimates?Are they better or worse than your estimates from part (b)?(e)  Justify the following statement: “For large values of x, P (x) is not a good approximation of g(x).” 

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