Determine where f(x) = arctau(x^2 - 2x) is increasing. The graph of f(x) is show

Question

Determine where f(x) = arctau(x^2 - 2x) is increasing. The graph of f(x) is shown and the table gives values of g(x) and g'(x). a) Given h(x) = f(g(x)). find h'(1). b) Given k(x) = g(f(x)), find k'(3). c) Given l(x) = g(g(x)), find l(1). d) Given m(x) = Squareroot f(x), find m'(1). The length of the day in Boulder (Latitude 40 degree N) can be modeled approximately by l(t) = -3 cos (2 pi/365(t + 10) + 12 where l is given in hours and t is the day of the year. a) Evaluate l(355); fully interpret the result in the context of this problem, including units. b) Evaluate l'(265); fully interpret the result in the context of this problem, including units. c) Calculate when l'(t) is largest. Explain. The U.S. gross domestic product can be modeled by P(t) = 4.351 e^0.0368 t where P is given in billions of dollars and t is years since 1790. a) Find P(224); fully interpret the result in the context of this problem, including units. b) When was the GDP one trillion dollars? c) How many years does it take for the GDP to double? d) What is P'(224)? Again, fully interpret (including units).

Explanation / Answer

A1) given f(x)=arctan(x2-2x)

differentiate with respect to x

f '(x)=[1/(1+(x2-2x)2)]*(2x-2*1)

f '(x)=[2(x-1)/(1+(x4+4x2-4x3))]

f '(x)=[2(x-1)/(x4-4x3+4x2+1)]

f(x) increasing when f '(x)>0

[2(x-1)/(x4-4x3+4x2+1)]>0

=>x>1

f(x) is increasing in (1,)

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