If u(x) = f(x) + ig(x) is a complex-valued function of a real variable x and the real and imaginary parts f(x) and g(x) are differentiable functions of x, then the derivative of u is defined to be u'(x) = f '(x) + ig'(x). Use this together with ex + iy = x iy = x(cos(y) + i sin(y)) to prove that if F(x) = erx , then F '(x) = r x when f = a + bi is a complex number.

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

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answer: d. $20

step-by-step explanation:

just took the test

$12. that is my answer.