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Hi all, I'm reviewing for an upcoming exam and so I'll be asking for clarification on a few things throughout the evening.
Solved problems in spoilers.
1. Differentiate f(x) = (2e^x) - (e^-x)/7
I know that the answer is f'(x) = (2e^x) + (e^-x)/7, but I'm not clear on why. With terms separated by + or -, you just take the derivative of one + or - the derivative of the other, correct? And the first term falls under the "d/dx of e^x is e^x" rule, yes?
So with the second term, do I use the quotient rule?
quotient rule is (u'v-uv')/v^2
So I end up with something like [(e^-x)*7] / 49, right?
2. f(x) = 2e^(3x+1) - 17
f'(x) = 6e^(3x+1)
Since f'(e^x) = f'(x) * e^x
3. f(x) = ln (x^2+1)
f'(x) = 2x/(x^2 +1)
I know that d(lnx) = 1/x
Why is it 2x * 1/(x^2+1) and not 2x + 1/(x^2+1) ?
4. A demand function is p = 400 - 2q, where q is the quantity of good sold for price $p. Which of the following is an expression for the total revenue, R, in terms of q?
I just too R = p*q and substituted the function for p to get R = (400-2q)*q Right?
5. The figure below* shows the graphs of marginal cost and marginal revenue. What production level could maximize the profit?
*The graph is a concave-up parabola labeled $/unit (MC) that intersects with a horizontal line labeled MR at x=1000 and x=3000
I hope that's clear, otherwise I can rehost the image if necessary.