QUESTION 15. [7 points] Find. 2 ) HINT: split in 3 fractions, make every fraction a POWER, and then use power rule for integrals (along with sum, constant term rules). As an example: the first term is x 2". The next? Wow, it is x2, amazing, what about the last one? 3x-7, yay Hint: FTC gives In(x2) dx = F(2) - F(1) = ...; where F(x) = f 2 In(x) dx = Now use Parts: f'(x) = 2 and g(x) = In(x), so f(x) = 2x and g'(x) = 2, now continue and get F, then finish the computation: F(2) - F(1), BTW what is In(1)? c ) I VI2 - 2 dx Hint: Use a SUB: u = x2 - 2, dy = 2x, or 2 = rdu, now your new integral is just Ju-1/24", NOW use power rule for the integrals, and replace u by its definition ... 2ul/2 + C=...QUESTION 12. [7pts] Consider f(x) = 2 - 3e3/2. use the below steps to draw THE graph OF f. a) fIND DOMAIN OF f. Hint: since x * 0, we get D = (-0o, 0) U (0, co). yay b) Find VA, HA and the IMPORTANT limits. Hints: do the limits at infinities: and get: y = -1, and for VA do both lim at 0 (L, R) and get x = 0 is a VA. c) find the 1st derivative and CPs Hint: 9e3/* x-2 d) When is f increasing? decreasing? Local max, min- any?. e) When is f C Up, C Down? f) Any inflection points? page 9 of 16 9 g) Graph f using previous page work.QUESTION 11. [4 pts] Find: (no plugging in a calc to guess) a) lim 1 x-0+ xVI + 4x Use L'Hopital as it is bad: co - co, so Just go to the same denominator: 1-v1+4x *V1+4x . yay another way: rationalize! e3x + 3ex - 14 b) lim x-+In(2) ex - 2 It is 0/0, try L'HopitalQUESTION 2. The table gives info on f(x), g(x), f'(x) and g'(x) for some x . Consider h(x) = (fog) (x). 1 2 3 4 f(x) 1 g (ac) 3 2 f' (x) 3 4 g' (x) 4 4 3 2 Find the correct answer: A. h(2) = 1 , h'(2) = 12. D. h(2) = 8 , h'(2) = 6. B. h(2) = 8 , h'(2) = 9 E. h(2) = 8, h'(2) = 1. C. h(2) = 1 , h'(2) =6 F. h(2) = 4, h'(2) = 12. Answer: Chain rule says: h'(2) = f'(g(2))g'(2) = f'(2) x 4 = 3 x 4 = 12. Note h(2) = f(g(2)) = f(2) = 4, hiha SO: QUESTION 3. Find the derivative of f(x) = In 2" sin(2:x) (x+ 1)5ex-2 .. 5 A. In(2) + 2 cot(2x) D. In(2) - 2 cot(2x) + 2x x+1 B. - + 2 cot(2x) - 1 (2 + 1)5 - 2x 5 E. In(2) + 2 cot(2x) - 2x C. In(2) + 2 2 F. 2 5 cos (2.x) - 2x cos (2x) Answer:QUESTION 13. [7 pts] The number of insects in my dead brain is modeled by: It+1 = , It (1 - It) - hat for t = 0, 1, 2, ..., where h is the harvest factor. We need h 2 0 such that the Yield is maximum. a) give the updating function. Answer: f(x) = 2x(1 -x) - hx, yay b) Find Eq points. Some may depend on h. Hint: Solve x = f(x), get r; = 0 and x2 = 3-2h c) for what values of h is there a strictly positive Eq Point? Hint: did you ever solve: 3 - 2h > 0? d) For the strictly positive Eq P found above, say r* , find h such that Y(h) = hx* is maximal. Why is it a global max? Hint: Take the derivative of Y(h) = h3-2h as we did in class, and create a table, to find the LOCAL Max. Justify it is also a Global, looking at the endpoints of the domain. e) Find that max yield. Hint: Plug the above h you found in Y, of course you need a calc .. .. .. f) is the max yield sustainable? In other words: is your eq point stable? Use calculus! Hint: Compute f'(x), then solve If'(3 2h) |