please answer all the questions, show and explain each step.

Universit d'Ottawa | University of Ottawa page 2 of 8 1. (1 point) If f(z) = e arctan(z) then f'(1) = A0 C. en/d+e/2 E. 1 B. en/4 D. e/2en E. & Your answer: 2. (1 point) Which of the following is equal to the derivative of f(z) = 25"(@)? A, 25 cos(x) C. 2% sin(z)In(2) E. 2sin() B. 1In(2)25"@) cos(x) B i) F. sin(x) cos(x)25n@)~1 In(2) Your answer: 3. (1 point) Which of the following is equal to the derivative of g(x) = In(e23)? e' C. 3x2e\" 4 xde\" E. 1+43/x 513_3 2 3\\, T, .3 ) D. (3z 4+ z)e* In(e*x B. 3w B. e\"In(z) 3/ ( ) (%) Your answer: Universite d'Ottawa | University of Ottawa page 3 of 8 4. (1 point) Suppose f is a function and we know that tan(f(x)) + f(2) = 22 for all x in the domain of f. Then 2x 2x A. f' (a ) = sec(x) tan(x) + 1 D. f'(x) = sec(f (x) ) tan( f(a)) + 1 B. f'(x) = 2x - sec(x) tan(x) E. f'(x) = 2x - sec2 (f (x)) 2.x C. f'(a) = sec2 (ac ) + 1 F. f' (2 ) = - sec2 (f (ac) ) + 1 Your answer: 5. (1 point) Evaluate the following limit using l'Hopital's rule: sin(3x) lim x-+0 4x . 3/4 C. 0/0 E. 1 B. 0 D. 0 F. cos(3) /4 Your answer:Universit d'Ottawa | University of Ottawa 6. (1 point) If h(z) = Ve** + 23 then 2 2 A W(x)="+ \\/E C. W(z)= 2878 E. W(z)= 2 2v/e2x 1+ 3 2e** + 32 il B. Wx)= e - (z) e 1 o) D. H(2) = s F. h(z)= Your answer: 7. (1 point) Suppose we are given 3 + 62 22 P L3 6 fa)= AT payn AR g = Which one of the following statements is true of the value r=V3? It is not a critical point of f, but it is an inflection point. f has a local minimum there, but it is not a global minimum. f has a local maximum there, but it is not a global extremum. f has a local and global minimum there. f has a local maximum and a global minimum there. m =0 aQw > It is neither a critical point nor an inflection point of f. Your answer: page 4 of 8 Universit d'Ottawa | University of Ottawa page 5 of 8 8. (14+3+1=5 points) In this question, you will use the definition of the derivative to find the derivative of a function, and then you will check your answer using the quotient rule. (a) Give the definition of the derivative of a differentiable function f(z) at a point z in its domain. lim Answer: f'(z) = 2 (b) Using the definition of the derivative, find the derivative of f(z) = 3 z its domain. at a point x in x x (c) Compute the derivative of f(z) = using the quotient rule. z 3 Universit d'Ottawa | University of Ottawa page 6 of 8 9. (2 + 2 + 1 = 5 points) A population of snails for a nice restaurant grows at a logistic rate, and is harvested regularly. Denoting the fraction of snails harvested each month by the parameter h, the DTDS governing the growth of this population is given by Nt+1 3Nt(1 Nt) h,Nt, where t is in months, N, is the population (in thousands of snails) at time , and 0