1. [5 points] Find an equation of the line that passes through the points ( 3, 2) and (1, 10) = 2, b = 2 2 ( 3) = 8. Equation: y = 2x + 8.

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1 Mat 12 Final Eam Sample Answer Keys 1. [ points] Find an equation of te line tat passes troug te points 3, 2) and 1, 1) m = = 2, b = 2 2 3) =. Equation: y = Evaluate te it a) [ points] ) 3 + ) 3 + = 3. b) [ points] ) + 2) + 2 2) = Find te derivative of eac function. You need not simplify te result. a) [ points] f) = 3 2 f) = 2/3 1/2, f ) = 4 1/ /2 b) [ points] ft) = t2 + 2 t 1 f t) = 2tt 1) t2 2 = t2 2t 2 t 1) 2 t 1) 2 c) [ points] g) = g ) = d) [ points] gt) = e 2t ln1 7t 2 ) gt) = 2e 2t ln1 7t 2 ) e 2t 14t 1 7t 2

2 4. [1 points] Use te definition of derivative to find te derivative of f) = 2. No credit will be given wen te definition is not used). 2 + ) = 2 f ) f + ) f) + ) a) [ points] Find dy d if 2 y y 3 = 2 2y + 2 dy d y3 3y 2 dy d =, dy d = y3 2y 2 3y. 2 b) [ points] Find te equation for te tangent line to te curve 2 y y 3 = 2 at te point 2, 1). m = = 3 2 = 3 2. Tangent line equation: y = ), y = For te function f) = a) [ points] find orizontal asymptotes f) = ± ± =. b) [ points] make a sign diagram for te derivative f ) = ) 3) = 2, 2 + 9) ) 2 f ) > wen 3 < < 3; f ) < wen < 3 or > 3. c) [ points] find all relative maimum and minimum values Relative minimum is at = 3, f 3) = 1 3, relative maimum is at = 3, f3) =

3 7. [1 points] Maimum Profit: A furniture store can sell 3 cairs per week at a price of $7 eac. Te manager estimates tat for eac $3 price reduction se can sell two more cairs per week. Te cairs cost te store $7 eac. If stands for te number of $3 price reductions, find te price of te cairs and te quantity tat maimize te profit. Price is p) = 7 3, te quantity sold is q) = Revenue is R) = p) q) = 7 3)3 + 2) = Cost is C) = 7q) = Profit is P ) = R) C) = We maimize te profit by finding its derivative and CNs. P ) = CNs: P ) is defined everywere, P ) = wen = 3 wic is te only CNs of P ). P ) = 12 <. Hence profit as a relative maimum at = 3. Tis maimum is absolute b/c te grap of te profit is a parabola opened down. p3) = 7 9 = 1, q3) = 3 + = 3. Answer: te price of te cairs tat maimize te profit is $1 and te quantity is 3.. [1 points] Two cars start moving from te same point. One travels east at 4 mi/ and te oter travels nort at 3 mi/. At wat rate is te distance between te cars increasing two ours later? Let t) be te distance traveled by te first car and yt) be te distance traveled by te second car; d dy = 4, dt dt = 3. Te distance between cars is dt) = 2 + y 2. dt = d ) dy 1 + 2y = d 2 + y 2 dt dt 2 + y 2 dt + y dy ) dt After tree ours = 3 4 mi, y = 3 3 mi, d = = 3 = 1 mi. Ten dt = ) = 3 dt = mi/. = = 2 3

4 9. [1 points] A bank offers % compounded continuously. How soon will a deposit triple? Leave answer in eact form. P e.t = 3P, e.t = 3, ln e.t = ln 3, t = 12. ln 3 years. 1 1 ln 3 t = ln 3, t =, 1. [1 points] Find te area under te curve y = 2 e 3 wen 2. Leave answer in eact form. Te area is A = 2 2 e 3 d. Substitution u = 3, du = 3 2 d, u) =, u2) =. Ten A = 1 3 e u du = 1 3 eu = e For te demand function d) = and supply fuction s) =. 2 find a) [1 points] te market demand level te positive value of at wic te demand function intersects te supply fuction) =. 2,. 2 = 2, 2 = 2, A = =. b) [1 points] te consumer s surplus at te market demand level found in part a). Te market price is B = s) = 1. Consumer s surplus is d) B) d =.2 2 ) d = ] [ 3 = = 3 = [1 points] Te population of a town is increasing at te rate of t e t/2 people per year, were t is te number of years from now. Find te average gain in population during te net si years. Leave your answer in eact form. Te average gain in population during te net si years is 1 t e t/2 dt = 4 t e t/2 dt

5 By parts: u = t, du = dt, dv = e t/2 dt, v = 2e t/2. = 2t e t/2 2e t/2 dt = 12e 3 4e t/2 = 12e 3 4e 3 1) = e For te function f, y) = e 2y ln a) [1 points] find te domain domain is {, y) > } b) [1 points] find partials f and f y. f = e 2y ln + e 2y f y = f y = 2e 2y ln + 1 ). = e 2y ln + 1 ) 14. [1 points] If a company s profit function is P, y) = 2y 2 2 3y y + 72 tousand dollars, find ow many of eac unit and y sould be produced in order to maimize te profit. P = 2y 4+4 = y = 2 2, P y = 2 y +1 =, 3y +9 =, =, + 1 = = 3, y = 4. CP is 3, 4). P = 4, P yy =, P y = 2, D = 24 4 = 2 >, P <. Relative maimum. So profit is maimized wen = 3, y = [1 points] Use Lagrange multipliers to find te maimum value of te function f, y) = y subject to te constraint 3 + y = 12. [Hint: Find CP. Te maimum value of te function is attained at CP.] F, y, λ) = y + 3λ + yλ 12λ F = y + 3λ =, F y = + λ =, F λ = 3 + y 12 = λ = y 3 =, y = 3, F λ = =, = 2, y =. CP is 2, ). Te maimum value of te function is f2, ) = 12.