Problem Total Points Score. Total 100

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Evangeline Opal Bishop
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1 Your Name Solutions Instructor Name Your Signature Problem Total Points Score Extra Credit (5) Total 100 This test is closed notes and closed book You may not use a calculator In order to receive ull credit you must show your work Be wary o doing computations in your head Instead write out your computations on the exam paper PLACE A BOX AROUND YOUR FINAL ANSWER to each question where appropriate I you need more room use the backs o the pages and indicate to the reader that you have done so Raise your hand i you have a question Geometric Formulas sphere V 4 3 r3 A 4 r 2 cylinder V r 2 h cone V 1 3 r2 h A r p r 2 + h 2

2 when 1 (15 points) Consider the unction (x) 1 Hey! domain R except # (x 1)3 We have computed or you x 2 0 (x) (x 1)2 (x +2) x 3 and 00 (x) 6x 6 x 4 For ull credit show your work (a) Find the intervals where (x) isincreasinganddecreasing 0 when X I and z o is increasing on C x undeined when X 0 tpotsign and decreasing on ( go ) 2) UCOP ) (b) Find the intervals where (x) isconcaveupandconcavedown " o when x I " undi X0 { 0 + sign Fei " (c) Classiy all critical points o (x) doesnt change I : neither ig cc up on ( IP ) and cc down on to o)u( 0 1) signs on either side o # 1 2 : local Max " t 2) co ( Note to NOI a critical point the domain o ad ) because it isnt in

3 2 (15 points) Evaluate the ollowing limits [Note: You should be careul to apply L Hospital s rule only when appropriate] (a) lim x!0 ln(x +1) 1 e 5x orm % Imo III (b) p lim x ln x x!0 + q orm 0 rim # dye III III Hot 42 T orm I YZ a liw 2 0 Tito 3/2 x o+ sin (c) lim! 1 cos S 1nF Fsa Fo

4 D pts only 3 (10 points) Show that the point on the curve y 1 with x>0thatisclosesttothepoint x (0 0) is the point (1 1) For ull credit you must provide an argument showing that an absolute minimum is attained at the stated point 7 Point picture good " : minimize distance C squared 2+405*75 G) 2+ ( T ) domain ( o A) D G) ; so 3 0 or x4 10 cvit : ±1 + :# 2 10 Xl is in our domain sign i First Derivative Test implies D has a local xl It has an abie minimum (here because is decreasing increasing on the or right every hn#thedtmaanidn lathe Irishwoman " 5 :IaI :p :! * so So a local minimum is an absolute minimum

5 4 (20 points) Sketch the graph o a unction (x) thathastheollowingproperties (x) isdeinedorallx except x 1 ( 3) 1 ( 2) 2 (0) 0 and (2) 2 (x) hasaverticalasymptoteatx 1 lim x! 1 (x) 0andlim x!1 It (x) 3 0 (x) iszeroatx 2andx 0ispositiveorx< 2 2 <x< 1andx>0 and is negative elsewhere too 00 (x) iszeroatx 3 x 2andx 2ispositiveorx< 3 2 <x< 1and ll µ 1 <x<2 and is negative elsewhere ±t±* On your graph mark each point o inlection with a box and classiy all o the critical points I px Fedora local a min I local minimum I r + III

6 5 (15 points) (a) Find the linearization o (x) sin(x) atx /4 ( ty4) sin ( %) Fyz ( D Cos X F ( My) cos # % point : ( Ei N%) ; slope : mv% eq o line : y ryz Fyz ( ET) arise : Lay Ez + FE ( *ED (b) Use your linearization to approximate sin raction Leath ) E + Ez tpted Ez ( I + 10 Express your answer as a single Ez #

7 Each 6 (15 points) Air is being slowly released rom a spherical balloon At time t 0 the radius o the balloon is observed to be 10 cm and the radius is observed to be decreasing at the rate o 1 cm/s (a) Determine the rate o change o volume o the balloon when r 10 at to r1o cm Pylugih dr dq4 # (1072t ) 1 Tt cg qqgg 400 T cm% Goal " Finddavqwhennw ) dv d4tr2d_r at (b) Assuming the rate o change o volume remains constant how long will it take to empty the balloon? at t 0 r1o ; So V cm3 when to 0 z 400 ICMYS ( ie second the volume V is decreasing by 400 # cm3) How long to empty? Ee4:ioi EE

8 I a) 7 (10 points) For each o the ollowing scenarios draw the graph o a unction (x) with domain all o R that has a derivative at every point and that satisies the desired criteria (a) The unction attains an absolute maximum value and has a local minimum but does not attain an absolute minimum value mate t [ ~ local min not absolute (b) The unction has a critical point but at no point has a local minimum or maximum value tangent : thorizontal critical point [5 points extra credit:] Formally state the Mean Value Theorem and use it to prove that or all real numbers a and b where a<b I G) is cohts (b a) apple sin b sin a apple b a on [ aid and dierentiable on Ca b) then there is a C in (a) b) So that Pick cxtsinx or all R Now (c) F(b)Cab a So G) cosx MVTHM says Also is continuous there is c in ( a b) so that : t dierentiable Cos c Sirkka ) or ( b a) b a cos c Sin b Sina But E Cos C E 1 So : ( b I sinb Sina E ( b a)

Section 3.4: Concavity and the second Derivative Test. Find any points of inflection of the graph of a function.

Unit 3: Applications o Dierentiation Section 3.4: Concavity and the second Derivative Test Determine intervals on which a unction is concave upward or concave downward. Find any points o inlection o the