Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least

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1 Problems to practice for FINAL. Below is the graph of a function () At which of the marked values ( and ) is: (a) () greatest = (b) () least = (c) () the greatest = (d) () the least = (e) () = = (f) () = = Identif the interval(s) in terms of ( and ) on which (g) () is positive ( ) ( ) (h) () is negative ( ) (i) () is positive ( ) (j) () is negative ( ) (k) () is increasing and concave up ( ) (l) () is increasing and concave down ( ) (m) () is decreasing and concave up ( ) (n) () is decreasing and concave down ( )

2 . Evaluate (if infinit, specif whether it is + or ) (a) sin = (b) = (c) (d) (e) (f) = = sin = ln = (g) ( ) = 6. Evaluate (if infinit, specif whether it is + or ) sin (a) sin 5 = 5 (b) = (c) tan ln = + (d) = (e) +5 6 =6 (f) (g) (h) (i) (j) (k) (l) (m) +5 = = = = () =7,where () = + () =4,where () = = = (n) cos = (o) = (p) (ln ) = (q) = ½ 7 if + if ½ 7 if + if

3 (r) µ + = (s) = + (t) µ + = 8 +5 (u) µ sin = +cos = ³ cos sin = = sin 4cos = 4 (v) (sin ) tan ln = ln (sin ) tan ln sin = tan ln sin = cot = = = sin cos csc = sin cos = 4. Find if (a) + = Solution: + + =, Solution is: = +. (b) = Solution: + 9 =., Solution is: = 9 (c) + =Solution: =., Solution is: = +9

4 5. Find the derivative for: (a) = tan Solution: = tan tan + sec (b) =sin( ) Solution: = cos ( ) (c) =4 Solution: = (ln 4) 4 (d) =sin Solution: = r 9 (e) =ln( +)Solution: = + (f) =sin(cos) Solution: = sin cos (cos ) (g) =sin (cos ) Solution: sin = cos (h) = sin Solution: = sin (i) =tan(ln) Solution: = tan (ln )+ (j) = sin Solution: =(cos ln ) sin (k) = sin Solution: =(cos) +(sin) (l) = sin ln Solution: = ln cos ln sin (m) = 4 Solution: =4 + 4 (n) = cos () Solution: = cos sin (o) = (p) = Solution: = ( ) 4 (sin )(cos) Solution: = ³ (q) =(sin) Solution: =((sin) ) (r) =sin (5) Solution: = 5 5 (s) =cos Solution: = sin ³ (t) =ln (u) + Solution: = µ cos sin sin ln (sin )+ cos sin µ + + = +5Solution: =while = +5 (v) =( +sin) Solution: = ( +sin) (cos +) (w) = +Solution: = ++ () =sin () Solution: =cos sin +

5 6. Find the tangent line to the curve cos () = at the point p p. r r cos () = : = ; = sin ()( + ) = sin () = +sin() = +sin() sin () : µ r = ; = = p +sin p p p p sin p p p r µ r = r = r = p +sin p p sin p = p + p p p 7. Find the derivative of = 7 4 ( + ) 4 (HINT: Easier if ou use logarithmic differentiation). ln = ln + ln 7 + ln ( ) 4ln + µ = 7 4 ( + ) 4 + ( ) 8 + = ½ +5 + if 8. Suppose the function () is defined as () = +4 if function continuous on the interval ( + )? = + = =.Forwhatvalueof is this 9. Evaluate: Z (a) + =tan + Z (b) + = ln + + (c) (d) (e) (f) (g) Z Z + = =69 + = 5 6 =8 Z + =ln + + Z = + Z = +

6 Z (h) 4 ( 4 +) = (i) Z cos = (j) Z 9 4 = 8 (k) Z = (l) Z = (m) Z sin + = sin + (n) Z sin sin + = + (o) Z sin = sin (p) Z sin = 8 (q) Z +9 = 9 (r) Z 4+9 = 8

7 . Graph the function = = Determine the asmptotes (if an), intercepts (if nice), where the function is increasing/decreasing and concavit. Identif the points of etrema and in ection points (if an). LABEL ALL OF THESE IMPORTANT POINTS. (a) Horizontal asmptote is? (b) Eplain (in our own words wh there is no vertical asmptote) (c) Show that the first derivative can be simplified to = (d) Identif the points where the first derivative is zero (e) Identif the interval(s) where the function is increasing (f) Identif the interval(s) where the function is decreasing (g) Show that the second derivative can be simplified to = h = 4 i + (h) Show that the points where the second derivative is zero are approimatel: and 5 (i) Identif the interval(s) where the function is concave up (j) Identif the interval(s) where the function is concave down (k) Graph the function

8 . For the functions below: (a) Find the roots of () (b) Find the horizontal and vertical asmptotes of () (c) Determine the critical points and the inflection points of the function () (d) Identif the interval(s) where the function is increasing or decreasing. (e) Where is the function () concave up, and where is it concave down? (f) What are the relative etrema of the function ()? (g) Sketch the graph of () showing the most important features.

9 () = () = () = ( )(+) 4 () = () = () = () = () = () = () = Position function of a particle is () = 4 4 +for () = 4 () = 6 : (a) Find the position, the velocit and the accceleration at the time = () = ft, () = ft/s and () = 4 ft/s

10 (b) At what time did the particle stop? () = 4=, Solution is: = =6 seconds (c) At what time interval(s) is the particle s velocit increasing? () = () : Alwas. Find the absolute maimum and the absolute minimum of () = + on the interval [ 7] ( +) () = + : Critical points: = == Between and 7: () = =54 (Min) : (7) = 56 =467 (Ma) 4. A rectangle is to be inscribed in a right triangle having sides of length 6 in, 8 in, and in. Find the dimensions of the rectangle with greatest area assuming the rectangle is positioned as in the accompaning figure. (see below) 8 6 = 8 : = 6 (8 ) : in [ 8] 8 () = = 6 8 (8 ) =6 8 8 = 6 (8 ) :Critical point: =4; = 8 : () = ; (4) = ; (8) = : =4; = 5. A bo-shaped wire frame consists of two identical wire squares whose vertices are connected b four straight wires of equal length. If the frame is to be made from a wire of length 96 in, what should the dimensions be to obtain a bo frame of greatest volume? (see figure above) length = =8 +4 =96: = =4 where is in [ ] 4 Volume = = = (4 ) : () =4 () = 48 6 =6 (8 ) Critical points: =; =8in () = ; MAX: (8) = 5 in ; and () =

11 6. A closed clindrical can (with top and bottom) is to hold cm of liquid. How should we choose the height and the radius to minimize the amount of material needed to manufacture the can? = = : = : is in ( ) = + : Surface () = + = + () = 4 = 4 5 r 5 Critical points: =; and = =55 µ = : + = r Ãr! 5 = : 5 =9955 : µ = : + = 7. The U.S. Postal Service will accept a bo for domestic shipment onl if the sum of its length and girth (distance around) does not eceed 8 in. What dimensions will give a bo with a square end the largest possible volume? = = 8 = 4 + : = 8 4 () = (8 4) 8. Epress the number as a sum of two nonnegative numbers whose product is as large as possible. = + : = = = ( ) 9. Two nonnegative numbers multipl to 5. How small can their sum be? = 5 : = 5 = + = + 5. Find the largest possible value of +, if and are the lengths of the sides of a right triangle whose hpothenuse is 5 inches long. = + h i + = 5 : = p 5 : is in Find the ma of () = + p 5 on the interval 5 h 5 i. A conical water tank with verte down has a radius of ft at the top and is 4 ft high. If water flows into the tank at a rate of ft min, how fast is the depth of the water increasing when the water is 6 ft deep? related rates see our book/notes

12 . An open bo is to be made from a 6-inch b -inch piece of cardboard b cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain a bo with the largest volume? = (6 )( ) = for 8 () = = for = = : () = = µ µ : = 6 µ µ µ µ = 75 9 : = 8 : (8) = Answer: bo with largest volume is of the size 6 =9 b =.. A farmer has 4 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? + = 4 : = 4 = = (4 ) = 4 : () = 4 4 : = 4 = : 4 () = (75) = () (4 ()) = : rectangle is () = 4. Find the point on the line = 4 that is closest to the origin! between ( ) and ( ) = () =( ) +( ) = + = +( 4) = +( 4) () = 6 = for =6 : =(6) 4= 8 q = (6 ) +( 8 ) Find the point on the line = +6that is closest to the origin! What is the shortest distance (up to two decimal places)? between ( ) and ( ) = () =( ) +( ) = + = +( +6) = +( +6)= +4=for = 4 : =( 4) + 6 = q = ( 4 ) +( ) A piece of wire cm long is going to be cut into several pieces and used to construct the skeleton of rectangular bo with a square base. What are the dimensions of the bo with the largest volume. see previous prbs. 7. Find the largest possible value of +5, if and are the lengths of the sides of a right triangle whose hpothenuse is 5 inches long. see previous prbs

13 8. How should two nonnegative numbers be chosen so that their sum is and the sum of their squares is as small as possible? + = : = in [ ] = + = + ( ) 9. Sand pouring from a chute forms a conical pile whose height is alwas equal to the diameter. If the height increases at a constant rate of 5 ft/min, at what rate is sand pouring from the chute when the pile is 8 ft high? related rates see our book/notes. A truck driving over a flat interstate at a constant speed of 5 mph gets 4 miles per gallon. Fuel costs $.9 per gallon. The truck loses a tenth of a mile per gallon in fuel efficienc for each mile per hour increase in speed. Drivers are paid $7.5 per hour in wages and benefits. Fied costs for running the truck are $. per hour. A trip of miles is planned. What speed minimizes operating epenses? = : = : distance = = = miles at = 5mph (5) = 9 + (75 + ) 4 5 at 5 mph () = 4 (9) + ( 5) Find minimum of () for speed in interval [5 9] 57 (75 + ) = Sand is being dumped on a pile in such a wa that it alwas forms a cone whose radius equals its height. If the sand is being dumped at a rate of cubic feet per minute, at what rate is the height of the pile increasing when there is cubic feet of sand in the pile? related rates see our book/notes. A -ft plank is leaning against a wall. If a certain instant the bottom of the plank is ft from the wall and is being pushed toward the wall at the rate of 6 in/s, how fast is the acute angle that the plank makes with the ground increasing? related rates see our book/notes. A stone dropped into a still pond sends out circular ripple whose radius increases at a constant rate of ft/s. How fast is the area enclosed b the ripple increasing at the end of seconds? related rates see our book/notes 4. A baseball diamond is a square 9 ft on a side. A runner travels from home plate to first base at ft/sec. How fast is runner s distance from second base changing when the runner is halfwa to first base? related rates see our book/notes 5. A highwa patrol plane flies one mile above a straight section of rural interstate highwa (speed it 55 mph) at a stead ground speed of miles per hour. The pilot sees an oncoming car and, with radar, determines that the line-of-sight distance from the plane to the car is.5 miles and that the distance is decreasing at a rate 6 miles per hour. Should the driver of the car be given a ticket for speeding? Eplain it to the judge. related rates see our book/notes

14 6. A girl flies a kite at a height of ft, the wind carring the kite horizontall awa from her at a rate of 5 ft/sec. How fast must she let out her string when the kite is 5 feet awa from her? : + = : =5ft/s When = 5 = p + 5 = 4 = 58 ft = or = = 5 (5) = ft/s 58 Z 7. Let () = () where is the function graphed below (NOTE: The graph of is made up of straight lines and semicircles.) Evaluate ( 5), (),and (4) - see previous prbs

15 Z 8. Let () = () where is the function graphed below (NOTE: The graph of is made up of straight 5 lines and a semicircle.) Which is the smallest: ( 4) ( ) or (5)? Justif our answer. see previous prbs 9. Let () = Z lines and a semicircle.) - - () where is the function graphed below (NOTE: The graph of is made up of straight Which is the smallest: ( 5) ( ) or (5)? Justif our answer. see previous prbs 4. What is the total area between the function () = and the ais on the interval Total Area = = = = Z Z + ( ) µ + Z Z µ µ ( ) µ + µ µ =4

16 Z 4. Let () = () where is the function graphed below (NOTE: The graph of is made up of straight lines and semicircles.) 4 Evaluate ( 5), (),and (4) Z 5 Z - - µz 5 () = ( 5) = () = = 5 µ (5) ( ) + (5) () = () = = Z () = Z 5 Z () + () µ() () +()()=5 Z () + () 5 (4) = = Z 4 () = µ5 + Z () + Z µ+ () () + Z 4 () + () µ+ 4 = 4 4. A particle moves with acceleration () = along an - ais and has velocit = at time =sec. Find the displacement and the distance traveled b the particle during the interval 5 () = Z () = () = + : () = so = Z Z µ () = () = = 6 + µ µ dsiplacement is (5) () = 6 + =5 6 + = µ µ = = velocit is zero: () = = ( 4) = : =and =4: Change of direction at = 4 total distance = (5) (4) + (4) () = µ µ µ µ = = 7

17 4. A model rocket is fired verticall at ground level upward from rest. Its acceleration for the first three seconds is () =6 at which time the fuel is ehausted and it becomes a freel "falling" bod. Fourteen seconds later, the rocket s parachute opens, and the (downward) velocit slows linearl to 8 ft/s in 5 s. The rocket then "floats" to the ground at that rate. (a) Determine the position function s and the velocit function v (for all times t). Sketch the graphs of s and v. During the first seconds () =6; () = +, but since () = (from rest); So : the velocit function is () = ; at =is () = = 7 ft/s The position function is then: () = +, and given that it is fired from the ground () = so = and () = () =6 ft/s : () = ft/s : () = 8 ft/s () = ft/s : () = ft/s : () = 7 ft/s () = ft : () = ft : () = 7 ft Net fourteen seconds: from =to =7with acceleration = gravit, so we have () = ft/s and () = + so 7 = () +, so = 66, or () =66 ft/s, and at =7: (7) = 66 (7) = 78 ft/s The position function is then () = ; and given that () = 7, wehave 7 = 66 () 6 +,so, = 684 and () = Then: 7 () = ft/s : () = ft/s : (7) = ft/s () = 66 ft/s : () = 7 ft/s : (7) = 78 ft/s () = ft : () = 7 ft : (7) = 94 ft From =7to =(5s), rocket s parachute opens, and the (downward) velocit slows linearl from 78 to 8 ft/s. For 7 : (7) = 78 ft/s and () = 8 ft/s: so the slope of this linear function (derivative of velocit) is acceleration () = 8 ( 78) 7 =and () = + (a linear function). Plug it in: 78 = (7) +, Solution is: = 7 or () = 7 ft/s The position function is then () =6 7 +, and since we know that (7) = 94 we have 94 = 6 (7) 7 (7) +, Solution is: = 8564 and therefore () = () =ft/s : (7) = ft/s : () = ft/s () = 7 ft/s : (7) = 78 ft/s : () = 8 ft/s () = ft : (7) = 94 ft : () = 44 ft After =seconds, the velocit is constant () = 8 ft/s., hence () =,and () = 8 + Plug in =to get : 44 = 8 () +,so = 8 and () = () =ft/s : () = ft/s () = 8 ft/s : () = 8 ft/s () = ft : () = 44 ft

18 All together: 6 for for 7 () = for 7 for for 66 for 7 () = 7 for 7 8 for - 5 for 66 6 () = 684 for for for 5 (b) At what time does the rocket reach its maimum height? (Give our answer correct to one decimal place.) obviousl - the height is maimum when () =: (between 7) 66 =so = 8 = s 6 (c) What is that height? (Give our answer correct to the nearest whole number.) (4) = 66 (4) 6 (4) 684 = ft (d) At what time does the rocket land? (Give our answer correct to one decimal place.) rocket lands when () =(after =s), so =,so = 4 = s A projectile is launched upward from ground level with an initial speed of 56 ft/s. () = ft (ground level), () = 56 ft/s (initial velocit), and () = ft/s (gravit) () = + 56 () = (a) How long does it take for the projectile to reach its highest point? at highest point () =so + 56 = or =8s

19 (b) How high does the projectile go? (8) = (8) = 4 ft (c) When will the projectile reach the height of 88 feet? () = 88 for 88 = twice = 5(on the wa up) and =(onthewadown) (d) What is the speed of the projectile when it reaches 88 feet? (5) = 96 ft/s (Up)and () = 96 ft/s (Down) (e) What is the acceleration of the projectile when it reaches 88 feet? (5) = () = ft/s () 45. Position function of a particle is () = 4 4 +for (a) Find the position, the velocit and the accceleration at the time = () = 4 4 +: () = () = 4: () = () = 6 : () = 4 (b) At what time did the particle stop? () =for 4=or = (c) At what time interval(s) is the particle s velocit increasing? () when 46. The position (in feet) of a particle is given b = () = 6 +9,where is in seconds. (a) Find the velocit function () = +9= 4 + =( ) ( ) (b) What is the velocit after 4 s? (4) = (4) (4) + 9 = 9 ft/s (c) When is the particle at rest? () =: =sor =s (d) When is the particle moving forward (that is in the positive direction)? () ;for and for (e) Find the total distance traveled b the particle during the first five seconds? ³ ³ distance for = to =: () () = () 6() +9() () 6() +9() =4 ³ ³ distance for = to =: () () = () 6() +9() () 6() +9() = 4 ³ ³ distance for = to =: (5) () = (5) 6(5) +9(5) () 6() +9() = total distance =(4ft in positive)+(4ft in negative)+(ft positive) =8ft (f) Find the acceleration at time =4seconds. () = 6 (4) = 6 (4) = ft/s

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