Applications of differential calculus

Chapter 22 Applications of differential calculus Sllabus reference: 7.4, 7.5 Contents: A B C Properties of curves Rates of change Optimisation

648 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) OPENING PROBLEM A skatepark designer proposes to build a bowl with cross-section given b h() =0:014 4 0:196 3 +1:039 2 2:471 +2:225 metres, 0 6 6 7. Things to think about: a How high is the wall of the bowl? b Is the lowest point eactl in the middle of the bowl? c How steep are the sides? What units could we use to measure this? h() 7 We saw in the previous chapter how differential calculus can be used to help find the equation of a tangent to a curve. There are man other uses, however, including the following which we now consider: ² properties of curves (decreasing and increasing, stationar points) ² rates of change ² optimisation (maima and minima) A PROPERTIES OF CURVES In this section we consider some properties of curves which can be established using derivatives. These include intervals in which curves are increasing and decreasing, and the stationar points of functions. INCREASING AND DECREASING INTERVALS The concepts of increasing and decreasing are closel linked to intervals of a function s domain. Some eamples of intervals and their graphical representations are: Algebraic form Geometric form > 2 >2 2 2 6 4 <4 2 6 6 4 2 6 <4 4 4 2 4 2 4

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 649 On an interval which is increasing, an increase in produces an increase in. On an interval which is decreasing, an increase in produces a decrease in. increase in decrease in increase in increase in Suppose S is an interval in the domain of f(), sof() is defined for all in S. ² f() is increasing on S if f(a) <f(b) for all a, b 2 S such that a<b. ² f() is decreasing on S if f(a) >f(b) for all a, b 2 S such that a<b. For eample: =2 = 2 is decreasing for 6 0 and increasing for > 0. Important: People often get confused about the point =0. The wonder how the curve can be both increasing and decreasing at the same point when it is clear that the tangent is horizontal. The answer is that increasing and decreasing are associated with intervals, not particular values for. We must clearl state that = 2 is decreasing on the interval 6 0 and increasing on the interval > 0. We can deduce when a curve is increasing or decreasing b considering f 0 () on the interval in question. For the functions that we deal with in this course: ² f() is increasing on S if f 0 () > 0 for all in S. ² f() is strictl increasing if f 0 () > 0 for all in S. ² f() is decreasing on S if f 0 () 6 0 for all in S. ² f() is strictl decreasing if f 0 () < 0 for all in S. INCREASING AND DECREASING FUNCTIONS Man functions are either increasing or decreasing for all 2 R. We sa these are either increasing functions or decreasing functions.

650 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) For eample: =2 =3-1 1 =2 is increasing for all. =3 is decreasing for all. Eample 1 Find intervals where f() is: a increasing b decreasing. (-1, 3) = f() Self Tutor (2,-4) a f() is increasing for 6 1 and for > 2 since f 0 () > 0 on these intervals. b f() is decreasing for 1 6 6 2. (-1, 3) = f() (2,-4) Sign diagrams for the derivative are etremel useful for determining intervals where a function is increasing or decreasing. The critical values for f 0 () are the values of for which f 0 () =0 or f 0 () is undefined. When f 0 () =0, the critical values are shown on a number line using tick marks. When f 0 () is undefined, the critical values are shown with a vertical dotted line. We complete the sign diagram b marking positive or negative signs, depending on whether f 0 () is positive or negative, in the intervals between the critical values. Consider the following eamples: ² f() = 2 f 0 () =2 which has sign diagram - + 0 decreasing increasing DEMO ) f() = 2 is decreasing for 6 0 and increasing for > 0.

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 651 ² f() = 2 f 0 () = 2 which has sign diagram DEMO + - increasing 0 decreasing ) f() = 2 is increasing for 6 0 and decreasing for > 0. ² f() = 3 f 0 () =3 2 which has sign diagram DEMO + + 0 increasing for all ( never negative) ) f() is increasing for all. ² f() = 3 3 +4 f 0 () =3 2 3 =3( 2 1) =3( + 1)( 1) which has sign diagram 4 DEMO + - -1 1-1 1 + increasing decreasing increasing ) f() is increasing for 6 1 and for > 1, and decreasing for 1 6 6 1. Eample 2 Self Tutor Find the intervals where f() = 3 +3 2 +5 is increasing or decreasing. f() = 3 +3 2 +5 ) f 0 () = 3 2 +6 ) f 0 () = 3( 2) f 0 () =0 when =0or 2 So f 0 () has sign diagram - + - 0 2 So, f() is decreasing for 6 0 and for > 2, and increasing for 0 6 6 2. Remember that f() must be defined for all on an interval before we can classif the interval as increasing or decreasing. We must eclude points where a function is undefined, and need to take care with vertical asmptotes.

652 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) EXERCISE 22A.1 1 Find intervals where the graphed function is: i increasing ii decreasing: a b c 3 (-2, 2) (3,-1) -2 2 d (2, 3) e =3 f g h =4 i (5, 2) (6, 5) (1, -1) (2, 4) (2, -2) 2 Find intervals where f() is increasing or decreasing: a f() =2 +1 b f() = 3 +2 c f() = 2 d f() = 3 e f() =2 2 +3 4 f f() = 3 6 2 g f() = 2 3 +4 h f() = 4 3 +15 2 +18 +3 i f() =2 3 +9 2 +6 7 j f() = 3 6 2 +3 1 STATIONARY POINTS A stationar point of a function is a point such that f 0 () =0. It could be a local maimum, local minimum, or horizontal inflection. TURNING POINTS (MAXIMA AND MINIMA) Consider the following graph which has a restricted domain of 5 6 6 6. = f() B(-2, 4) -2 2 C(2,-4) D(6, 18) A(-5,-16 _Qw )

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 653 Aisaglobal minimum as it is the minimum value of on the entire domain. Bisalocal maimum as it is a turning point where the curve has shape and f 0 () =0 at that point. Cisalocal minimum as it is a turning point where the curve has shape and f 0 () =0 at that point. Disaglobal maimum as it is the maimum value of on the entire domain. For man functions, a local maimum or minimum is also the global maimum or minimum. For eample, for = 2 the point (0, 0) is a local minimum and is also the global minimum. HORIZONTAL OR STATIONARY POINTS OF INFLECTION It is not alwas true that whenever we find a value of where f 0 () =0 we have a local maimum or minimum. For eample, f() = 3 has f 0 () =3 2 and f 0 () =0 when =0: The -ais is a tangent to the curve which actuall crosses over the curve at O(0, 0). This tangent is horizontal but O(0, 0) is neither a local maimum nor a local minimum. It is called a horizontal inflection (or infleion) as the curve changes its curvature or shape. SIGN DIAGRAMS horizontal Consider the graph alongside. local maimum inflection 1 The sign diagram of its gradient function is shown directl beneath it. We can use the sign diagram to describe the stationar points of the function. -2 local minimum + - + + -2 1 3 local local maimum minimum horizontal inflection 3 Stationar point Sign diagram of f 0 () near = a Shape of curve near = a local maimum + - a =a local minimum - + a =a horizontal inflection or stationar inflection + + - - a or a =a or =a

654 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) Eample 3 Self Tutor Find and classif all stationar points of f() = 3 3 2 9 +5: f() = 3 3 2 9 +5 ) f 0 () =3 2 6 9 =3( 2 2 3) =3( 3)( +1) which has sign diagram: + - -1 3 + So, we have a local maimum at = 1 and a local minimum at =3. f( 1) = ( 1) 3 3( 1) 2 9( 1) + 5 =10 f(3) = 3 3 3 3 2 9 3+5 = 22 There is a local maimum at ( 1, 10). There is a local minimum at (3, 22). If we are asked to find the greatest or least value on an interval, then we should alwas check the endpoints. We seek the global maimum or minimum on the given domain. Eample 4 Self Tutor Find the greatest and least value of 3 6 2 +5 on the interval 2 6 6 5. First we graph = 3 6 2 +5 on 2 6 6 5. In this case the greatest value is clearl at the local maimum when d d =0. Now d d =32 12 =3( 4) d and =0 when =0 or 4: d So, the greatest value is f(0)=5 when =0. The least value is either f( 2) or f(4), whichever is smaller. Now f( 2) = 27 and f(4) = 27 ) least value is 27 when = 2 and when =4.

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 655 EXERCISE 22A.2 1 The tangents at points A, B and C are horizontal. a Classif points A, B and C. b Draw a sign diagram for the gradient function f 0 () for all. c State intervals where = f() is: i increasing ii decreasing. d Draw a sign diagram for f() for all. e A(-2, 8) -4 5 B Comment on the differences between the sign diagrams found above. C(3, -11) 2 Consider the quadratic function f() =2 2 5 +1. a Use quadratic theor to find the equation of the ais of smmetr. b Find f 0 () and hence find such that f 0 () =0. Eplain our result. = f() 3 For each of the following functions, find and classif the stationar points, and hence sketch the function showing all important features. a f() = 2 2 b f() = 3 +1 c f() = 3 3 +2 d f() = 4 2 2 e f() = 3 6 2 +12 +1 f f() =4 3 4 At what value of does the quadratic function f() =a 2 + b + c, a 6= 0, have a stationar point? Under what conditions is the stationar point a local maimum or a local minimum? 5 f() =2 3 + a 2 24 +1 has a local maimum at = 4. Find a: 6 f() = 3 + a + b has a stationar point at ( 2, 3). Find: a a and b b the position and nature of all stationar points. 7 Find the greatest and least value of: a 3 12 2 for 3 6 6 5 b 4 3 2 + 3 for 2 6 6 3 8 A manufacturing compan makes door hinges. The have a standing order filled b producing 50 each hour, but production of more than 150 per hour is useless as the will not sell. The cost function for making hinges per hour is: C() =0:0007 3 0:1796 2 +14:663 + 160 dollars where 50 6 6 150. Find the minimum and maimum hourl costs, and the production levels when each occurs. B When we first introduced derivative functions, we discussed how RATES OF CHANGE d d gives the rate of change in with respect to. If increases as increases, then d d will be positive. If decreases as increases, then d d will be negative.

656 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) TIME RATES OF CHANGE There are countless quantities in the real world that var with time. For eample: ² temperature varies continuousl ² the height of a tree varies as it grows ² the prices of stocks and shares var with each da s trading. Varing quantities can be modelled using functions of time. For eample, we could use: ² s(t) to model the distance travelled b a runner ds dt or s 0 (t) is the instantaneous speed of the runner. It might have units metres per second or m s 1. ² H(t) to model the height of a person riding in a Ferris wheel dh or H 0 (t) is the instantaneous rate of ascent of the person in the Ferris wheel. dt It might also have units metres per second or m s 1. ² C(t) to model the capacit of a person s lungs, which changes when the person breathes. dc dt or C 0 (t) is the person s instantaneous rate of change in lung capacit. It might have units litres per second or L s 1. EXERCISE 22B.1 1 Find: a dm dt if M = t 3 3t 2 +1 b dr dt if R =(2t +1) 2 2 a If A is measured in cm 2 and t is measured in seconds, what are the units for b If V is measured in m 3 and t is measured in minutes, what are the units for da dt? dv dt? Eample 5 The volume of air in a hot air balloon after t minutes is given b V =2t 3 3t 2 +10t +2 m 3 where 0 6 t 6 8. Find: a the initial volume of air in the balloon b the volume when t =8 minutes c dv dt d the rate of increase in volume when t =4 minutes. Self Tutor

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 657 a When t =0, V (0)=2m 3 So, initiall there was 2 m 3 of air in the balloon. b When t =8, V (8) = 2(8) 3 3(8) 2 + 10(8) + 2 = 914 m 3 in the balloon. c After 8 minutes there was 914 m 3 in the balloon. dv dt =6t2 6t +10 m 3 min 1 d When t =4, dv dt = 6(4)2 6(4) + 10 =82m 3 min 1 Since dv > 0, V is increasing. dt Hence, V is increasing at 82 m 3 min 1 when t =4. 3 The number of bacteria in a dish is modelled b B(t) =0:3t 2 +30t + 150 thousand, where t is in das, and 0 6 t 6 10. a Find B 0 (t) and state its meaning. b Find B 0 (3) and state its meaning. c Eplain wh B(t) is increasing over the first 10 das. 4 The estimated future profits of a small business are given b P (t) =2t 2 12t + 118 thousand dollars, where t is the time in ears from now. a What is the current annual profit? b Find dp and state its units. dt dp c What is the significance of dt? d For what values of t will the profit: i decrease ii increase on the previous ear? e What is the minimum profit and when does it occur? f Find dp when t = 4, 10 and 25. What do these figures represent? dt 5 Water is draining from a swimming pool such that the remaining volume of water after t minutes is V = 200(50 t) 2 m 3. Find: a the average rate at which the water leaves the pool in the first 5 minutes b the instantaneous rate at which the water is leaving at t =5minutes. 6 When a ball is thrown, its height above the ground is given b s(t) =1:2+28:1t 4:9t 2 metres, where t is the time in seconds. a From what distance above the ground was the ball released? b Find s 0 (t) and state what it represents. c Find t when s 0 (t) =0. What is the significance of this result? d What is the maimum height reached b the ball?

658 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) e Find the ball s speed: i when released ii at t =2s iii at t =5s. State the significance of the sign of the derivative at these values. f How long will it take for the ball to hit the ground? 7 The height of a palm tree is given b H =20 18 metres, where t is the number of t ears after the tree was planted from an established potted juvenile tree, t > 1: a b How high was the palm after 1 ear? Find the height of the palm at t =2, 3, 5, 10, and 50 ears. c Find dh dt and state its units. d At what rate is the tree growing at t =1, 3, and 10 ears? e Eplain wh dh dt growth? > 0 for all t > 1. What does this mean in terms of the tree s GENERAL RATES OF CHANGE Other rate problems can be treated in the same wa as those involving time. However, we must alwas pa careful attention to the units of the quantities involved. For eample: ² the cost of manufacturing items has a cost function C() dollars associated with it. dc d or C0 () is the instantaneous rate of change in cost with respect to the number of items made. In this case dc has the units dollars per item. d ² the profit P () in making and selling items is given b P () =R() C() where R() is the revenue function and C() is the cost function. dp d or P 0 () represents the rate of change in profit with respect to the number of items sold. Eample 6 The cost of producing items in a factor each da is given b a b C() =0:000 13 3 {z +0:002 2 } + 5 + 2200 cost of labour raw material costs fied or overhead costs such as heating, cooling, maintenance, rent Find C 0 (). Find C 0 (150). Interpret this result. c Find C(151) C(150). Compare this with the answer in b. Self Tutor

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 659 a C 0 () =0:000 39 2 +0:004 +5 b C 0 (150) = $14:38 This is the rate at which the costs are increasing with respect to the production level when 150 items are made per da. c C(151) C(150) ¼ $3448:19 $3433:75 ¼ $14:44 This is the actual cost of making the 151st item each week, and is similar to the answer from b. tangent ( b answer) chord ( c answer) 1 150 151 C(151) C(150) EXERCISE 22B.2 1 Find: a dt dr if T = r 2 100 r b da dh if A =2¼h + 1 4 h2 2 If C is measured in pounds and is the number of items produced, what are the units dc for d? 3 The total cost of running a train from Paris to Marseille is given b C(v) = 1 200 000 5 v2 + euros where v is the average speed of the train in km h 1. v a Find the total cost of the journe if the average speed is: i 50 km h 1 ii 100 km h 1. b Find the rate of change in the cost of running the train at speeds of: i 30 km h 1 ii 90 km h 1. 4 The cost function for producing items each da is C() =0:000 072 3 0:000 61 2 +0:19 + 893 dollars. a Find C 0 () and eplain what it represents. b Find C 0 (300) and eplain what it estimates. c Find the actual cost of producing the 301st item. 5 Seablue make denim jeans. The cost model for making pairs per da is C() =0:0003 3 +0:02 2 +4 + 2250 dollars. a Find C 0 (): b Find C 0 (220). What does it estimate? c Find C(221) C(220). What does this represent? d Find C 00 () and the value of when C 00 () =0. What is the significance of this value?

660 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 6 Alongside is a land and sea profile where sea hill lake the -ais is sea level. The function = 1 10( 2)( 3) km gives the height of the land or sea bed relative to sea level. a b c Find where the lake is located relative to the shore line of the sea. Find d d and interpret its value when = 1 2 and when =11 2 km. Find the point at which the lake floor is level, and the depth at this point. 7 The resistance to the flow of electricit in a metal is given b R =20+ 1 10 T 1 200 T 2 ohms, where T is the temperature in o C of the metal. a Find the resistance R, at temperatures of 0 o C, 20 o C and 40 o C. b Find the rate of change in the resistance at an temperature T. c For what values of T does the resistance increase as the temperature increases? 8 The profit made in selling items is given b P () =5 2000 2 10 000 dollars. a Graph P () using technolog and determine the sales levels which produce a profit. b Find P 0 () and hence find such that the profit is increasing. 9 The cost of producing items is given b C() =0:002 3 +0:04 2 +10 + 3000 dollars. If each item sells for $30, find: a the revenue function R() b the profit function P () c P 0 () d P 0 (50) and eplain the significance of this result. C OPTIMISATION There are man problems for which we need to find the maimum or minimum value for a function. We can solve such problems using differential calculus techniques. The solution is often referred to as the optimum solution and the process is called optimisation. WARNING The maimum or minimum value does not alwas occur when the first derivative is zero. It is essential to also eamine the values of the function at the endpoint(s) of the domain for global maima and minima.

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 661 For eample: d d =0 d d =0 = f() =a =p =b The maimum value of occurs at the endpoint = b. The minimum value of occurs at the local minimum = p. TESTING OPTIMAL SOLUTIONS If one is tring to optimise a function f() and we find values of such that f 0 () =0, there are several tests we can use to see whether we have a maimum or a minimum solution: SIGN DIAGRAM TEST If near to = a where f 0 (a) =0 the sign diagram of f 0 () is: ² + - a we have a local maimum ² - we have a local minimum. a + GRAPHICAL TEST If the graph of = f() shows: ² we have a local maimum ² we have a local minimum. OPTIMISATION PROBLEM SOLVING METHOD Step 1: Step 2: Step 3: Draw a large, clear diagram of the situation. Construct a formula with the variable to be optimised (maimised or minimised) as the subject. It should be written in terms of one convenient variable, sa. You should write down what restrictions there are on. Find the first derivative and find the values of when it is zero. Step 4: If there is a restricted domain such as a 6 6 b, the maimum or minimum ma occur either when the derivative is zero or else at an endpoint. Show using the sign diagram test or the graphical test, that ou have a maimum or a minimum situation. Use calculus techniques to answer the following problems. In cases where finding the zeros of the derivatives is difficult ou ma use a graphics calculator or graphing package to help ou. GRAPHING PACKAGE

662 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) EXERCISE 22C 1 The cost of making tennis racquets each da is given b C() = 2 20 + 120 dollars per racquet. How man racquets should be made per da to minimise the cost per racquet? 2 When a stone is thrown verticall upwards its height above the ground is given b h(t) =49t 9:8t 2 metres. Find the maimum height reached. 3 A small business which emplos workers earns profit given b P () = 3 + 300 + 1000 pounds. How man workers should be emploed to maimise the profit? 4 A manufacturer can produce fittings per da where 0 6 6 10 000. The production costs are: ² E1000 per da for the workers ² E2 per da per fitting ² E 5000 per da for running costs and maintenance. How man fittings should be produced dail to minimise costs? 5 For the cost function C() = 720 + 4 +0:02 2 dollars and revenue function R() =15 0:002 2 dollars, find the production level that will maimise profits. 1 6 The total cost of producing blankets per da is 4 2 +8 +20 dollars, and for this production level each blanket ma be sold for (23 1 2) dollars. How man blankets should be produced per da to maimise the total profit? 7 The cost of running a boat is $ v2 10 per hour where v km h 1 is the speed of the boat. All other costs amount to $62:50 per hour. Find the speed which will minimise the total cost per kilometre. Eample 7 A 4 litre container must have a square base, vertical sides, and an open top. Find the most economical shape which minimises the surface area of material needed. open Self Tutor Step 1: cm cm cm Let the base lengths be cm and the depth be cm. The volume V = length width depth ) V = 2 ) 4000 = 2... (1) fas 1 litre 1000 cm 3 g

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 663 Step 2: The total surface area A = area of base +4(area of one side) = 2 +4 µ 4000 = 2 +4 fusing (1)g 2 ) A() = 2 + 16 000 1 where >0 Step 3: A 0 () =2 16 000 2 Step 4: ) A 0 () =0 when 2 = Sign diagram test If =10, A 0 16 000 (10) = 20 100 =20 160 = 140 16 000 2 ) 2 3 = 16 000 The minimum material is used to make the container when =20 and = 4000 20 2 =10. 0 ) = 3p 8000 = 20-20 + If =30, A 0 16 000 (30) = 60 900 ¼ 60 17:8 ¼ 42:2 So, 20 cm 10 cm 20 cm is the most economical shape. 8 9 An open rectangular bo has a square base and a fied inner surface area of 108 cm 2. a Eplain wh 2 108 2 +4 = 108. b Hence show that = : 4 c Find a formula for the capacit C of the container, in terms of onl. d Find dc. Hence find when dc d d =0: e What size must the base be in order to maimise the capacit of the bo? Radioactive waste is to be disposed of in full enclosed lead boes of inner volume 200 cm 3. The base of the bo has dimensions in the ratio 2:1. h cm a What is the inner length of the bo? b Eplain wh 2 h = 100. cm

664 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) c Eplain wh the inner surface area of the bo is given b A() =4 2 + 600 cm2. d Use technolog to help sketch the graph of =4 2 + 600 : e Find the minimum inner surface area of the bo and the corresponding value of. f Sketch the optimum bo shape showing all dimensions. 10 Consider the manufacture of clindrical tin cans of 1 L capacit where the cost of the metal used is to be minimised. This means that the surface area must be as small as possible. a Eplain wh the height h is given b h = 1000 ¼r 2 cm. r cm b Show that the total surface area A is given b A =2¼r 2 + 2000 h cm cm 2. r c Use technolog to help ou sketch the graph of A against r. d Find the value of r which makes A as small as possible. e Sketch the can of smallest surface area. Eample 8 A square sheet of metal 12 cm 12 cm has smaller squares cut from its corners as shown. What sized square should be cut out so that when the sheet is bent into an open bo it will hold the maimum amount of liquid? 12 cm Self Tutor 12 cm cm (12-2) cm Let cm b cm squares be cut out. Volume = length width depth = (12 2) (12 2) = (12 2) 2 = (144 48 +4 2 ) =4 3 48 2 + 144 ) V 0 () =12 2 96 + 144 = 12( 2 8 + 12) = 12( 2)( 6) ) V 0 () =0 when =2 or 6 However, 12 2 must be > 0 and so <6 ) =2 is the onl value in 0 <<6 with V 0 () =0. Conclusion: + - 2 0 The resulting container has maimum capacit when 2 cm 2 cm squares are cut from its corners. 6

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 665 11 Sam has sheets of metal which are 36 cm b 36 cm square. He wants to cut out identical squares which are cm b cm from the corners of each sheet. He will then bend the sheets along the dashed lines to form an open container. a Show that the capacit of the container is given b V () =(36 2) 2 cm 3. b What sized squares should be cut out to produce the container of greatest capacit? 36 cm 36 cm 12 An athletics track has two straights of length l m and two semi-circular ends of radius m. The perimeter of the track is 400 m. a Show that l = 200 ¼ and hence write down the possible values that ma have. b Show that the area inside the track is A = 400 ¼ 2. c What values of l and produce the largest area inside the track? 13 Answer the Opening Problem on page 648. m l m 14 A beam with rectangular cross-section is to be cut from a log of diameter 1 m. The strength of the beam is given b S = kwd 2, where depth w is the width and d is the depth. a Show that d 2 =1 w 2 using the rule of Pthagoras. b Find the dimensions of the strongest beam that can width be cut. 15 A water tank has the dimensions shown. The capacit of the tank is 300 kl. a Eplain wh 2 = 100. b Use a to find in terms of. c Show that the area of plastic used to make the tank is given b A =3 2 + 800 1 m 2. d Find the value of which minimises the surface area A. e Sketch the tank, showing the dimensions which minimise A. m open top 3 m m

666 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) REVIEW SET 22A 1 Find intervals where the graphed function is increasing or decreasing. a b c =-3 =4 (-1, 3) 10 (4, -7) 6 2 Consider the function f() = 3 3. a b c Determine the -intercept. Find the coordinates of an local maima, local minima or horizontal inflections. Hence, sketch the graph of the function. 3 Consider the function f() = 3 4 2 +4. a Find all aes intercepts. b Find and classif all stationar points. c Sketch the graph of = f() showing features from a and b. 4 An open bo is made b cutting squares out of the corners of a 24 cm b 24 cm square sheet of tinplate. What size squares should be removed to maimise the volume of the bo? 5 A factor makes thousand chopsticks per da with a cost of C() =0:4 2 +1:6 + 150 dollars. Packs of 1000 chopsticks sell for $28. Find the production level that maimises dail profit, and the profit at that production level. REVIEW SET 22B 1 Find the maimum and minimum values of 3 3 2 +5 for 1 6 6 4. 2 Consider the function f() =2 3 3 2 36 +7. a Find and classif all stationar points. b Find intervals where the function is increasing and decreasing. c Sketch the graph of = f(), showing all important features. 3 An astronaut standing on the moon throws a ball into the air. The ball s height above the surface of the moon is given b H(t) =1:5+19t 0:8t 2 metres, where t is the time in seconds. a Find H 0 (t) and state its units. b Calculate H 0 (0), H 0 (10) and H 0 (20). Interpret these values, including their sign. c How long is the ball in the air for?

APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 667 4 For the function shown opposite, draw a sign diagram of: a f() b f 0 (). = f() -7 5 (-1, -9) 5 end view A rectangular gutter is formed b bending a 24 cm wide sheet of metal as illustrated. Where must the bends be 24 cm made to maimise the capacit of the gutter? 6 A 200 m fence is to be placed around a lawn which has the shape of a rectangle with a semi-circle on one of its sides and dimensions as shown. a Find the epression for the perimeter of the lawn in terms of r and. b Find in terms of r. c Show that the area of the lawn A can be written as A = 200r r 2 2+ ¼ 2. d Find the dimensions of the lawn which maimise the area of the lawn. r REVIEW SET 22C 1 f() = 3 + A + B has a stationar point at (1, 5). a Find A and B. b Find the nature of all the stationar points of f(). 2 Consider the function f() = 3 + 2 +2 4. a State the -intercept. b State where the function is increasing or decreasing. c Find the positions and nature of an stationar points. d Sketch the cubic, showing the features found in a, b and c. 3 The cost per hour of running a barge up the Rhein is given b C(v) =10v + 90 v euros, where v is the average speed of the barge. a Find the cost of running the barge for: i two hours at 15 km h 1 ii 5 hours at 24 km h 1. b Find the rate of change in the cost of running the barge at speeds of: i 10 km h 1 ii 6 km h 1. c At what speed will the cost be a minimum?

668 APPLICATIONS OF DIFFERENTIAL CALCULUS (Chapter 22) 4 A manufacturer of open steel boes has to make one with a square base and a volume of 1 m 3. The steel costs $2 per square metre. a If the base measures mb m and the height is m, find in terms of. b Hence, show that the total cost of the steel is C() =2 2 + 8 dollars. c Find the dimensions which minimise the cost of the bo. m open m 5 The cost of running an advertising campaign for das is C() = 6900 + 950 98 000 pounds. Research shows that after das, 7500 people will have responded, bringing an average profit of $17 per person. a Show that the profit from running the campaign for das is 1 666 000 P () = 120 600 950 pounds. b Find how long the campaign should last to maimise profit. 6 Find the maimum and minimum values of = 1 3 3 + 2 3 for 4 6 6 4.