MATHEMATICAL METHODS (CAS) Derivatives and rates

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1 MATHEMATICAL METHODS (CAS) Application Task 011 Derivatives and rates Teachers may select questions from this document to develop a task for their students. Teachers may choose to increase or reduce the amount of scaffolding provided to make the task more or less openended or challenging. The task should have 3 components of increasing complexity. Component 1: Constructing a graph of a function from properties of its derivative Consider x= a, x= band x c abc,, Rand a< b< c. Also consider the polynomial =, where { } function f : R R with properties that include, but are not limited to, the following: f ( a) = f ( c) = 0 f ( a) = f ( b) = 0 > for x< a f ( x) 0 the degree of the polynomial is less than 5 a. Consider a family of curves that satisfies these conditions. i. On the same set of axes, sketch the graphs of three members of a family of curves that satisfy these properties. ii. For this family of curves, write an equation for the rule of f in terms of one or more parameters, which may include, but need not be limited to, a, b and c. y= f x for this family of curves. iii. On a separate set of axes, sketch the graph of ( ) b. Consider a different family of curves that satisfies the conditions. i. For this alternative family of curves, write an equation for the rule of f in terms of one or more parameters, which may include, but need not be limited to, a, b and c. ii. Sketch the graphs of three members of the alternative family. y= f x for this alternative family of curves. iii. On a separate set of axes, sketch the graph of ( ) c. What additional information is needed to uniquely determine a particular family of curves? Explain your answer. d. What additional information is needed to uniquely determine a particular member of one of your family of curves? Explain your answer. e. Select one member of your family of curves from part (a) above. i. For your chosen curve, find k, the average rate of change of f in the interval [a, b]. f q = k. ii. For your chosen curve, find a real number, q, such that ( ) Page 1 of 14

2 Component : Rates of change Consider a particle P which is moving on a plane such that its path traces a curve that is modelled by 1 the graph of the function g: R + R, g( x) = x, where x is the horizontal distance of the particle, in metres, from the origin. The diagram below shows that, at t seconds, the coordinates of the particle are 50 t 50 t, g, where 0 t < m, and m is the least upper bound of t. 5 5 a. What are the initial coordinates of P? b. Show that the velocity of the particle parallel to the x-axis (the rate of change of the x-coordinate), dx, is 0. ms 1. c. Find the velocity of the particle parallel to the y-axis (the rate of change of the y-coordinate), dy, at t. d. Let z be the distance OP of the particle from the origin at time, t. i. Sketch the graph z against t, labelling any endpoint(s) and turning point(s) with their coordinates, and labelling any asymptote with its equation. ii. iii. What is the value m, the least upper bound of t? Explain why t cannot exceed this value. What is the minimum value of z, and what are the cartesian coordinates of P at which z is a minimum? e. Consider the tangent to the graph of g at P. ( ) i. Find the equation of the tangent at any point, ( ) Paga. ii. Find the coordinates of R and S, the points of intersection of the tangent with the x and y axes, respectively. iii. Find the area of the triangle OSR. Show that this area is constant for all values of a. iv. Let c be the length of the line segment RS. Find, correct to two decimal places, dc, the rate of change in the length of RS, when x =. Page of 14

3 Component 3: Related rates of change In an industrial process, a liquid drains from a conical filter into a cylindrical vat. The filter is designed so that the liquid will drain at a constant rate of 5 L/minute (5000 cm 3 /min or m 3 /min), as illustrated in the diagram. The cone and the cylinder have the same diameter, of 150 cm. The height of the cone is 00 cm, and the cylinder has a height of 100 cm. Note that 1 L = 1000 cm 3 = m 3 a. At the start of the process, the cone is filled to the top with liquid, and the cylinder is empty. i. Qualitatively sketch a graph of the height of water in the cone against time. On the same set of axes, sketch a graph of the height of water in the cone against time. Explain why each graph has the shape that you have sketched. ii. Correct to the nearest minute, how long will it take for all of the liquid to drain through the filter? iii. iv. What is the domain and range of each graph in part (i) above? To what fraction of its maximum capacity will the vat be filled when all of the liquid has drained into it? b. At time t minutes after the liquid has started draining, the height of the liquid in the vat is h cm and the depth of liquid in the filter is x cm. i. Find the volume of liquid in the vat at time t minutes. ii. Find the volume of liquid in the filter at time t minutes. c. When the vat contains 5π litres of liquid, what is the rate of change in the height of the liquid in the vat? d. When the vat contains 5π litres of liquid, what is the rate of change in the depth of the liquid in the filter? The cylindrical vat is replaced with a new tank, the sides of which are the surface of revolution of the x parabola with equation y= 100, a x 30, as shown in the diagram below, where x and y are 4 measured in cm. x y = Page 3 of 14

4 e. Consider the circular top and bottom of the tank. i. Find the value a, and hence find the area, A 1, of the bottom of the tank. ii. What is the height, l cm, of the tank? f. When the depth of liquid in the tank is h cm, find, in terms of h, the area, A, of the circular surface of the water. When the depth of liquid in the tank is h cm, the volume, V m 3, is given by 1 V = (( h) A 100 A1). g. Find, in terms of h, the volume of liquid in the tank when the depth of liquid in the tank is h metres. h. What is the maximum capacity of the tank, in litres? i. When liquid is draining into the tank at a rate of 5 L/min, what is the rate of change in the depth of liquid, in cm/min, correct to two decimal places, when the liquid has been draining for 5 minutes? s: next page Page 4 of 14

5 SAMPLE SOLUTIONS Component 1: Constructing a graph of a function from properties of its derivative Consider x= a, x= band x c abc,, Rand a< b< c. Also consider the polynomial =, where { } function f : R R with properties that include, but are not limited to, the following: f ( a) = f ( c) = 0 f ( a) = f ( b) = 0 > for x< a f ( x) 0 the degree of the polynomial is less than 5 a. Consider a family of curves that satisfies these conditions. i. On the same set of axes, sketch the graphs of three members of a family of curves that satisfy these properties. (many other possibilities exist) An alternative could be... ii. For this family of curves, write an equation for the rule of f in terms of one or more parameters, which may include, but need not be limited to, a, b and c. (many other possibilities exist) A possible family of curves could have the rule f ( x) = p( x a) ( x q)( x c), where p R + and a< b< q< c Or, a more specific case, the graphs shown in part (i), f ( x) = px ( x 1)( x ), p { 1,3,5 } Other possibilities could include f ( x) x ( x 1)( x c), c {,3,4} iii. On a separate set of axes, sketch the graph of y f ( x) (many other possibilities exist) = etc. = for this family of curves. Or, for the alternative b. Consider a different family of curves that satisfies the conditions. i. For this alternative family of curves, write an equation for the rule of f in terms of one or more parameters, which may include, but need not be limited to, a, b and c. (many other possibilities exist) Alternative families of curves could have the rule Page 5 of 14

6 f ( x) = p( x a) ( x q)( x c), where p R + and a< b< q< c ( ) ( ) f x = p x a ( x q) ( x c) etc. Or, a more specific case, the graphs shown below, f ( x) = px 3 ( x 1)( x ), p { 1,3,5 } Other possibilities, could include f ( x) = ( x+ a) 3 ( x 1)( x ), a {, 1,0} etc. ii. Sketch the graphs of three members of the alternative family. (many other possibilities exist) iii. On a separate set of axes, sketch the graph of y f ( x) (many other possibilities exist) = for this alternative family of curves. c. What additional information is needed to uniquely determine a particular family of curves? Explain your answer. Additional information needed to uniquely determine a particular family includes the degree of the polynomial and whether each stationary point is a maximum, minimum, or an inflection. d. What additional information is needed to uniquely determine a particular member of one of your family of curves? Explain your answer. A particular member of a family could be uniquely determine if n points on the curve are known, for a polynomial of degree n. e. Select one member of your family of curves from part (a) above. i. For your chosen curve, find k, the average rate of change of f in the interval [a, b]. Sample calculation Page 6 of 14

7 ii. For your chosen curve, find a real number, q, such that f ( q) = k. Sample calculation. In this particular case, there are 3 places where the instantaneous rate of change is equal to the average rate of change. This is an application of the mean value theorem. Component : Rates of change Consider a particle P which is moving on a plane such that its path traces a curve that is modelled by 1 the graph of the function g: R + R, g( x) = x, where x is the horizontal distance of the particle, in metres, from the origin. The diagram below shows that, at t seconds, the coordinates of the particle are 50 t 50 t, g, where 0 t < m, and m is the least upper bound of t. 5 5 a. What are the initial coordinates of P? Solution For t = 0, x = = 10 and y = =. Coordinates are 10, b. Show that the velocity of the particle parallel to the x-axis, dx, is 0. ms 1. Solution 50 t dx x= = 10 0.t. Therefore 0. 5 =, as required. Page 7 of 14

8 c. Find the velocity of the particle parallel to the y-axis (the rate of change of the y-coordinate), dy. Solution 5 y = = 550 ( t) 1. Therefore 50 t dy 5 = 5( 50 t) = 50 t ( ) d. Let z be the distance OP of the particle from the origin at time, t. i. Sketch the graph z against t, labelling any endpoint(s) and turning point(s) with their coordinates, and labelling any asymptote with its equation. Solution 50 t 5 Using Pythagoras theorem: z = x + y. Therefore z = t The graph is shown in the screen-shot. Note that when t = 0, z is undefined. The domain of the graph is t [0,50). Therefore, only the left branch of the graph, fromt = 0, should be included in the solution. Also note, z( ) = + = ii. What is the value m, the least upper bound of t? Explain why t cannot exceed this value. m = 50. t [0,50) because t > 50 would corresponds to the particle tracing a curve with 1 1 equation y =, x < 0. However, the path of the particle traces a curve y =, x > 0 only. x x iii. What is the minimum value of z, and what are the cartesian coordinates of P at which z is a minimum? The minimum value of z occurs at t = 45, when the coordinates of P are (1, 1). Using Pythagoras theorem, the minimum value of z is therefore. ( ) e. Consider the the graph of g at, ( ) Paga. i. Find, in terms a, the equation of the tangent at P. Page 8 of 14

9 dy 1 = dx x 1 At ( ag, ( a )), m = a Substitute into y y = m( x x ) y = ( x a) a a 1 y= x+, where 0 < a 10 a a ii. Find, in terms of a, the coordinates of R and S, the points of intersection of the tangent at P with the x and y axes, respectively. 1 0 = x +, therefore coordinates of R are R (,0 a ) a a 1 y = (0) +, therefore coordinates of S are S 0, a a a iii. Find the area of the triangle OSR. Show that this area is constant for all values of t. 1 A = a = a Triangle OSR has a constant area of m. This can be confirmed using dynamic geometry on CAS. iv. Let c be the length of the line segment RS. Find, correct to two decimal places, dc, the rate of change in the length of RS, when x =. c= ( a) + a, and t a =. Therefore, t 10 c = t 50 t If x =, then =. Therefore, t = dc When t = 40, 0.19 =. The rate of change is 0.19 m/s. Page 9 of 14

10 Component 3: Related rates of change In an industrial process, a liquid drains from a conical filter into a cylindrical vat. The filter is designed so that the liquid will drain at a constant rate of 5 L/minute (5000 cm 3 /min or m 3 /min), as illustrated in the diagram. The cone and the cylinder have the same diameter, of 150 cm. The height of the cone is 00 cm, and the cylinder has a height of 100 cm. Note that 1 L = 1000 cm 3 = m 3 a. At the start of the process, the cone is filled to the top with liquid, and the cylinder is empty. i. Qualitatively sketch a graph of the height of water in the cone against time. On the same set of axes, sketch a graph of the height of water in the cone against time. Explain why each graph has the shape that you have sketched. Since the volumes are changing at a constant rate, the height of liquid in the cone will be changing with an increasingly negative rate of change (i.e. decreasing faster with time), until the volume is zero. The height of liquid in the cylinder is increasing at a constant rate, from zero height, until all of the liquid has drained from the filter. ii. Correct to the nearest minute, how long will it take for all of the liquid to drain through the filter? 1 ( ) 3 3 dv V = π cm 1000cm /L and = 5L/min 3 Page 10 of 14

11 t = 1 π ( 75 ) L 5L/min t 36 min (or 3h 56 min) iii. What is the domain (correct to the nearest minute) and range of each graph in part (i) above? The domain for both graphs is 36 min. Range of height of cone, in cm is [0,00]. Range of height of cylinder, in cm is 1 π 3 00 h = 3 iv. 3 ( 75) 00 cm π ( 75) = h 00 0, 3 To what fraction of its maximum capacity will the vat be filled when all of the liquid has drained into it? The cone and cylinder both have the same radius, so if the cylinder was 00 cm tall, it would fill to 1 3 capacity. However, since it is only 100 cm tall, it will fill to of its maximum capacity. 3 b. At time t minutes after the liquid has started draining, the height of the liquid in the vat is h cm and the depth of liquid in the filter is x cm. i. Find the volume of liquid in the vat at time t minutes. V = 5t L (or V = 5000t cm 3, or V = 0.005t m 3 ), where [0, 36] mins, correct to nearest minute. ii. Find the volume of liquid in the filter at time t minutes. ( ) π V = 5 t = ( 115 π 5 t) 1000 L (or the equivalent amount in cm3 or m3) c. When the vat contains 5π litres of liquid, what is the rate of change in the height of the liquid in the vat? dh dh dv =... eqn (1) dv π ( 75) h 45π V = = h L, therefore dv 45π dh 8 = L/cm and dh 8 dv = 45π cm/l... eqn () dv = 5 L/min... eqn (3). Substituting eqns () and (3) in (1), dh 8 8 = 5 = cm/min. Note that dh is independent of h. 45π 9π d. When the vat contains 5π litres of liquid, what is the rate of change in the depth of the liquid in the filter? Page 11 of 14

12 dx dx dv =... eqn (1) dv π rx r 75 3 V = L, and by similar triangles, = = L/cm, therefore x πx 9x 3πx dv 9π x dx V = = L, therefore = L/cm, and = dx dv 9π x dv = 5 L/min... eqn (3). Substituting eqns () and (3) in (1), dh 8 8 = 5 = cm/min. Note that dh is independent of h for a cylinder. 45π 9π... eqn () The cylindrical vat is replaced with a new tank, the sides of which are the surface of revolution of the x parabola with equation y= 100, a x 30, as shown in the diagram below, where x and y are 4 measured in cm. e. Consider the circular top and bottom of the tank. i. Find the value a, and hence find the area, A 1, of the bottom of the tank. a 0= a = 0. Therefore the radius of the bottom is 0 cm. A1 = π 0 = 400π cm ii. What is the height, l cm, of the tank? 30 l = 100 = 15cm 4 f. When the depth of liquid in the tank is h cm, find, in terms of h, the area, A, of the circular surface of the water. r h = A = π 4( h+ 100) = 4π( h+ 100) Page 1 of 14

13 When the depth of liquid in the tank is h cm, the volume, V m 3, is given by 1 V = (( h) A 100 A1). g. Find, in terms of h, the volume of liquid in the tank when the depth of liquid in the tank is h metres. V = 1 (( h) 4 π( h+ 100 ) π) cm 3 1 or V = (( h) 4 π( h+ 100 ) π) 1000 L π V = ( h + 00h) h. What is the maximum capacity of the tank, in litres? 35 Maximum capacity: V = π ( ) = π L 4 i. When liquid is draining into the tank at a rate of 5 L/min, what is the rate of change in the depth of liquid, in cm/min, correct to two decimal places, when the liquid has been draining for 5 minutes? dh dh dv =... eqn (1) dv When the tank has been filling for 5 minutes, π 5 5 = ( 00 ) h + h, therefore h = 7.90 cm π dv V = ( h + 00h) L, therefore.177 dh = L/cm and dh 1 = cm/l... eqn () dv.177 dv = 5 L/min... eqn (3). Substituting eqns () and (3) in (1), dh 1 = 5=.30cm/min (or 3.0 mm/min).177 Marking scheme next page Page 13 of 14

14 APPLICATION TASK MARK ALLOCATION and CRITERIA MAPPING Note: criteria mapping, to complete the last column of this table for the task that teachers compile, will be discussed at the PD sessions. Outcome Criteria Marks available 3 Outcome 1 Define and explain key concepts as specified in the content from the areas of study, and apply a range of related mathematical routines and procedures. 15 marks 1 Appropriate use of mathematical conventions, symbols and terminology Definition and explanation of key concepts 6 Question/ Marks 3 Accurate application of mathematical skills and techniques 6 Outcome Apply mathematical processes in non-routine contexts and analyse and discuss these applications of mathematics. 0 marks 1 Identification of important information, variables and constraints Application of mathematical ideas and content from the specified areas of study Analysis and interpretation of results 8 Outcome 3 Select and appropriately use technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem solving, modelling or investigative techniques or approaches. 5 marks 1 Appropriate selection and effective use of technology Application of technology 3 TOTAL MARKS: 40 Page 14 of 14