e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)

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1 Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion. (Total 7 marks). A television screen, BC, of height one metre, is built into a wall. The bottom of the television screen at B is one metre above an observer s eye level. The angles of elevation ( Aˆ OC, Aˆ OB ) from the observer s eye at O to the top and bottom of the television screen are and radians respectively. The horizontal distance from the observer s eye to the wall containing the television screen is metres. The observer s angle of vision ( Bˆ OC ) is radians, as shown below. 1 (i) Show that = arctan arctan. Hence, or otherwise, find the eact value of for which is a maimum and justify that this value of gives the maimum value of. (iii) Find the maimum value of. Find where the observer should stand so that the angle of vision is 15. (17) (Total marks). Sketch and label the curves y = for, and y = 1 ln for 0 <. () Find the -coordinate of P, the point of intersection of the two curves. If the tangents to the curves at P meet the y-ais at Q and R, calculate the area of the triangle PQR. () (6) (d) Prove that the two tangents at the points where = a, a > 0, on each curve are always perpendicular. (Total 14 marks) 4. Let f () = ln 5, 0. 5 < <, a, b; (a, b are values of for which f () is not defined). (i) Sketch the graph of f (), indicating on your sketch the number of zeros of f (). Show also the position of any asymptotes. Find all the zeros of f (), (that is, solve f () = 0). () Find the eact values of a and b. 1

2 Chapter 0 Application of differential calculus 014 Find f (), and indicate clearly where f () is not defined. (d) Find the eact value of the -coordinate of the local maimum of f (), for 0 < < 1.5. (You may assume that there is no point of infleion.) (e) Write down the definite integral that represents the area of the region enclosed by f () and the -ais. (Do not evaluate the integral.) () (Total 16 marks) 5. Let f ( ) ( 1), Sketch the graph of f (). (An eact scale diagram is not required.) On your graph indicate the approimate position of (i) (iii) each zero; each maimum point; each minimum point. (i) Find f (), clearly stating its domain. Find the -coordinates of the maimum and minimum points of f (), for 1 < < 1. (7) Find the -coordinate of the point of infleion of f (), where > 0, giving your answer correct to four decimal places. () (Total 1 marks) 6. The point B(a, b) is on the curve f () = such that B is the point which is closest to A(6, 0). Calculate the value of a. 7. A rectangle is drawn so that its lower vertices are on the -ais and its upper vertices are on the No GDC! curve y = e. The area of this rectangle is denoted by A. Write down an epression for A in terms of. Find the maimum value of A. 8. A particle moves along a straight line. When it is a distance s from a fied point, where s > 1,

3 Chapter 0 Application of differential calculus 014 (s ) the velocity v is given by v =. (s 1) Find the acceleration when s =. (Total 4 marks) 9. Given that y = e find d y ; d the eact values of the -coordinates of the points of infleion on the graph of y = justifying that they are points of infleion. e, 10. A closed cylindrical can has a volume of 500 cm. The height of the can is h cm and the radius of the base is r cm. Find an epression for the total surface area A of the can, in terms of r. Given that there is a minimum value of A for r > 0, find this value of r. 11. A normal to the graph of y = arctan ( 1), for 0, has equation y = + c, where c. Find the value of c. 1. A family of cubic functions is defined as f k () = k k +, k +. Epress in terms of k (i) f k () and f k (); the coordinates of the points of infleion P k on the graphs of f k. Show that all P k lie on a straight line and state its equation. (6) () Show that for all values of k, the tangents to the graphs of f k at P k are parallel, and find the equation of the tangent lines. (Total 1 marks) 1. André wants to get from point A located in the sea to point Y located on a straight stretch of beach. P is the point on the beach nearest to A such that AP = km and PY = km. He does this by swimming in a straight line to a point Q located on the beach and then running to Y.

4 Chapter 0 Application of differential calculus 014 When André swims he covers 1 km in 5 5 minutes. When he runs he covers 1 km in 5 minutes. If PQ = km, 0, find an epression for the time T minutes taken by André to reach point Y. dt 5 5 Show that 5. d 4 dt (i) Solve 0. d Use the value of found in part (i) to determine the time, T minutes, taken for André to reach point Y. (iii) Show that minimum. d T 0 5 d 4 and hence show that the time found in part is a (11) (Total 18 marks) 14. The radius and height of a cylinder are both equal to cm. The curved surface area of the cylinder is increasing at a constant rate of 10 cm /sec. When =, find the rate of change of the radius of the cylinder; the volume of the cylinder. 15. Car A is travelling on a straight east-west road in a westerly direction at 60 km h 1. Car B is travelling on a straight north-south road in a northerly direction at 70 km h 1. The roads intersect at the point O. When Car A is km east of O, and Car B is y km south of O, the distance between the cars is z km. Find the rate of change of z when Car A is 0.8 km east of O and Car B is 0.6 km south of O. 16. The following diagram shows the points A and B on the circumference of a circle, centre O, and radius 4 cm, where A ÔB =. Points A and B are moving on the circumference so that is increasing at a constant rate. 4

5 Chapter 0 Application of differential calculus 014 O A B Given that the rate of change of the length of the minor arc AB is numerically equal to the rate of change of the area of the shaded segment, find the acute value of. 17. The volume of a solid is given by 4 V = πr πr h. At the time when the radius is cm, the volume is 1 cm, the radius is changing at a rate of cm/min and the volume is changing at a rate of 04 cm /min. Find the rate of change of the height at this time. 18. A man PF is standing on horizontal ground at F at a distance from the bottom of a vertical wall GE. He looks at the picture AB on the wall. The angle BPA is. E B A P D F G Let DA = a, DB = b, where angle Pˆ DE is a right angle. Find the value of for which tan is a maimum, giving your answer in terms of a and b. Justify that this value of does give a maimum value of tan. (Total 9 marks) 19. The curve y = + 4 has a local maimum point at P and a local minimum point at 5

6 Chapter 0 Application of differential calculus 014 Q. Determine the equation of the straight line passing through P and Q, in the form a + by + c = 0, where a, b, c. 0. The diagram shows a trapezium OABC in which OA is parallel to CB. O is the centre of a circle radius r cm. A, B and C are on its circumference. Angle O ĈB = θ. O r A C B Let T denote the area of the trapezium OABC. Show that T = r (sin θ + sin θ). For a fied value of r, the value of T varies as the value of θ varies. Show that T takes its maimum value when θ satisfies the equation 4 cos θ + cos θ = 0, and verify that this value of T is a maimum. Given that the perimeter of the trapezium is 75 cm, find the maimum value of T. (6) (Total 15 marks) 1. Give eact answers in this part of the question. The temperature g (t) at time t of a given point of a heated iron rod is given by g (t) = ln t, where t > 0. t Find the interval where g (t) > 0. Find the interval where g (t) > 0 and the interval where g (t) < 0. (d) Find the value of t where the graph of g (t) has a point of infleion. Let t* be a value of t for which g (t*) = 0 and g (t*) < 0. Find t*. 6

7 Chapter 0 Application of differential calculus 014 (e) Find the point where the normal to the graph of g (t) at the point (t*, g (t*)) meets the t-ais. (Total 18 marks). A curve has equation y = 8. Find the equation of the normal to the curve at the point (, 1).. The following diagram shows an isosceles triangle ABC with AB = 10 cm and AC = BC. The verte C is moving in a direction perpendicular to (AB) with speed cm per second. C Calculate the rate of increase of the angle A B C ÂB at the moment the triangle is equilateral. 4. An airplane is flying at a constant speed at a constant altitude of km in a straight line that will take it directly over an observer at ground level. At a given instant the observer notes that the 1 1 angle is radians and is increasing at radians per second. Find the speed, in 60 kilometres per hour, at which the airplane is moving towards the observer. Airplane km Observer 5. The function f is defined by f () =, for > 0. (i) Show that f () = ln Obtain an epression for f (), simplifying your answer as far as possible. (i) Find the eact value of satisfying the equation f () = 0 Show that this value gives a maimum value for f (). 7

8 Chapter 0 Application of differential calculus 014 Find the -coordinates of the two points of infleion on the graph of f. (Total 1 marks) 6. Air is pumped into a spherical ball which epands at a rate of 8 cm per second (8 cm s 1 ). Find the eact rate of increase of the radius of the ball when the radius is cm. 7. Find the -coordinate of the point of infleion on the graph of y = e, A rectangle is drawn so that its lower vertices are on the -ais and its upper vertices are on the curve y = sin, where 0 n. Write down an epression for the area of the rectangle. Find the maimum area of the rectangle. (Total marks) 9. Let f : e sin. Find f (). There is a point of infleion on the graph of f, for 0 < < 1. Write down, but do not solve, an equation in terms of, that would allow you to find the value of at this point of infleion. (Total marks) 0. Consider the function f ( ), where n Show that the derivative f ( ) f ( ). Sketch the function f (), showing clearly the local maimum of the function and its horizontal asymptote. You may use the fact that 1n lim 0. Find the Taylor epansion of f () about = e, up to the second degree term, and show that this polynomial has the same maimum value as f () itself. (Total 1 marks) 1. A curve has equation y + y =. Find the equation of the tangent to this curve at the point (1, 1). 8

IB Practice - Calculus - Differentiation Applications (V2 Legacy)

IB Math High Level Year - Calc Practice: Differentiation Applications IB Practice - Calculus - Differentiation Applications (V Legacy). A particle moves along a straight line. When it is a distance s from