3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH. In Isaac Newton's day, one of the biggest problems was poor navigation at sea.

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1 BA01 ENGINEERING MATHEMATICS 01 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's day, one of the biggest problems was poor navigation at sea. Before calculus was developed, the stars were vital for navigation. Shipwrecks occured because the ship was not where the captain thought it should be. There was not a good enough understanding of how the Earth, stars and planets moved with respect to each other. Calculus (differentiation and integration) was developed to improve this understanding. Differentiation and integration can help us solve many types of real-world problems. We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.). Derivatives are met in many engineering and science problems, especially when modelling the behaviour of moving objects. Our discussion begins with some general applications which we can then apply to specific problems. NOTES: a. There are now many tools for sketching functions (Mathcad, LiveMath, Scientific Notebook, graphics calculators, etc). It is important in this section to learn the basic shapes of each curve that you meet. An understanding of the nature of each function is important for your future learning. Most mathematical modelling starts with a sketch. b. You need to be able to sketch the curve, showing important features. Avoid drawing x-y boxes and just joining the dots. c. We will be using calculus to help find important points on the curve. 57

2 BA01 ENGINEERING MATHEMATICS 01 3 types of turning points: a. Maximum point Use d y dx <0 sign: ve b. Minimum point Use d y dx >0 sign: +ve c. Inflection point Use d y dx = 0 58

3 BA01 ENGINEERING MATHEMATICS 01 The kinds of things we will be searching for in this section are: x-intercepts Use y = 0 NOTE: In many cases, finding x-intercepts is not so easy. If so, delete this step. y-intercepts Use x = 0 Guide to solve the problem: dy a. Determine stationary point, 0 dx Find the value of x. b. Find the value of y, when x? c. Determine the nature of stationary point, d y? dx 59

4 BA01 ENGINEERING MATHEMATICS 01 Example 1: Find the stationary point of the curve y x x 6 and sketch the graph. Solution: Determine stationary point, dy dx = dy dx 0 Find the value of x, Find the value of y, when 1 x y x x 6 60

5 BA01 ENGINEERING MATHEMATICS 01 So, stationary point is ( ) Determine the nature of the point, So, the point ( ) is a minimum point. y - intercept, when x 0 ( ) ( ) x-intercept, when y 0 y x x 6 x x 6 0 ( x )( x 3) 0 and 61

6 BA01 ENGINEERING MATHEMATICS 01 y x -6 (1/,-5/4) Example : Find stationary point for the curve and determine the nature of the point. Solution: Determine stationary point, 6

7 BA01 ENGINEERING MATHEMATICS 01 ( ) and When when ( ) ( ) ( ) ( ) The stationary points are ( ) and ( ) Nature : 63

8 BA01 ENGINEERING MATHEMATICS 01 At point ( ) when x=3 At point ( ) when x= ( ) ( ) ( ) ( ) Stationary point ( ) stationary point ( ) is a minimum point is a maximum point Sketch graph x-intercept when y=0 ( ) and 64

9 BA01 ENGINEERING MATHEMATICS 01 y-intercept when x=0 ( ) ( ) y (-3,54) x (3,54) 65

10 BA01 ENGINEERING MATHEMATICS 01 Example 3: Find stationary point for the curve y 3 x 4 and determine the nature of the point. Solution: dy Determine stationary point, 0 dx dy dx = Find the value of x, Find the value of y, when x 0 y 3 x 4 66

11 BA01 ENGINEERING MATHEMATICS 01 So, stationary point is (, ). Determine the nature of the point, d y dx So, the point (, ) is a 67

12 BA01 ENGINEERING MATHEMATICS 01 Example 4: Find the stationary point on the curve y x x 4 3 and determine their nature. 68

13 BA01 ENGINEERING MATHEMATICS 01 Example 5: Determine whether the equation inflection and sketch the graph of ( ). f x 4x 5x 6 is a maximum, minimum or 69

14 BA01 ENGINEERING MATHEMATICS 01 Exercise 3.1 a. Find the coordinates of stationary points of the curve y x x and determine their nature. b. Find the coordinates of stationary points of the curve their nature. y x 4 and determine c. Find the coordinates of stationary points of the curve y x x determine their nature and sketch the graph. d. Find turning point for the curve 3 y x x x and determine maximum point and minimum point for the curve. 70

15 BA01 ENGINEERING MATHEMATICS RECTILINEAR MOTION ONE OF THE most important applications of calculus is to motion in a straight line, which is called rectilinear motion. In this matter, we must assume that the object is moving along a coordinate line. The object that moves along a straight line with position s = f(t), ds has corresponding velocity v, and its acceleration dt dv d s a dt dt. If t is measured in seconds and s in meters, then the units of velocity are meters per second, which we abbreviate as m/sec. The units of acceleration are then meters per second per second, which we abbreviate as m/sec². s=0 -The particle at the beginning - The particle returns back to O again v=0 -the particle is instantaneously at rest -maximum displacement a=0 -constant velocity -the particle is begin t=0 -initial velocity -initial acceleration 71

16 BA01 ENGINEERING MATHEMATICS 01 Example 1: An object P moving on a straight line has position interval time of t seconds. Find s t t 3 8 in meter and the a. The velocity of P at the time t. b. The acceleration of P at the time t. c. The velocity of P when t 3s. d. The displacement of P during the 6 th seconds of moving. Solution a. Given that s t t 3 8, So ds v dt v 16t 3t b. We know that dv d s a dt dt, a 16 6t c. When t 3s, v 16t 3t 16(3) 3(3) ms 1 7

17 BA01 ENGINEERING MATHEMATICS 01 d. During 6s means t 6s, 3 st 6s 8(6) (6) m 3 st 5s 8(5) (5) m s s t s 6 t m Example : A car moves along a straight line so that the displacement, s in meter and t in 3 seconds passes through O is given by s t 6t 5t. Find a. The displacement of the car when t 3s. b. The velocity of the car when its acceleration is c. The time when the car returns back to O. 1ms. Solution: a. When t 3s. 3 s t 6t 5t 3 3 6(3) m 5(3) 73

18 BA01 ENGINEERING MATHEMATICS 01 b. Value of t when its acceleration is Given, a 1ms a 6t 1 So, 6t 1 1 6t 1 1 6t 4 4 t 6 t 4s 1ms. c. The time when the car returns back to O, s 0m, 3 s t 6t 5t 3 t t t t( t 6t 5) 0 t 0, t 6t 5 0 ( t 1)( t 5) 0 t 1 0 t 1s t 5 0 t 5s 74

19 BA01 ENGINEERING MATHEMATICS 01 Example 3: A car is moving at a straight line with position s in meter and the time, t in seconds. Find the: 1, where its displacement, 3 3 s t 9t a. Displacement, s of the car after 3 seconds. b. Displacement, s of the car in fourth second. c. Velocity, v of the car when the acceleration, a 10 m/ s d. Acceleration of the car at the time t = seconds. 75

20 BA01 ENGINEERING MATHEMATICS 01 Exercise: 1. A car moves from a static condition in a straight line. Its displacement s meter from a fixed point after t second is given by s t t a. Acceleration of the car when t 4s. b. The time when the velocity is c. Displacement in the 4 th second. 1 39ms., find. A ball is rolling along a straight ground from the fixed point O and its displacement, s meter and time t second is given by 3 s t t 0t. a. Calculate the acceleration of the ball when it begins to move. b. Calculate the acceleration when t 3s. c. Find the displacement when the ball is instantaneously at rest. d. Find displacement of the ball during 4th seconds. 3. A particle move in linear and the distance, S from fixed point 0, t seconds after passing point 0, is given by 3 s t t t Calculate a. Distance in the 5 th second. b. Values of t when the particles stop for a while. c. Velocity when acceleration is zero. d. Acceleration when t 4s. 76

21 BA01 ENGINEERING MATHEMATICS RATE OF CHANGE If variables both vary with respect to time and have a relation between them, we can express the rate of change of one in terms of the other. We need to differentiate both sides w.r.t. (with respect to) time d dt. If y f x, then respectively. dy dy dx where dy dt dx dt dt and dx dt are the rates of change of y and x Example 1: If the radius of a circle increase at the rate of when the radius is 10cm cms, find the rate of area at the instant Solution: Given, Find, dr dt 1 5 da? dt r 10cm We know that the area of a circle is, A r Differentiate A with respect to r, da r dr 77

22 BA01 ENGINEERING MATHEMATICS 01 Using the formula, da da dr dt dr dt 1 r, when r 10cm 5 (10) 5 4 cm s 1 78

23 BA01 ENGINEERING MATHEMATICS 01 Example : An open cylinder has a radius of 0cm. Water is poured into the cylinder at the rate of cm s. Find the rate of increase of the height of the water level. v r h r= 0cm dv dt 3 40cm s dh Find,? dt 1 79

24 BA01 ENGINEERING MATHEMATICS 01 Example 3: The radius of a circle is increasing at a rate of 5 cm / m. Find the rate of increase of the da area of the circle at the instant when its radius is1cm. (Ans: 10 cm / m ) dt 80

25 BA01 ENGINEERING MATHEMATICS 01 Example 4: 3 1 The volume of a spherical balloon is decreasing at a rate of 4 cm s. Find the rate of dj decrease of its radius when its radius is 5cm. (Ans: 0.04 cm / s dt ) 81

26 BA01 ENGINEERING MATHEMATICS 01 Exercise: The formulae for the volume and surface area of a sphere of radius r are V r and A 4 r respectively. When r 5cm, V is increasing at the rate of 10cm s. dj Find the rate of increase of A at that instant. (Ans: 0.03 cm / s dt, da 1.8 cm / s ) dt. Given that the height of a cylinder is 3cm and the radius increase at the rate of 0.cm/s. Find the rate of change of its volume when the volume is 8

27 BA01 ENGINEERING MATHEMATICS OPTIMIZATION PROBLEM Important! The process of finding maximum or minimum values is called optimisation. We are trying to do things like maximise the profit in a company, or minimise the costs, or find the least amount of material to make a particular object. These are very important in the world of industry. Example 1: If the sum of height, h cm and radius, r cm of a cone is 15cm. What is the maximum volume of the cone? Solution: Substitute (1) into () ( 1) () ( ) 83

28 BA01 ENGINEERING MATHEMATICS 01 84

29 BA01 ENGINEERING MATHEMATICS 01 Example : A wire with length meters is bent to form a rectangular with a maximum area. Find the measurement of its sides. Solution: 85

30 BA01 ENGINEERING MATHEMATICS 01 Example 3: y x The rectangular above has perimeter of 40 meters. Given the perimeter equation is x y 40. Find a. Find the equation of the area in term x. b. Find the maximum area of the rectangle. 86

31 BA01 ENGINEERING MATHEMATICS 01 POLITEKNIK KOTA BHARU JABATAN MATEMATIK, SAINS DAN KOMPUTER BA 01 ENGINEERING MATHEMATICS PAST YEAR FINAL EXAMINATION QUESTIONS 1. A particle is moving along a straight line where s is the distance travelled by the particle in t seconds. Find the velocity and acceleration of the particle by using the following equations. a. b. c. s 6t t 3 3 s 8t 48t 7t s 64t 16t 4. For the curve of y x x 3 6, find a. The coordinates of all the turning points. b. The maximum and minimum points. c. Sketch the graph for the above curve. 3. A particles moves along a straight line such that its distance, s meter from a fixed point O is given by s t t a. Find the velocity of the particles after seconds and the acceleration after 4 seconds. b. Find the acceleration when the velocity is 9 m/ s. 87

32 BA01 ENGINEERING MATHEMATICS The curve is given as y x x Find a. The coordinates of the stationary points. b. The nature of the points. 5. A particle moves in a straight line. Its displacement, s meter, from the fixed point O after t second is given by 3 s t 3t 8t. a. Find the velocity when time is 3 and 5 seconds. b. Find the time when the velocity is 1 8ms. 6. A balloon is filled with air at the rate of 4 3 formed. Find the rate of change of the radius when the radius is 4cm ( v r ) 3 3 0cm per minute until a sphere is 7. A car moves from a static condition from point O with displacement s meter on a straight road in time t seconds is given by s t t a. Find the total displacement when the car is instantaneously at rest. b. Find the velocity when the acceleration is c. Find the initial acceleration. 3ms. d. Find the acceleration when it goes back to point O. 8. i. Find the stationary points for ii. Given 3 f ( x) x 6x 9x 3 y x x and determine the maximum points.. Determine the nature of the stationary point and hence sketch the graph of f( x ). 88

33 BA01 ENGINEERING MATHEMATICS i. Find the stationary point and sketch the graph, if any of the function 3 y x x x ii. The curve 3 y x x x has stationary points. Find the points and state the condition of the points. 10. i) A cone shape container that placed upside down having a base radius 1cm and height 0cm. Then one boy pouring some water to the cone which is xcm height, the volume of that cone is 3 V cm, prove that V 3 3 x. (Given volume of cone, V r h ) ii) The water is flow out of the cone through the hole at the cone peak. Find the nearest alteration for V when x decrease from 5cm to 4.98cm. 89

To find a maximum or minimum: Find an expression for the quantity you are trying to maximise/minimise (y say) in terms of one other variable (x). dy Find an expression for and put it equal to 0. Solve

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