MATHEMATICS FOR ENGINEERING TUTORIAL 4 MAX AND MIN THEORY

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1 MATHEMATICS FOR ENGINEERING TUTORIAL 4 MAX AND MIN THEORY This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning by example. The approach is practical rather than purely mathematical. On completion of this tutorial you should be able to do the following. Define a maximum and minimum point of a function. Define a turning point. Define a point of Inflection. Use differential calculus to determine these points. Use differential calculus to determine if a point is a maximum or minimum. Solve practical problems by applying the theory. Students must be familiar with basic differentiation and algebra including the solution of quadratic equations. D.J.Dunn 1

2 MAXIMA AND MINIMA 1. INTRODUCTION There are many engineering problems where a value peaks or dips. For example the power of an engine under constant throttle conditions peaks at a certain speed. Another example is a cylindrical canister to contain a certain volume. The amount of material needed can be minimised by choosing the correct diameter and length. This tutorial is about finding the critical values that make a function reach a maximum or minimum value and differential calculus is the way to do it.. MAXIMUM POINT Consider the function y = 10x x Plotting y against x from x = 0 to x = 5 gives the following graph. Figure 1 From the graph we can clearly see that there is a peak at point A and that this is the maximum value of y. The question is, what is the value of x and y at this maximum point? At point A the gradient is horizontal and hence zero so if we equate dy/ to zero we can find the point. Here is the method. y = 10x x dy/ = 10 4x Equate to zero 10 4x = 0 4x = 10 x = 10/4 =.5 and hence y = 10 x.5 x.5 = = 1.5 This result is confirmed by the graph. D.J.Dunn

3 WORKED EXAMPLE No.1 What is the largest area of a rectangular field that can be enclosed by a fence with a perimeter of 00 m? SOLUTION Let the rectangle be x m long and y m wide. The area enclosed is A = x y.. (1) The perimeter is x + y = 00 m. () We must eliminate one variable, say y, from equation (1) by substitution from equation. y = 00 - x y = 100 x... (3) Substitute (3) into (1) A = x (100 x) = 100x x Plotting A against x reveals that the maximum area is 500 m when x = 50 m. Prove this by differentiation. A = 100x x Figure da = 100 x For a max or min da/ = 0 hence 100 x = = x x = 100/ = 50 m A=100x x A = = = 500 mm D.J.Dunn 3

4 3. MINIMUM POINT Consider the function y = x - 4x Plotting y against x from x = 0 to x = 4 gives the following graph. Figure 3 From the graph we can clearly see that there is a minimum value of y at B. The question is, what is the value of x and y at this minimum point? At point B the gradient is horizontal and hence zero. If we equate dy/ to zero we can find the point. dy/ = x 4 Equate to zero x 4 = 0 4 = x x = 4/ = Substitute x = into the function to find the value of y. y = x - 4x = 8 = -4 This result is confirmed by the graph. D.J.Dunn 4

5 SELF ASSESSMENT EXERCISE No.1 1. Determine the maximum area and the lengths of the sides of a rectangle that can be enclosed by a rectangular perimeter of 8 m length.. The velocity of a missile is related to time by the equation v = t 4t. Calculate the time at which the velocity is a minimum and determine this minimum. 3. The current in a device is related to time by the equation i = 4t 8t. Find the time at which the current the minimum and determine this minimum. 4. Find the value of u that makes q a maximum when they are related by the equation q = 10u 5u +. Find this maximum value of q. D.J.Dunn 5

6 4. TURNING POINTS and POINTS of INFLECTION Consider the function y = x 3 5x +5x +. The graph for x = 0 to x = 4 is shown below. Figure 4 At point A the graph changes from up to down and at B it changes from down to up. They look like a maximum and minimum point but strictly speaking they are not as the value of y exceeds this value later on (and is smaller for negative values of y). Such points are called turning points and they may be found in the same way as a max or min point. Examining the graph we see the turning points occur at about x = 0.6 and.7 but we need to use calculus to find them precisely. The important thing to note is that at A and B the gradient of the graph is horizontal so the value of dy/ must be zero. This enables us to find the value of x and y at these points. Here is how to do it. 3 dy y = x 5x + 5x + = 3x 10x + 5 At the turning points dy/ is zero so equate to zero as follows. dy = 3x 10x = 3x 10x + 5 This is a quadratic equation and we must solve it to find the two values of x. QUADRATIC EQUATION ax + bx + c = 0 b ± b 4ac x = a In this case a = 3, b = -10 and c = 5 so solving we get b ± x = b a 4ac ( 10) ± x = ( 10) ± = ± = 6 40 D.J.Dunn 6

7 There are two possible solutions because all positive numbers have a positive and a negative square root. 40 = ± ± ± 6.34 x = = x = = = or x = = = Hence the turning points occur at x =.71 and The graph tells us which is A and B. Note that without the graph you could not be sure which is the maximum and which a minimum. We need further studies to find out how to do this. POINT OF INFLECTION A point of inflection is where the tangent at a point crosses the graph and it can be shown that at these points the second derivative is zero. For the equation y = x 3 5x +5x + plotted on figure 4, there is such a point at C. This may be determined from the second derivative. d y = 6x 10 = 0 x = 10/6 = An important point about this is that you can have a point of inflection where the gradient is zero and so they could be confused with a maximum or minimum point. SELF ASSESSMENT EXERCISE No. 1. Find the values of x where the maximum and minimum occur for the following function. Also find the point(s) of inflection. y = x 3-6x + 9x + 10 Sketch the graph and determine which is the maximum and which the minimum points. (Answers x = 1 gives a max and x = 3 is a min and x = is a point of inflection). Find the values of x where y is a maximum or minimum for the following function. Also find the point(s) of inflection. y = x 3-5x - 8x (Answers x = 4 gives a min and x = is a max x = is a point of inflection) D.J.Dunn 7

8 WORKED EXAMPLE No. A cylindrical vessel is to be made from thin metal plate to contain 15 litres (0.015 m 3 ) of liquid. Find the dimensions of the cylinder that make the surface area a minimum including the two ends. SOLUTION We need to set up two equations, one for the volume and one for the surface area. Volume = A L = πd L/ = πd L/4.. (1) Surface Area = A = wall + ends (see diagram) Figure 5 A = πdl + πd /4 = πdl +πd /. () Now we decide whether to first find L or D by max or min theory. Let us decide to find D x From (1) find L in terms of D L = =...(3) πd πd Substitute this into () 0.06 πd A = πd x + πd -1 πd A = 0.06D +...(4) For a minimum area the differential coefficient da/dd = 0. Differentiate with respect to D D.J.Dunn 8

9 da = 0.06D dd 0.06D = πd = D 3 + πd Equate to zero 0.06 = πd D D = ( ) 1 3 = 0.673m 0 = 0.06D 0.06 = D π 3 + πd We are not sure that this value gives a minimum value of Area without plotting the graph but assuming it is we can now find the value of L from (3). L = 0.67 m (i.e. equal to the diameter. Now we can find the area from () or (4). A = m D.J.Dunn 9

10 SELF ASSESSMENT EXERCISE No.3 1. A rectangular trough is to be made from a sheet of metal 5 m long and 4 m wide by cutting a square of side x m from each corner and turning up the ends and sides. Show that the volume is given by V = 4x 3-18x + 0x. Find the value of x that makes the volume a maximum. (Answer m). The power transmitted by a pulley belt system is given by the following equation. P = (Fv ρav µθ )(1 e ) P is the power v is the velocity of the belt. ρ is the density of the belt material ρ = 100 kg/m 3 A is the cross sectional area of the belt A = 800 x 10-6 m F is the maximum force in the belt F = 500 N µ is the coefficient of friction µ = 0.3 θ is the angle of contact θ = 1. radian Determine the velocity that makes the power a maximum and calculate this power. (Answers v = 60.4 m/s and P = 19.7 kw) 3. The power developed by a Pelton Wheel water turbine is given by the following equation. P = mu(v u)(1 kcos θ) P is the power m is the mass flow rate m = 40 kg/s u is the velocity of the buckets v is the velocity of the water jet v = 0 m/s k is the blade friction coefficient k = 0.98 θ is the angle of deflection θ = 165 o Determine the velocity of the bucket that will make the power a maximum and calculate the maximum power. (Answers 0 m/s and 31.1 kw) D.J.Dunn 10

11 5. DETERMINING IF A POINT IS A MAXIMA OR MINIMA Consider the equation V = 4x 3-18x + 0x from question 1 previous. The equation is plotted below (in red) and as predicted the maximum point occurs at x = Differentiating the equation we dv = 1x 36x + 0 This equation represents the gradient of the graph at any point x. If we plot this as dv well (in blue), we can see that as expected, the gradient is zero at the max and min points. Figure 6 If we differentiate a second time we shall get an equation representing the gradient of the new graph, i.e. the gradient of the gradient. When we do this we write it as follows. d V = 4x 36 Plotting this graph (in green) produces a straight line and reveals that where the d V original function was a maximum, the value of is negative and where it is a minimum, the value is positive. This is always the case and so we can determine if a d V point is a maximum or minimum by evaluating at that point. D.J.Dunn 11

12 WORKED EXAMPLE No.3 Find the turning points of the following function and determine which is the maximum and which the minimum. y = x 3 0x + 4x SOLUTION Differentiate once dy = 6x Equate to zero Solve using the quadratic equation - b ± x = 40x + 4 b 4ac a 6x 40 ± = 40x + 4 = 0 40 ± x = = or Differentiate again. d y = 1x - 40 a = 6 b = -40 c = 4 (-40) (4)(6)(4) 40 ± 1504 = ()(6) 1 d y Evaluate when x = and = Since this is positive the point x = must be a minimum. d y Evaluate with x = 0.10 and = Since this is negative, the point must be a maximum. The graphs of the functions show these answers are correct. Figure 7 D.J.Dunn 1

13 SELF ASSESSMENT EXERCISE No.4 Find the turning points of the following functions and determine which is the maximum and which is the minimum point. 1. y = x 3 1x + 10x (Answers 3.58 Min and 0.47 max). p = 4q 3 0q + q +10 (Answers Min and 0.05 max) D.J.Dunn 13