leaving certificate Active Maths Ordinary Level - Book Old Syllabus Strand 5 m = y - y x - x πr πr Oliver Murphy
Editors: Priscilla O Connor, Sarah Reece Designer: Liz White Layout: Compuscript Illustrations: Compuscript, Denis M. Baker, Rory O Neill Oliver Murphy, Folens Publishers, Hibernian Industrial Estate, Greenhills Road, Tallaght, Dublin 4, Ireland Acknowledgements The author and publisher wish to thank Thinkstock for permission to reproduce photographs. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from the publisher. The publisher reserves the right to change, without notice, at any time, the specification of this product, whether by change of materials, colours, bindings, format, text revision or any other characteristic.
Contents Chapter Functions and Graphs.... The Idea of a Function.... The Definition of a Function...4. Domain and Range...4.4 Three Ways of Depicting Functions...5.5 Periodic Functions...8.6 Linear Graphs...4.7 Quadratic Graphs...45.8 Cubic Graphs...5.9 Reciprocal Graphs and Asymptotes...5 Chapter Differential Calculus...57. Calculus...58. Names... and More Names...6. Differentiating by Rule...6.4 Searching for Points with Special Slopes...6.5 The Product Rule...64.6 The Quotient Rule...65.7 The Chain Rule...66.8 Turning Points...67.9 Rates of Change...7 Answers...75
chapter Functions Statistics and Graphs Learning Learning Outcomes Outcomes In this this chapter chapter you you will willlearn learn: In ÂÂ The idea and definition of a function ÂÂ The range and periods of periodic functions ÂÂ How to draw and interpret different kinds of graphs: linear, quadratic and cubic ÂÂ How to graph reciprocal graphs and asymptotes
. THE IDEA OF A FUNCTION YOU SHOULD REMEMBER... A function is like a machine. You put a number in, it is transformed by the machine, and another number emerges. For example, the illustration shows the function f : x x + 7. This function has a job to do. It doubles and adds seven. Whatever number you put in will be doubled and then 7 will be added. KEY WORDS If you put in, 7 will come out. This can be illustrated in any of three different ways: f() = 7 or (,7) or by plotting the point (,7) on a graph. is called the first component (the input). 7 is called the second component (the output). If you put in, will come out. This can be written in three ways: f() = or (,) or by plotting the point (,) on a graph. The first component is ; the second component is. If you put 6 in, 5 will come out. (,7) If you put in (5x +), the function will double this and add 7. So, you will get out (5x + ) + 7 = x + 9. Here, then is a graph showing the three points (,7), (,) and ( 6, 5) and linking them with a straight line. 4 This, too, can be written in three ways: f( 6) = 5 or ( 6, 5) or by plotting ( 6, 5) on a graph. The first component is 6; the second component is 5. We can write f(5x + ) = x + 9. 6 4 ( 6, 5) (,) 4 6 8 The first components are the x-coordinates. It would be wrong to think that functions are something which we meet in the field of Mathematics, but not in everyday life. On the contrary, you meet functions throughout your normal daily life. Your remote control for the television performs a function. If you press one number, say, you get RTE. If you press another (say 5) you get BBC. If you press another (say 9) you get UTV. These could be represented by the set of couples: (,RTE ), (5,BBC ), (9,UTV),... Telephone numbers are another function. Functions and graphs The second components are the y-coordinates.
If you ring 9, you get directory enquiries. If you ring 9, you get the speaking clock. Here are some couples of this function: (9, Directory Enquiries), (9, Speaking Clock),... Even the bar codes on goods in a supermarket perform a function, which determines price. When the check-out person swipes through one bar code, a certain price comes through. When (s)he swipes through another, a different price comes through. A more tenuous and subtle function might be the amount of study you do in Maths and the mark you get in the Leaving. Do you agree with the following function?: (Regular hard work, A), (Moderate work, B), (No work at all, E),... Can you think of any functions which you meet in your life?. the DEFinition OF A Function Take the set of couples: {(,4), (,6), (,8), (4,)}. This set of couples is a function. When you put in, out comes 4; when you put in, out comes 6, etc. But take the set of couples: {(,), (,5), (,), (,)}. When the machine gets up-and-running, what happens when you put in the number? Will the machine send out 5 or will it send out? We cannot tell. We say that this set of couples is not a function, because two of the first components are the same. This brings us to a formal definition of a function: function Functions and graphs For example, {(,5), (7,7), (5,5), (,7)} is a function, since all first components (, 7, 5, and ) are different. It doesn t matter that some of the second components are the same. But the set of couples: {(5,), (7,7), (5,5), (,)} is not a function since two of the first components are the same.. domain and range domain range For example, take the function f = {(,7), (,), ( 6, 5)}. The domain of f = {,, 6}. The range of f = {7,, 5}. 6 f 7 5 4
.4 THREE WAYS OF DEpicting Functions A function which squares a number and then adds, can be depicted in three different ways: f(x) = x + or f : x x + or y = x +. These all mean the same thing: that if you put in a number, you will get out the number squared plus two. If the domain is {,, }, what is the range? The couples of this function are (,), (,) and (,). The range is {, } f Worked Example. f is a function defined by f : x x. The domain of f is {,,,, }. Find the range. Solution f() = () =. This couple is (,). The range = the set of second components f() = () =. This couple is (,). = {,, } f() = () =. This couple is (, ). f( ) = ( ) =. This couple is (,). f( ) = ( ) =. This couple is (,). Worked Example. f : x 5x k is a function. (i) If f( ) =, find k. (ii) Find the value of x for which f(x + 5) = 6. Solution (i) f( ) = 5( ) k = 5 k = k = + 5 k = 6 k = 6 (ii) f(x) = 5x 6 (from (i)) f(x + 5) = 6 5(x + 5) 6 = 6 5x + 9 = 6 5x = 5 x = 7 Worked Example. The diagram shows part of the graph of the function y = ax + b. Find the values of a and of b. Solution The couple (, 8) is on the graph. Therefore, f( ) = 8 a( ) + b = 8 a + b = 8 (, 8) y (,7) x Functions and graphs a b = 8... equation I 5
The couple (,7) is on the graph. Therefore, f() = 7 a() + b = 7 a + b = 7... equation II We solve the simultaneous equations I and II in the usual way: I: a b = 8 II: a + b = 7 Add! a = 5 a = 5 II: a + b = 7 (5)+ b = 7 + b = 7 b = Answer: a = 5, b = Exercise. Functions and graphs. State, giving a reason, if these are functions or not. (i) {(,4), (,6), (5,7)} (ii) {(,), (,7), (,5)} (iii) {(4,4), (,4), (,9)} (iv) {(,9), (,), (,7), (, 6)} (v) {(,9), (9,), (,), (, 7) (vi) {(,), (,)} (vii) {(,), (,), (,4)} (viii) {(,), (,), (,)} (ix) {(, ), (,), (,), (, )} (x) {(,), (,), (,), (,), (4,)}. f : x x + 6 is a function. Find the values of the following: (i) f(5) (ii) f(). f : x 5x is a function. Find the values of the following: (i) f(4) (ii) f( ) (iii) f() 4. f : x x is a function. Find the values of the following: (i) f(7) (ii) f(5) (iii) f( ) 5. f : x x is a function. Find the values of the following: (i) f() (ii) f( ) (iii) f() 6. f : x x + is a function. Find the values of the following: (i) f() (ii) f() (iii) f() Investigate if f() + f() = f(). 7. f : x 4x is a function. Find: (i) The value of f() (ii) The value of f ( ) (iii) The value of k if f(k) = 9 8. f : x x is a function. Find: (i) The value of f ( ) (ii) The value of n if f(n) = (iii) The value of x if f(x) = x 9. f : x x + x 4 is a function. Find: (i) The value of f(4) (ii) The value of f( 6) (iii) The value of k if f(6) = kf() 6
. f : x x + k is a function. Find the value of k if f() =.. f : x ax 9 is a function. Find the value of a if f(7) = 5.. f : x x + k is a function. Find the value of k if f() =.. f : x x + x is a function. Find: (i) The value of f(). f is a function defined by f(x) = x 6 x for x (i) Evaluate f(6). (ii) Find the two values of x for which f(x) =. (iii) Show that there is no value of x for which f(x) = x.. The diagram shows part of the graph of the function y = ax + b. (ii) The value of f(5) (iii) The value of f ( _ ) y (,8) Investigate if f ( _ ) = f(). 4. f : x x + x k is a function. Find the value of k if f() =. (,) x 5. f : x x x is a function. Find the two values of x for which f(x) =. 6. f : x x x + k is a function. (i) Find the value of k if f(4) =. (ii) Find a value of x (other than 4) for which f(x) =. Find the values of a and b.. The diagram shows part of the graph of the function y = x + ax + b. (,) 7. f : x x + 5 is a function. (i) Find the value of f(). (ii) Find f(x + ). (iii) If f(x + ) = 6, find x. 8. g : x 4x 7 is a function. (i) Find the value of g( ). (ii) Find g(x + ). (iii) If g(x + ) = 7, find x. 9. f is a function such that f(x) = x 4. (i) Evaluate f() f( ). (ii) Find the value of k if f(k) f( k) = 66.. f : x x + and g : x x 4 are two functions. (i) Evaluate f(4). (ii) Evaluate g(4). (iii) Verify that f( ) = g( ). (iv) Find a value of k, other than for which f(k) = g(k). (, 6) Find the values of a and b 4. The diagram shows part of the graph of the function y = ax + b. y 4 (,) 4 5 Find the values of a and b. (5, 4) x Functions and graphs 7
5. The diagram shows part of the graph of the function y = x + ax + b. 7. The diagram shows part of the graph of the function y = ax + bx. y y (,5) (,) x (,p) (, 6) (i) Find the values of a and b. (ii) If (,p) is a point on the graph, find the value of p. 6. The diagram shows part of the graph of the function y = f(x) = ax + bx. x y O 4 (4,) (,) 4 x (,k) (i) Find the values of a and b. (ii) Find the value of k. 8. The diagram shows part of the graph of the function y = p + qx x. y 4 5 6 7 8 9 x 4 (, 4) (i) Find the values of a and b. (ii) Show that f( + x) = f( x). (n, ) (i) Find the values of p and q. (ii) Find the value of n, where n <. (iii) Show that f(5 + x) = f(5 x). Functions and graphs.5 PEriodic Functions periodic functions 6 5 4 4 5 6 7 8 9 8
What is the range of this function? We can see from the graph that the highest output (or y-value) is 5 and the lowest is. Therefore, the range of this function is the set of real numbers between and 5, inclusive. This is written {x x 5, x R} or [, 5]. The second shorter notation is usually used, since it is so handy and neat. 7 6 5 4 4 5 6 7 8 9 4 4 The period of this periodic function is and the range is [, 7]. 6 5 4 8 7 6 5 4 4 5 6 7 8 9 4 The period of this periodic function is 7 and the range is [, 4]. If you were asked to find f(5), how could you find it? Since the period is, we can say that f(5) = f(4) = f() = f() = f() = f() =, from the graph. Exercise. Write down the period and range of these periodic functions:. 4 5 6 7 9.. 5 4 4 5 6 7 8 9 4 5 Functions and graphs 9
4. 5 5 5 5 5. 4 5 6 7 8 6. 6 4 5 5 7. 4 4 5 6 7 8 9 8. 4 4 5 6 7 8 9 9. Functions and graphs. 4 5 6 8 8 6 4
. 8 9 9 8 7 6 45 54. 4 5 6 7 8 9 4 5 6 7 8 4. 8 4 8 4 4 8 6 4 4. 4 4 5. 5 5 5 5 5 7. Here is part of the graph of a periodic function f. (i) What is the range? (ii) What is the period? (iii) Find f(4). (iv) Find f(5). 6. 5 4 4 5 6 7 4 4 5 6 7 8 Functions and graphs 4
8. Here is a section of the graph of a periodic function f. (i) What is the range? (ii) What is the period? (iii) Find f ( _ ). (iv) Find f(). 8 4 4 5 6 4 9. Here is part of the graph of a periodic function f. (i) What is the range? (ii) What is the period? (iii) Find f(44) and f( 44) (iv) Is f(x) = f( x) for all x R? Give a reason for your answer. 6 5 4 4 5 6 7 8 9. Here is a section of the graph of a periodic function f. 5 (i) What is the range? (ii) What is the period? (iii) Find f(95). (iv) Find f(85). 4. Here is a section of the graph of a periodic function f. (i) What is the period? (ii) What is the range? (iii) Find f(.5). (iv) Find three values of x, where 4 < x < 5 such that f(x) =. 4 4 5 6 7 8 9 Functions and graphs. Here is a section of the graph of a periodic function f. (i) Write down the period and the range. (ii) What is the value of f()? (iii) Write down two values of x for which f(x) = 4. (iv) Find the least value of x, where x > 5, for which f(x) =.5. 5 4 4 5 6 7 8 4
.6 LinEar graphs Any graph which consists of a single straight line is called a linear graph. Worked Example.4 A car is accelerating uniformly. At time t seconds after it passes through a point p, its velocity v (in metres per second) is given by the formula (a) Draw the graph of this function for t. (b) Estimate from your graph: (i) The velocity at t =.6 seconds (ii) The time when the velocity is 9 m/s (iii) The range of time for which v 5 Solution v = t + (a) To find the velocity, we double the time and add. Hence, here is a table to help us do these calculations: t 4 6 8 t 4 8 6 v 6 4 8 Couples: (,) (,6) (4,) (6,4) (8,8) (,) Here, then, is the linear graph, joining these points: 9 5 9 5 5 4 5 6 7 8 9.6 8.5 (b) (i) Draw a line from t =.6 to the graph, then draw a horizontal line to the v-axis. The reading is approximately 9 m/s. (ii) Draw a line from v = 9 to the graph and then down to the t-axis. The reading is approximately 8.5 seconds. (iii) Draw parallel lines from v = and v = 5 to the graph and then to the t-axis. Shade in between them. The corresponding values of t are: 4 t 6.5 Functions and graphs 4
Exercise.. Draw a graph of the linear function f : x x in the domain x 4, x R. Use your graph to estimate (i) f(.) (ii) The value of x for which f(x) = 6. Draw a graph of the linear function f : x x in the domain x, x R. Use your graph to estimate (i) The value of f(x) when x =.8 (ii) The value of x for which f(x) = (iii) The range of values of x for which f(x) > 5. The time t (in minutes) for which a whole trout should be cooked in the oven is given by t = (m + ) where m is the mass (in kg) of the trout. (i) Copy and complete the following table and hence draw the graph: Mass (m) 4 5 6 Time (t) 6 (ii) Estimate the time taken to cook a.4 kg trout. (iii) A trout is cooked for _ 4 hours. Estimate its mass. (iv) For what values of m is t <? 6. The conversion formula for changing Celsius (C) readings to Fahrenheit (F) is F =.8C +. Functions and graphs. Use the same scales and axes to draw the two graphs y = x and y = 8 4x in the domain x 4. What is the point of intersection of the two graphs? 4. A car passes a point P. Its speed v (in m/s) after that is given by the function: v = 5 t where t is the time in seconds. Draw a graph of v for t 5. Use your graph to estimate (i) The speed at t =. (ii) The time when the speed is m/s (iii) The speed as the car passes P (iv) The time when the car stops (i) Copy and complete the following table and hence draw the graph: Celsius 4 Fahrenheit (ii) Estimate the temperature in degrees Fahrenheit when it is 8 Celsius. (iii) The temperature in Athens is Fahrenheit. Estimate this in degrees Celsius. (iv) The weather forecast is for temperatures in the range C 5. What is the range of expected temperatures in Fahrenheit? 7. The number of minutes (t) which a turkey of mass m kilograms should be cooked for is given by the formula: t = 5(m + ) (i) Draw the graph of t for the values m. (ii) For how long should a turkey of mass 7. kg be cooked? (iii) A turkey was cooked for 5 _ 4 hours. What was its mass? (iv) What are the masses of turkeys whose cooking times are between and 4 hours? 44
8. C = 5 (F ) is the formula for converting degrees Fahrenheit (F) to degrees Celsius (C). 9 (i) Copy and complete the following table, giving the values to the nearest integer. Fahrenheit 4 6 8 Celsius 8 (ii) Draw a graph to illustrate this data. (iii) Estimate the temperature in degrees Celsius when it is 77 Fahrenheit. (iv) Estimate the temperature in degrees Fahrenheit when it is 5 Celsius. (v) If C 5, estimate the range of values of F..7 Quadratic graphs Any graph of the form y = ax + bx + c is called a quadratic graph. All such graphs have the shape of a parabola, which is shaped like a wineglass. If a is a negative number then the wineglass is upside-down. y = ax + bx + c (if a is positive) y y = ax + bx + c (if a is negative) y x x The points (a and b) at which these graphs cut the x-axis mark the roots of the quadratic equation ax + bx + c =. Worked Example.5 Draw the graph of the function f : x x x 7 in the domain x, x R. Find, from your graph: (i) The value of f(.5) (ii) The values of x for which f(x) = (iii) The minimum value of f(x) and the value of x at which it occurs (iv) The values of x for which x x 7 Solution x x 7 x 4 4 6 7 7 7 7 7 7 7 y 9 7 6 4 Points: (,9), (, ), (, 7), (, 6), (,), (,4) Functions and graphs 45
5 y 7 5. x.5..5 5 7 7.5 5 (i) Draw a line from x =.5 on the x-axis, up to the graph and across to the y-axis. The reading is approximately 7. Therefore f(.5) = 7. (ii) Draw lines east and west from y = on the y-axis. The corresponding readings on the x-axis are x =.5,.. (iii) The minimum value of f(x) is approximately 7.5 at x =.. (iv) x x 7 f(x) [Since f(x) = x x 7] y [So, we look for parts of the graph where the y-value is negative]. x.9 [The values of x where the graph is below the x-axis] Worked Example.6 Using the same scales and axes, draw the graphs of the functions f : x 4 x x and g : x x in the domain 4 x, x R. Use your graph to estimate: (i) The range of values of x for which 4 x x < (ii) The solutions of the equation x + 4x 5 = (iii) The value of Solution Functions and graphs f(x) g(x) x 4 4 4 4 4 4 4 4 4 4 x 8 6 4 4 6 x 6 9 4 4 9 f(x) 4 4 5 4 4 Points ( 4, 4) (,) (,4) (,5) (,4) (,) (, 4) (, ) x 4 x 8 6 4 4 6 g(x) 9 7 5 5 Points ( 4,9) (,7) (,5) (,) (,) (, ) (, ) (, 5) 46
8 6 4 4..9.7.9..7 4 6 8 (i) 4 x x < f(x) < x <. or x >. [The values of x for which the graph is below the x-axis] (ii) Change both sides of the equation until you end up with f(x) [i.e. 4 x x ] on one side: x + 4x 5 = x + x.5 = [Dividing both sides by ].5 x x = [Multiplying both sides by ] 4 x x =.5 [Adding.5 to both sides to give us f(x) on the left] f(x) =.5 x =.9 or.9 (iii) x = [Again, we will try to get 4 x x on one side] x = = x [Subtracting x from both sides] x = 4 x x [Adding x to both sides to give f(x) on the right] g(x) = f(x) We want the x-value of the points of intersection of the two graphs. These are x =.7 and x =.7. Since is a positive number, we can say that =.7 Functions and graphs 47
Exercise.4 Functions and graphs. Draw the graph of the function f : x x x 5 in the domain x 4, x R. Find, from your graph: (i) The value of f(.) (ii) The values of x for which x x 5 = (iii) The values of x for which x x 5 < (iv) The minimum value of f(x). Draw the graph of the function y = x 4x + 5 in the domain x 4. Use your graph to find: (i) The value of y when x =.5 (ii) The values of x if y =. Draw the graph of the function y = x 6x + 7 in the domain x 6. Use your graph to find: (i) The value of y when x =.5 (ii) The values of x if y = 4. Draw the graph of the function f : x x x 4 in the domain x 5, x R. Find, from your graph: (i) The values of f(4,5) (ii) The solutions to x x 4 = (iii) The approximate solutions to x x 4 = 5. Draw the graph of the function y = x + x 6 in the domain 4 x, x R. Use your graph to find: (i) The value of y when x =.5 (ii) The values of x if y = (iii) The approximate values of x for which y = 6. Draw the graph of the function y = x x 5 in the domain x 4, x R. Use your graph to find: (i) The value of y when x =.8 (ii) The values of x if y = (iii) The values of x for which y = 4 7. Draw the graph of the function f : x x x 7 in the domain x, x R. Find, from your graph: (i) The values of x for which x x 7 = (ii) The values of x for which x x = (iii) The range of values of x for which x x 7 (iv) The minimum value of f(x) and the value of x at which it occurs 8. Draw the graph of the function f : x 4 + x x in the domain x, x R. Find, from your graph: (i) The values of x for which 4 + x x = (ii) The values of x for which f(x) > (iii) The solution set of + x x = (iv) The maximum value of f(x) 9. Draw the graph of the function f : x 6 x x in the domain x, x R. Estimate, from your graph, the values of x for which: (i) 6 = x + x (ii) (6 x ) = x (iii) 6 x(x + ) 48
. Draw the graph of the function f : x x x 4 in the domain x, x R.. Use the same scales and axes to draw the graphs of the two functions f: x 4x + 7x and g : x x + 5 in the domain x, x R. Use your graph to estimate: Use your graph to estimate: (i) The values of x for which x = x + 4 (ii) The values of x for which x x 5 = (iii) The solution set of x x < (iv) The minimum value of f(x) and the value of x at which it occurs (i) The value of x for which g(x) = (ii) The values of x for which f(x) = (iii) The values of x for which f(x) = g(x) (iv) The range of values of x for which g(x) > f(x). Draw the graph of the function f : x 4x + 6x 7 in the domain x, x R. 4. Draw the graph of f : x 5 x x in the domain 5 x, x R. Using your graph: Find, from your graph: (i) Estimate the maximum value of f(x). (ii) Draw the axis of symmetry of the graph and write down its equation in the form x = k. (i) The value of f(.8) (ii) The values of x for which x + x = (iii) The range of values of x for which x(x + ) < 7 (iv) A negative value of x for which f(x) = f(). Copy and complete the following table for the function f : x 4 x x : x.5 f(x) 4 Draw the graph of y = f(x) in the domain x.5, x R. Use your graph to find: (i) The values of x for which x + x = 4 (ii) The range of values of x for which x + x < 4 (iii) The maximum value of f(x) (iv) The values of x which satisfy 4x + x 5 = (v) The least value of n N for which f(x) = n has no solution (iii) Find the values of f(k + ) and f(k ). (iv) Draw the graph of the line y = x and hence solve the equation f(x) = x. 5. (a) Solve the equation x 6x = correct to one decimal place, using the quadratic formula b ± b 4ac. a (b) Draw the graph of the function y = x 6x in the domain x 7, x R. (i) Use the graph to estimate the solutions to x 6x =. (ii) Estimate the values of x for which x 6x >. 6. (a) Solve the equation 5x x 4 = correct to one decimal place, using the quadratic formula b ± b 4ac a (b) Draw the graph of the function y = 5x x 4 in the domain x, x R. (i) Use the graph to estimate the solutions to 5x x 4 =. (ii) Estimate the values of x for which 5x x 4 <. Functions and graphs 49
7. Using the same scales and axes, graph the two functions: f : x 5 x x in the domain x.5, x R and g : x 6x x in the domain x, x R. f(x) represents the height (in km) of a foreign rocket which is launched at 4. p.m. (x = ). g(x) represents the height (in km) of a guided missile which is launched at 5. p.m. (x = ) to intercept the foreign rocket. Use the graphs to estimate: (i) The maximum height of the foreign rocket (ii) The time at which the guided missile intercepts the foreign rocket (iii) The height at which the collision occurs.8 cubic graphs The graph of any function of the form f : x ax + bx + cx + d, a is called a cubic graph. Its shape usually looks like a one-humped camel something like this: But if a (the coefficient of x ) is a negative number, the camel looks west, like this: Worked Example.7 Draw the graph of the cubic function f : x x x 4x + in the domain x 4, x R. Estimate from your graph: Functions and graphs (i) The values of x for which f(x) = (iv) The values of x for which f(x) is decreasing (ii) The value of f(.5) (v) The solutions of x = (x + x ) (iii) The minimum value of f(x) where x > Solution x 4 x 8 8 7 64 x 8 8 8 4x 8 4 4 8 6 + y 7 4 7 7 Points (, 7) (,) (,) (, 4) (, 7) (, ) (4,7) 5 5.9..5.8 4 6 5 5
(i) We want the values of x where the graph cuts the x-axis. They are x =.,. and.. (ii) Draw a line from x =.5 to the graph and then to the y-axis. The reading is approximately 6. (iii) The minimum value of f (x), when x > is 7 (when x = ). (iv) We read a graph, as we read English or Irish, from left to right. If the graph is going up, we say that the function is increasing. When the graph is going down, we say that the function is decreasing. In this case the graph is decreasing between.5 and. That is to say, the values of x for which f(x) is decreasing are:.5 < x < (v) x = (x + x ) x = x + 4x 6 x x 4x + 6 = x x 4x + = 5 (subtracting 5 from both sides in order to get f(x) on the left) f(x) = 5 y = 5 Draw the straight line at y = 5. It cuts the graph at x =.9,.,.8: Answer decreasing.5 Exercise.5. Draw the graph of the function f : x x + x 8x 4 in the domain x, x R. 4. Draw the graph of the function f : x x + x x in the domain 4 x, x R. Estimate from your graph: (i) The values of x for which f(x) = (ii) The values of x for which f(x) =. Draw the graph of the function f : x x + 4x + x 6 in the domain 4 x, x R. Estimate from your graph: (i) The values of x for which f(x) = (ii) The values of x for which f(x) < (iii) The values of x < for which f(x) >. Draw the graph of the function f : x x x 4x + in the domain x 4, x R. Estimate from your graph the solutions to these three equations: (i) x x 4x + = (ii) x x 4x = (iii) x = (x + x 4) Estimate from your graph the values of x for which: (i) f(x) = (ii) f(x) is both negative and increasing. (iii) x + x x = Estimate, also, the value of f(.5). 5. Draw the graph of the function f : x x x + 6 in the domain 4 x 4, x R. Estimate from your graph the values of x for which: (i) f(x) = (ii) f(x) is both positive and increasing. (iii) x > x Functions and graphs 5
6. Draw the graph of the function f : x + x + x x in the domain x 4, x R. Estimate from your graph: (i) The maximum value of f(x) when x > (ii) The solutions of the equation + x + x x = (iii) The value of f(.7) 7. The diagram shows part of the graph of the function f(x) = x x 4x +. y 5 5 5 Use the graph to estimate: (i) The value of f(.) (ii) The values of x for which x x 4x = 4 (iii) The values of x for which x(x + ) = [(x + ) 5] x 9. Draw the graph of y = x in the domain x, x R. Use your graph to estimate the value of (i) 4 (ii).. The diagram shows the graphs of the two functions y = x x and y = 4x 7. 5 5 (,) (, 5) 5 (, ) (, 5) 5 (,5) Use the graphs to solve the following equations: (i) x x = (ii) x x 4x + 4 =. Draw the graph of the function f : x x 7x + in the domain x, x R. Estimate from your graph the values of x for which f(x) = and hence estimate the value of 7.. A = {x x, x R}. f : x x x + 5 is defined on A. Copy and complete this table: Functions and graphs 8. (a) Show that is a root of the equation x 9x 9x + 8 =. (b) Use the factor theorem to find the other two roots. (c) Draw the graph of x x 9x 9x + 8 in the domain x 4, x R. Use your graph to verify the roots of x 9 x 9x + 8 = which you found in parts (a) and (b). x.5 f(x) 5 5.65 Draw a graph of y = f(x) and hence write down the number of solutions in A to each of these equations: (i) f(x) = (ii) f(x) = (iii) f(x) = 5 (iv) f(x) = Write down the range of values of k for which f(x) = k has exactly one solution in A. 5
. Draw the graph of the function f : x x 5x + in the domain x, x R. Using your graph: (i) Find the values of x > for which f(x) <. (ii) Solve f(x) = and hence estimate 5. (iii) By drawing a suitable line, solve the equation x 6x =. 4. A function f is defined as f : x x x 6x + 4 in the domain x 4, x R. Copy and complete the table: x 4 f(x) 8 Using your graph: (i) Find the values of x > for which f(x) <. (ii) By drawing a suitable line, solve the equation x x 8x + 9 =..9 REciprocaL graphs AND asymptotes Any graph of the form y = is called a reciprocal graph. There is a special problem associated with x + a these graphs. The problem is that there is no such number as. y So the denominator of a fraction must never be zero. x The word asymptote comes from three Greek words: a sum piptwn = not meeting together. The graph gets closer and closer to the asymptote but the two never meet together. x = a Worked Example.8 Draw the graph of y = in the domain 6 x, x R. Find the values of x for which y >. x + Solution x 6 5 4 y 4 * 4 5 * There is an asymptote at x =, since y is undefined when x =. Here is the graph: We always try to show how the graph gets very close to the asymptote. From the graph, we see that y > when < x <. 6 5 4 y x Functions and graphs 5
Exercise.6. A function is defined by y = x + Copy and complete this table: x.5 y.5 *.5 Draw the graph in the domain x, x R, showing the asymptote at x =.. A function is defined by y = Copy and complete this table: x 4. A function is defined by f(x) = x Draw the graph of y = f(x) in the domain _ x 6, x R. Find from your graph: (i) The value of x for which f(x) =.6 (ii) The range of values of x for which f(x) < _ 4 5. A function is defined by f(x) = x. (i) Evaluate f ( _ 4 ). (ii) Evaluate f ( _ ) + f ( _ ). Functions and graphs 54 x.5.5 4 y.5 * Draw the graph in the domain x 4, x R, showing the asymptote at x. For what values of x is y > _?. A function is defined by f(x) = x Copy and complete this table: x.5.5 4 f(x) * Draw the graph y = f(x) in the domain x 4, x R, showing the asymptote at x =. Find from your graph: (i) The value of x for which f(x) =.5 (ii) The range of values of x for which f(x) < Revision Exercises. (a) f(x) = 5x is a function. (i) Evaluate f(). (ii) If f(x) =, find the value of x. (b) Let g(x) = (x )(7 x) (iii) Prove that f( + x) + f( x) =. (iv) Draw the graph of y = f(x) in the domain x 7, x R, showing the asymptote at x =. (v) By drawing the line y = _ (x 4) or otherwise, solve the equation x = (x 4). 6. A function is defined by f(x) = x + 4. (i) Evaluate f ( _ 4 ). (ii) Prove that f(x 4) + f(x 4) = x. (iii) If f(k) + f() =, find k. (iv) For what value of x is f(x) undefined? (v) Sketch the graph of y = f(x) in the domain 8 x. 7. Draw the graphs of the two functions f(x) = and g(x) = x 4 x 4 in the domain x 8. Deduce, from your graph, the values of x in the domain for which: (i) f(x) = g(x) (i) Verify that g() = and find another value of x for which g(x) =. (ii) f(x) < g(x) (ii) Given that g(x + ) = ax + bx, find the values of a and b, where a, b R. (c) Draw the graph of y = x x in the domain x. Use your graph to solve the inequality x x <.
. (a) f(x) = 7x + k is a function. If f(5) =, find the value of k. (b) The diagram shows part of the graph of y = g(x) = x + ax + b. Draw the graph of y = g(x) for x 5. Estimate, from the graph, the values of x for which: (i) g(x) =.7 (ii) g(x) < x 4. (a) f(x) = x (,p) (i) Evaluate f(4) and f ( 4 ). (ii) Find the value of x if f(x) = 4. (,) (,) (b) The diagram shows part of the graph of a periodic function y = g(x). (i) Find the values of the real numbers a and b. (ii) Find the value of p, where p R. 4 (c) Draw the graph of y = x x in the domain x 4. Use your graph to find the values of x for which: (i) x x = (ii) x x > 4 5 6 7 8 9 4 5 6 (i) Write down the range. (ii) Write down the period. (iii) Estimate the value of g(.5).. (a) Write down the range and period of the periodic function, part of whose graph is illustrated below: (c) h(x) = x is a function. (i) For what value of x is h(x) undefined? 5 4 8 7 6 5 4 4 5 6 7 8 9 (ii) Show that h ( x x ) = x (iii) Show that h(x) h ( x ) = (x )(x + ) (x )(x ) (b) f(x) = x + bx + c is a function such that f( ) = 7 and f() =. Find the values of b and c and hence find a value of k (other than ) for which f(k) = f( ). (c) Let g(x) =, for x R, x 5. x 5 Copy and complete this table: x 4 4 4 4 4 4 g(x) 4 5. (a) Draw a graph of y = x for x, x R. Use your graph to estimate. (b) Using the same scales and axes draw the graphs of the two functions: f : x 8 + x x and g : x x + in the domain x 5, x R. Use your graph to estimate: (i) The range of values of x for which 8 + x x > (ii) The value of 7 Functions and graphs 55
6. If f(x) = x x +, complete the following table: x f(x) 7 (c) Draw the graph of g(x) = in the x + domain x 5, x R. Using the same scales and axes, draw the graph of h(x) = x. Show how your graph may be used to estimate the value of. Functions and graphs 56 (i) Draw the graph of f: x x x + in the domain x, x R. (ii) Use your graph to estimate the range of values of x for which the graph is decreasing. (iii) Using the same axes and the same scales, draw the graph of the function g : x x for x R. (iv) Write down the values of x at which the two graphs intersect. (v) Solve algebraically x x + = x. 7. (a) f(x) = x is a function defined for x >. (i) Evaluate f ( ). (ii) If f(6) + f() = f(k), find the value of k. (iii) Write as a single fraction f(x) f(x + ) and hence solve f(x) f(x + ) =. (b) The depth of water (d) (measured in metres) in a harbour is given by the formula d = t + 5t + 7, where t is the time (in hours) after. noon. Draw the graph of d in the domain 4 x. Use your graph to estimate: (i) The minimum depth in the harbour and the time at which it occurs (ii) The times at which the depth is metres (iii) The depth at.45 p.m. (iv) The length of time for which the depth is less than metres 8. (a) The domain of the function f : x x is {,,,, }. Find the range. (b) The diagram shows part of the graph of y = x + ax + b. P (,6) (,) Q (i) Find the value of a and b. (ii) Find the coordinates of P and Q. 9. (a) The diagram shows part of the graph of the periodic function p. Write down the range and period of the function. Find the value of p(7). 5 4 5 5 (b) Using the same scales and axes, draw the graphs of y = x and y = x for x. Show how your graphs may be used to estimate the value of _. (c) Let f(x) = (4 x) ( + x), x R. Let g(x) = 5x k, where k is a constant. (i) Write down the roots of f(x) =. (ii) If x = is a solution to f(x) + g(x) =, find the other solution.. (a) f(x) = 6(x 4) (i) Evaluate f( ). (ii) Find k, where k >, if f(k) = f( ). (b) g(x) = is defined for x R \ {5}. x 5 (i) Evaluate g(.5). (ii) Prove that g(x) + g( x) =. (iii) Draw a graph of y = g(x) in the domain x, x R, showing the asymptote at x = 5. (c) The diagram shows the graph of h(x) = x 6x +. 5 (i) Estimate the values of x for which h(x) = and hence estimate 6, explaining your answer. (ii) Show that h ( 6 ) + h ( 6 ) = h().
chapter Differential Calculus Learning Outcomes In this chapter you will learn: ÂÂ ÂÂ ÂÂ ÂÂ ÂÂ Differentiation from first principles Differentiation by rule The product, quotient and chain rules How to find turning points and determine their nature How to calculate rates of change
Differential Calculus. Calculus In the 7th century a giant step was taken in the field of Mathematics. This was the discovery of the calculus. Strangely, this new branch of Mathematics was invented by two men independently in two different countries: Sir Isaac Newton in England and Gottfried Wilhelm Leibnitz in Germany. Calculus is the study of the rate at which things change: cars that speed up and trains which slow down, comets that move faster as they whizz past the sun, spacecraft that gradually lose mass as they burn up petrol, populations that grow. This invention has revolutionised Mathematics and has enabled humankind to make the huge strides which our world has seen in the last three centuries. In Mathematics we use graphs to judge the rate at which things grow. If the graph is increasing, things are on the way up; if the graph is decreasing, things are on the way down. Now, we know that the slope of a line can be got, if we know the co-ordinates of two points on the line (x,y ) and (x,y ). FORMULA The slope is given by the formula y y x x. y SALES Jan INCREASING (x,y ) DECREASING Feb Mar Apr May (x,y ) x YOU SHOULD REMEMBER... KEY WORDS But how can we find the slope of a curve? The trouble about a curve is that the slope at every point is different. But we may ask What is the slope of this curve [say y = f(x)] at a particular point [say, (x,f(x))]? Newton s (and Leibnitz s) method was to look at the particular point through an imaginary magnifying glass, enlarged so much that the curve is more or less a straight line. y f(x) (x,f(x)) Take (x,f(x)) and a nearby point on the curve. Let h = the small distance between the two x-coordinates. x 58 The y-coordinates are, therefore, f(x) and f(x + h). The slope of the line joining these two points (x,f(x)) and (x + h, f(x + h)) is y y f(x + h) f(x) f(x + h) f(x) x x = = (x + h) x h Now, as we said, this is the slope of the line joining these points. If we want to get the true slope of the curve joining these two points, we will have to let h become smaller and smaller (and use a stronger and stronger magnifying glass to see it!). Newton got over this problem by inventing the limit. f(x + h) f(x) He simply said: Find and see what happens to it as h h gets smaller and smaller. f(x + h) f(x) (x,f(x)) x h x + h (x + h,f(x + h))
In this way we can get the instantaneous rate at which the graph is increasing. y Curve Tangent If you carry out this procedure, the task you are performing is called differentiating from first (x,f(x)) principles with respect to x. Never forget that the resultant outcome is the slope of the curve x at a particular point, or, to be more correct, the slope of the tangent to the curve at a point. The slope of the curve at a certain point and the slope of the tangent to the curve at that point are the same, as the diagram shows. Worked Example. Differential Calculus Differentiate f(x) = x from first principles with respect to x. Solution lim h f(x) = x f(x + h) = (x + h) = x + xh + h f(x + h) f(x) = (x + xh + h ) x = xh + h f(x + h) f(x) xh + h = = x + h h h + h) f(x) f(x h = lim (x + h) = x h [Letting h ] This means that the slope of the tangent at any point on the graph of y = x is given by twice-the-x-coordinate. For example, at (,4) the slope is () = 4. At (,), the slope is () =. At (,), the slope is () =. At (,), the slope is ( ) =, etc. 5 4 y x Worked Example. Differentiate x 5x + 7 from first principles. Hence write down the slope and the equation of the tangent to the curve y = x 5x + 7 at the point (,9). 5 y Solution f(x) = x 5x + 7 f(x + h) = (x + h) 5(x + h) + 7 = x + 6xh + h 5x 5h + 7 (,9) lim h f(x + h) f(x) = 6xh + h 5h f(x + h) f(x) = 6xh + h 5h = 6x +h 5 h h f(x + h) f(x) h = lim (6x + h 5) = 6x + () 5 = 6x 5 h 5 Tangent at (,9) x 59
Differential Calculus This means that the slope of the tangent at any point is given by six times the x-coordinate minus five. The slope at (,9) = 6() 5 = 7 The equation of this tangent is: y y = m(x x ) y 9 = 7(x ) y 9 = 7x 4 7x y = 5 Exercise.. Differentiate x + x + from first principles. Hence write down the slope of the tangent to the curve y = x + x + at the point (,6).. Differentiate x + x + 4 from first principles. Hence write down the slope of the tangent to the curve y = x + x + 4 at the point (,4).. Differentiate x + 7x + from first principles. Hence write down the slope of the tangent to the curve y = x + 7x + at the point where x =. 4. f(x) = x x + 6. Differentiate x x from first principles with respect to x. 7. Differentiate x + x 5 from first principles. Hence write down the slope of the tangent to the curve y = x + x 5 at the point (, ). 8. Differentiate x from first principles. Hence write down the slope of the tangent to the curve y = x at the point (, 4). 9. Differentiate + 6x x from first principles with respect to x. Hence write down the slope and the equation of the tangent to the curve y = + 6x x at the point (,9). (i) Differentiate f(x) from first principles with respect to x. (ii) Find the slope of the tangent to the curve y x x + at the point (,4). 5. (i) Differentiate x x + 7 from first principles. (ii) Show that (,5) is a point on the curve y = x x + 7. (iii) Find the slope of the tangent to the curve y = x x + 7 at the point (,5).. f(x) = x (i) Differentiate f(x) from first principles. (ii) Find the slope of the tangent to the curve y = x at the point (, ). (iii) Find the equation of this tangent.. Differentiate x x from first principles with respect to x. Hence find the slope and the equation of the tangent to the curve y = x x at the point (,7).. f(x) =. Differentiate f(x) from first principles.. NAMes... and MOre names When we differentiate a function f(x), the outcome is called any of the following: the slope of the tangent to the curve, the gradient, the derivative of f(x), the differentiation of f(x). There are three mathematical ways of writing this: 6
. DifferentiatinG BY rule You may be wondering if there is a quick way of differentiating, rather than going through those five arduous steps every time. The answer is: There is a quick way: Multiply the coefficient by the power of x and reduce the power by one. For example, if y = x 5 + 7x 4 + x + x + 4x + 9 then dy dx = 5x4 + 8x + 6x + x + 4 Again, if y = 5x 7x+ then dy dx = x 7 x and y are not the only possible variables. For example, if FORMULA y = x n dy = nxn dx This formula is found on page 5 of Formulae and Tables. s = 9t 5t+4 then ds = 8t 5 (which is called the derivative of s with respect to t). dt Finally, if f(x) = x 4x x then f (x) = 8x x (using Sir Isaac Newton s notation). Differential Calculus Worked Example. Differentiate the following by rule: (i) y = x 5x + x Solution (i) y = x 5x + x dy = x + x dx (ii) y = x (ii) y = x = x _ dy _ x = x _ = x dx = ( ) Exercise. Differentiate the following by rule:. y = x + 5x + 6. y = 4x + x +. y = 5x + x + 4. y = x + 8x + 5. y = x 5x + 4 6. y = 4x + 5x x + 7. y = x 4 + x x 8. y = x 4 + x 9. y = 5x x + x. y = 9x + 4. y = x. y = x + 5. y = x 5 + x + x 4. y = x + x 5. y = x 6. If f(x) = x + 5x 6, find df dx. 7. If g(x) = x 4 x +, find dg dx. 8. If A(x) = x x, find da dx. 9. If f(t) = 5t + 5t +, find df dt.. If s = t + t t, find ds dt. 6
Differential Calculus. If A = pr + pr, find da dr. [Remember that p is a constant]. If s = t + t 9, find the value of ds when t = 4. dt. If f(x) = x 4 + x 6, find df when x =. dx 4. y = 5x x + (i) Find dy dx. (ii) Find the slope of the tangent to the curve y = 5x x + at the point (,4). 5. y = x x + (i) Find dy dx. (ii) Find the slope of the tangent to the curve y = x x + at the point (, 7). (iii) Find the equation of this tangent. 6. If y = x 6x + 4, show that dy = when x =. dx 7. If y = 5x x +, show that dy = at (, 9). dx 8. y = x 6x + x + (i) Find dy dx. (ii) Find the slope of the tangent to the curve y = x 6x + x + at the point where x =. 9. f(x) = 4x 6x + 49. (i) Find df dx. (ii) Find the slope of the tangent to the curve y = f(x) at (,7). (iii) Find the equation of this tangent.. Find the equation of the tangent to y = x + 5x+ at (,8)..4 searching FOR POints WitH SPECial slopes Worked Example.4 Find a point on the curve y = x 8x + where the tangent is parallel with the x-axis. Solution If the tangent is parallel with the x-axis, then the slope =. dy dx = x 8 = x = 4 y = x 8x + = (4 ) 8(4) + = 5 Therefore the point is (4,5) 6
Worked Example.5 Find a point on the curve y = x 7x + 5 where the tangent makes an angle of 45 with the positive sense of the x-axis. Solution If the tangent makes an angle of 45 with the positive sense of the x-axis, then the slope = tan 45 =. dy dx = 4x 7 = 4x = 8 x = y = x 7x + 5 = () 7() + 5 = y (, ) Therefore, the point is (, ). 45 x Differential Calculus Exercise.. Show that the tangent to the curve y = x x + 5 at (, 4) is parallel with the x-axis.. Find a point on the curve y = x 6x + where the tangent is parallel with the x-axis.. Find a point on the curve y = x x + 5 where the tangent is parallel with the x-axis. 4. Find a point on the curve y = x x + where the slope is equal to. 5. Find a point on the curve y = x x + where the slope is equal to 6. 6. Find a point on the curve y = x x + where the tangent makes an angle of 45 with the positive sense of the x-axis. 7. The diagram shows part of the graph of y = x x. Find: (i) The equation of the tangent T (ii) The coordinates of the point p where T cuts the x-axis T (, ) 4 p 8. Show that the tangents to the curve y = x at (, 8) and at (, 8) are parallel. 9. Find two points on the curve y = x x where the tangents have zero slope.. Show that the tangents to y = x 5x + x + at x = and at x = 4 are parallel.. The diagram shows two tangents to the curve y = x x at the points (, ) and (, ). (i) Find their slopes. (ii) Find their equations. (iii) Investigate if they are perpendicular. (iv) Find their point of intersection. 5 4 4 5 (, ) (, ) 4 5. Find a point on the curve y = x + x + where the tangent is parallel to the line x y 5 =. 6
Differential Calculus.5 THE PRODUCT rule If you want to differentiate two functions which are multiplied, for example y = (5x + 7)(x + ), then you may not just differentiate each part. Instead, you must apply the Product Rule which appears on page 5 of Formulae and Tables. And here it is: Worked Example.6 Differentiate (5x + 7)(x + ). Solution y = (5x + 7)(x + ) u = 5x + 7 v = x + dy dx = u dv dx + v du = (5x + 7)() + (x + )(5) du dx dx = 5 dv dx = = x + 4 + x + 55 = x + 69 Worked Example.7 Find the value of the derivative of ( x )(x x 5) at the point (,). Solution y = ( x )(x x 5) u = x v = x x + 5 dy dx = u dv + v du dx dx = ( x )(x ) + (x x 5)( x) du dv = x dx dx = x = ( )( ) + ( 5)( ) when x = = () + ( 6)( ) = Exercise.4 Differentiate the following products:. (5x + 7)(x + ). (x )(7x + ). (x + )(5x + ) 4. (4x )(5x + ) 5. (x )(x + ) 6. y = (x + x + )(x + 5x + ) Find the value of dy if x =. dx 64
7. Evaluate the derivative of (x )(x + x ) at x =. 8. Evaluate the derivative of (x )(x + ) at x =. 9. If y = (x + )(x ), find the value of dy dx : (i) When x = (ii) When x = (iii) When x =.6 THE QUOtient rule. If y = x(x 7), find the value of dy dx : (i) By using the product rule (ii) By multiplying out first and then differentiating. (i) If y = x(x + )(x ), find dy dx. (ii) Show that the tangent to this curve at (,) has slope.. Find the coefficient of x in the derivative of (x 5x + )(7x + x ). If you are asked to differentiate a quotient, such as x 5 x, you cannot just differentiate the top and + then the bottom. Instead, you must use the Quotient Rule, which appears on page 5 of Formulae and Tables. And here it is: Differential Calculus Worked Example.8 (i) Differentiate x 5 x +. (ii) Hence find the slope of the tangent to y = x 5 x + Solution (i) y = x 5 x + = u v dy du v dx = dx u dv dx v = + )() (x 5)(x) (x (x + ) = (x + ) (6x x) (x + ) = x + 6x + x (x + ) = x + x + (x + ) (ii) Slope = dy dx = + x + x (x = + + + ) ( + ) = at (, 5). = [Letting x = ] 65
Exercise.5 Differential Calculus Differentiate the following using the quotient rule:. x + 5 x +. 5x + x +. x 4x x 4. x + 5. x + x + 6. 4x + 7. x 8. x 9. x. x + x. Find the value of the derivative of x + 4 4x + when x =.. Find the slope of the tangent to the curve y = x x at (,).. Find the value of the derivative of 5x x + when x =. 4. Find the slope and the equation of tangent to the curve y = x at the point (,). 5. Tangents are drawn to the curve y = x at (,) and at (, ). Verify that these tangents are parallel. 6. Let y = x + 5 x. Find the two values of x for which dy dx =. 7. Let f(x) = for x R, x. x (i) Find the derivative of f(x). (ii) Two tangents to y = f(x) make an angle of 45 with the x-axis. Find the co-ordinates of the points on the curve of y = f(x) at which this occurs. 8. f(x) = is defined for all real values x of x (except x = ). Show that dy > for all valid values of x. dx x 9. y = x. Find dy and show that dy dx dx > for all x R, x,.. Use both the product and quotient rules to evaluate the derivative of x (x + 4) x + when x =..7 THE CHain rule If you are asked to differentiate a function, which is itself raised to a power of n, for example, (x + x + ) 4, then you must use the Chain Rule. And here it is: So, if y = (x + x + ) 4 then dy dx = 4(x + x + ) (x + ) Worked Example.9 66 Find the slope of the tangent to the curve y = (x x 5) when x =. Solution y = (x x 5) dy dx = (x x 5) (x ) = (9 5) (6 ) [when x = ] = ()(5) = 5
Exercise.6 Differentiate the following using the chain rule:. (x + ) 5 6. (x ) (). (x + 7) 4 7. (x + ). (8x + ) 8. (x + x x + ) 4. (x + ) 9. (x + 7x 6) 8 5. (x + x 4 ) 7. (x 5 + ) 9. Evaluate the derivative of (x + ) 7 when x =.. Evaluate the derivative of (x + ) when x =.. Evaluate the derivative of (x ) when x =. 4. y = (x 5) 4. Find the value of dy when x =. dx 5. If y = (x 5), show that dy for all real dx values of x. 6. Find the slope of the tangent to the curve y = (x x ) 5 at (, ). 7. Find the slope and equation of the tangent to the curve y = (x ) at the point (,). 8. Differentiate: (i) x x + (ii) ( x x + ) 9. Differentiate: (i) x x + 4 (ii) ( x x + 4 ). Differentiate: (i) x + x (ii) ( x + x ) 4 (iii) x + x Differential Calculus.8 turning POints If you are cycling up and down over hilly ground, your bicycle will be in a horizontal position on two occasions: at the top of a hill and at the bottom of a valley. local maximum On a graph, the top of a hill is called a local maximum point and the bottom of a valley is called a local minimum point. If a point is either a local maximum or a local minimum, we call it a turning point. local minimum 67
Worked Example. Differential Calculus Find a local minimum point on the curve y = x 4x +. Solution y = x 4x + dy = x 4 = x = 4 x = dx y = () 4() + = 4 8 + = 7 The local minimum point = (,7) y (,7) 5 4 x Worked Example. Find the two turning points of the curve y = x x 9x + 5 and determine which is a local minimum and which is a local maximum. Hence draw a rough sketch of the graph, showing these features. Solution y = x x 9x + 5 dy dx = x 6x 9 = x x = (x + )( x ) = x = or If x = then y = ( ) ( ) 9( ) + 5 = : (,) If x = then y = () 9() + 5 = : (, ) Since is a greater number than, we can conclude that (,) is the local maximum and that (, ) is the local minimum. Here is a rough sketch of the graph: (,) 4 5 6 (, ) 68
Exercise.7. Find a minimum point on the curve y = x 4x +.. Find a minimum point on the curve y = x 6x + 7.. Find a minimum point on the curve y = x x + 4. 4. Find a maximum point on the curve y = + 8x x. 5. Find a maximum point on the curve y = + x x. 6. Find a minimum point on the curve y = x x +. 7. f(x) = x x + 75 is a function. Find the minimum value of f(x). 8. f(x) = x x is a function. Find the maximum value of f(x). 9. f(x) = x + bx + c is a function such that f() = and f(5) =. (i) Find the values of b and c. (ii) Find a minimum point on the curve y = f(x). (iii) Draw a rough sketch of the curve y = f(x).. f(x) = 4x x is a function. (i) Find a maximum point on the curve y = f(x). (ii) Draw a rough sketch of the curve y = f(x). (iii) Investigate if the tangents at (,4) and at (,4) are perpendicular.. Find a maximum and a minimum point on the curve y = x x.. Find a maximum and a minimum point on the curve y = x x x 4.. Find a local maximum and minimum point on the curve y = x x +. 4. Find a local maximum and minimum point on the curve y = x x 6x +. 5. Find a local maximum and minimum point on the curve y = + 5x + 6x x. 6. Find a local maximum and minimum point on the curve y = (x + ) ( x). 7. f(x) = ax + bx is a function such that f() = 6 and f() = 46. (i) Find the values of a and b. (ii) Find a local maximum and a local minimum point on the curve y = f(x). 8. Let A = xy and let x y = 4. (i) Write y in terms of x. (ii) Write A in terms of x. (iii) Find the minimum value of A. 9. f (x) ax + bx is a function such that f() = and f(4) = 64. (i) Find the values of a and b. (ii) Find the values of x for which f (x) =. (iii) Find the co-ordinates of points P and Q on the curve y = f(x) at which the tangents are perpendicular to the line x + 8y =.. Let f(x) = x 6x 5x +. (i) Find the values of x for which f (x) =. (ii) Determine the slope and equation of the tangent to y = f(x) at (,). (iii) Find the equation of a parallel tangent. Differential Calculus 69