Further differentiation and integration

7 Toic Further differentiation and integration Contents. evision exercise................................... 8. Introduction...................................... 9. Differentiation of sin x and cos x.......................... 4.4 Differentiation of (x + a) n............................... 4.5 Differentiation of (ax + n.............................. 46.6 The Chain ule.................................... 48.7 Integration of sin x and cos x............................ 5.8 Integration of (ax + n................................ 56.9 Integration of sin(ax + and cos(ax +..................... 59. Summary....................................... 6. Extended information................................ 6. eview exercise................................... 6. Advanced review exercise.............................. 6.4 Set review exercise.................................. 65 Learning Objectives Use further differentiation and integration. Minimum erformance criteria: ffl Differentiate k sin x, k cos x ffl Differentiate using the function of a function rule. ffl Integrate functions of the form f (x) = (x + q) n, n rational excet for -, f (x) = cos x and f (x) = sin x

8 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION Prerequisites You should alrea have a basic knowledge of differentiation and integration. min ffl. evision exercise Learning Objective ffi Identify areas that need revision evision exercise There is an on-line exercise at this oint which you might find helful. Q: Differentiate the following functions fi fl a) f(x) = x + x f (x) = (x + ) (x - ) c) f(x) = x x -6 d) f(x) = 4x +x- x Q: Find the following integrals a) 6x -x / x -4x 5 x c) z z- z dz Q: Evaluate the following a) c) ( + 4x) 4 (x + ) -t t dt cfl HEIOT-WATT UNIVESITY 6

.. INTODUCTION 9. Introduction A ball, attached to the end of a stretched sring, is released at time t =. The dislacement y (cm) of the ball from the x -axis at time t (seconds) is given by the formula y (t) = sin t We might wish to know the answers to the following questions: ffl What is the seed of the ball after seconds? ffl When is the ball first stationary? Since seed is the rate of change of distance with resect to time, the seed of the ball is given by the differential equation dt = d ( sin t) dt Thus in order to answer these questions we need to find the derivative of the sine function. You will learn how to do this and more in the following section of work. cfl HEIOT-WATT UNIVESITY 6

4 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION ffl. Differentiation of sin x and cos x Learning Objective ffi Differentiate sine and cosine functions The grah for y = sin x is shown here. fi fl Notice that the tangent to the curve is drawn at various oints. The value for the gradient of the tangent at these oints is recorded in the following table. x ß ß ß ß m T - When these oints are lotted and joined with a smooth curve the result is as follows. Since the gradient of the tangent at the oint (a, f (a) ) on the curve y = f (x) is f (a) then the above grah reresents the grah of the derivative of sin x Then for y = sin x it aears that = cos x (You could check this further by calculating gradients at intermediate oints). Q4: Stu the grah for y = cos x as shown here. cfl HEIOT-WATT UNIVESITY 6

.. DIFFEENTIATION OF SIN X AND COS X 4 Comlete the following table for the gradient of the tangents. x ß ß ß ß m T Plot the oints and join with a smooth curve. Q5: When y = cos x it aears that =? a) sin x cos x c) - sin x d) - cos x When f (x) = sin x then f (x) = cos x alternatively d (sinx) = cos x When f (x) = cos x then f (x) = - sin x alternatively d (cos x) = - sin x Note: x must be measured in radians Examles. Find f (x) when f (x) = cos x Solution When f (x) = cos x then f (x) = - sin x cfl HEIOT-WATT UNIVESITY 6

4 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION. Calculate d sin x - cos x + 4 x Solution d sin x - cos x + 4 x =cosx-(-sinx)+8x =cosx+sinx+8x Now try the questions in the following exercise. Exercise 6 min There is an on-line exercise at this oint which you might find helful. Q6: Find f (x) when a) f (x) = sin x f (x) = cos x c) f (x) = - 5 sin x d) f (x) = - 4 cos x e) f (x) = 5 sin x - cos x f) f (x) = cos x + sin x g) f (x) = x - 5 cos x h) f (x) = 7 sin x + cos x - 6 i) f (x) = x sin x - 5 x j) f (x) = 4- x cos x x Q7: Calculate the following a) c) d) d ( cos x) d (5 sin t - cos t) dt d 5 du u - sin u 7 cos + d Q8: Find the gradient of the curve with equation y = sin x at the following oints. a) x = x = ß 4 c) x = ß d) x = ß cfl HEIOT-WATT UNIVESITY 6

.4. DIFFEENTIATION OF (X + A) N 4 Q9: Find the gradient of the curve with equation y = sin x - cos x at the following oints a) x = x = ß 4 c) x = ß d) x = ß Q: Calculate f (x) for f (x) = sin x and hence show that f (x) has a maximum turning ß oint at, ß and a minimum turning oint at,- Q: Find the equation of the tangent to the curve y = sin x at x = ß Q: Find the equation of the tangent to the curve y = cos x + sin x at x = ß 4 Q: Show that the grah of y = x + sin x is never decreasing. Q4: a) Find the gradient of the tangents to the curve y=x + sin x at the oints where x = and x = ß Calculate the acute angle, in degrees, between these tangents. ound your answer to decimal lace. Q5: Find the equation of the tangent to the curve y = 4x ß -cosxatx= ß ffl.4 Differentiation of (x + a) n Learning Objective ffi Differentiate functions of the tye ( x + a ) n We can differentiate exressions such as ( x + 5 ) and ( x - 4 ) by exanding and differentiating term by term. fi fl Examles. Find f (x) when f (x) = ( x + 5 ) Solution We need to exand the exression first. f(x)=(x+5) = ( x+5)(x+5) =x +x+5 cfl HEIOT-WATT UNIVESITY 6

44 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION Now we can differentiate term by term f (x) =x+ = ( x+5). Find f (x) when f (x) = ( x - 4 ) Solution Again we need to exand the exression first. f(x) =(x-4) = ( x-4)(x-4)(x-4) = ( x-4) x -8x+6 =x -8x +6x-4x +x-64 =x -x +48x-64 Now we can differentiate term by term f (x) = x -4x+48 = x -8x+6 = ( x-4) Use the method above to differentiate the following functions. Factorise your answers. Q6: a) f(x)=(x+) f(x)=(x+) c) f(x)=(x-) d) f(x)=(x-) Are you beginning to see a attern to your answers? Make a rediction for the derivatives of the following functions without exanding the exressions. Q7: a) f(x)=(x+) 4 f(x)=(x+) 5 c) f(x)=(x+) 6 d) y=(x-) 4 e) y=(x-) 5 f) y=(x-) 6 In general we can write When f (x) = ( x + a ) n then f (x) = n ( x + a ) n-, (n Q ;a ) We can use this rule for more comlex functions as in the following examles. cfl HEIOT-WATT UNIVESITY 6

.4. DIFFEENTIATION OF (X + A) N 45 Examles. Differentiate f (x) = ( x-) 5 Solution First we must rewrite the exression for f (x). f (x) = ( x-) 5 = ( x-) -5 Now we can differentiate using the above rule. f (x) = - 5 ( x-) -6-5 = ( x-) 6. Find d 5 x+ Solution d 5 d x+ = ( x+)5/ = 5 ( x+)/ = 5 x+ Now try the questions in the following exercise. Exercise There is an on-line exercise at this oint which you might find helful. Differentiate the following. 5 min Q8: a) (x+6) 8 (x-) 5 c) (x+8) - d) (x-4) / e) f) g) h) ( x+) q ( x-6) 5 x q ( x+5) Q9: Find the value of f () and f () when f (x) = ( x - ) 4 cfl HEIOT-WATT UNIVESITY 6

46 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION Q: Find the equation of the tangent to the curve y = at the oint where x = ( x+) Q: Find d a) sin x + ( x+9) 8 d x-5 - cos x d c) 4 sin x + 4 x- d d) x 8 - ( x+4) 5 ffl.5 Differentiation of (ax + n Learning Objective ffi Differentiate functions of the tye ( ax + b ) n We now know that, for examle, when y = ( x + ) then = ( x+) Consider y = ( x + ) How will the coefficient affect the derivative? fi fl Examle Find when y = ( x + ) Solution We need to exand the exression first. y = ( x+) = ( x+)(x + )(x+) = ( x+) 4x +4x+ =8x +8x +x+4x +4x+ =8x + x +6x+ Now we can differentiate term by term. = 4x +4x+6 = 6 ( 4x +4x+) = 6(x+) = (x+) x Notice that the coefficient of rovides a factor of in the derivative. cfl HEIOT-WATT UNIVESITY 6

.5. DIFFEENTIATION OF (AX + B) N 47 Use this method to differentiate the following functions. Write your answers as in the examle above. Q: a) y=(x+) y=(5x-) c) y=(x+5) d) y=(4x-) Have you sotted a attern? Make a rediction for the derivatives of the following functions without exanding the exressions. Q: a) y=(x+) 4 y=(x+) 4 c) y=(5x+) 4 d) y=(5x-) 8 e) y=(7x+5) 4 You may have noticed that When f (x) = ( ax + b ) n then f (x) = n ( ax+ n- a =an ( ax + n-, (n Q ;a ) You can see how this rule is used in the following examles. Examles. Differentiate f (x) = ( 9x - 4 ) 5 Solution f (x) = 5 ( 9x - 4 ) 4 9 9 9 9 =45 ( 9x - 4 ) 4. Differentiate y = Solution ( 6x - 5 ) / First we must rewrite the exression for y y = ( 6x - 5 ) / = ( 6x - 5 ) -/ cfl HEIOT-WATT UNIVESITY 6

48 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION Now we can differentiate using the rule. =- ( 6x - 5 ) -4/ 6 =- ( 6x - 5 ) -4/ =- ( 6x - 5 ) 4/ Now try the questions in the following exercise. Exercise 4 min There is an on-line exercise at this oint which you might find helful. Differentiate the following Q4: a) ( x + 5 ) 7 (-4x) c) ( 6x + ) / d) (-x) 4 e) ( x + ) -5 f) ( 7x + ) 4 g) 5-4x h) 5x+6 ffl.6 The Chain ule Learning Objective ffi Differentiate comosite functions using the chain rule. Function notation h (x) = (ax + n is an examle of a comosite function. Let f (x) = ax + b and g (x) = x n then we can write h (x) = ( ax + b ) n = ( f (x) ) n =g(f(x) ) When f (x) = ax + b then f (x) = a When g (x) = x n then g (x) = nx n- We could also write that when g (f) = (f) n then g (f) = n (f) n- = n (ax + n- fi fl cfl HEIOT-WATT UNIVESITY 6

.6. THE CHAIN ULE 49 However, we alrea know that when y = (ax + n then - =an ( ax+n We can now write this in function notation in the following way. Whenh(x)=g(f(x))then h (x) = an ( ax + b ) n- =n ( ax+ n- a =g ( f (x) ) f (x) (from (.) and (.)) This result is known as the chain rule. Leibniz notation We can also write the chain rule in Leibniz notation. Again let y = (ax + n but this time let u=ax+b We can now write y = u n Since u = ax + b then du =a Since y = u n then du =nun- =n ( ax+ n- As before, we alrea know that when y = (ax + n then - =an ( ax + b )n We can rewrite this in Leibniz notation in the following way. When y = u n then - =an ( x+a)n =n ( x+a) n- a = du du (from (.) and (.4)) It will be useful to remember both forms of the chain rule. (However, you are not required to be able to rove either of them) The chain rule Function notation Leibniz notation h (x) =g ( f (x) ) f (x) = du du The chain rule allows us to differentiate many tyes of comosite functions as you can see in the following examles. Note that either function or Leibniz notation can be used. cfl HEIOT-WATT UNIVESITY 6

5 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION Examle : Bracketed function raised to a ower. q Find h (x) when h (x) = x +6x Solution (using function notation) q h (x) = x +6x / = x +6x Let f (x) = (x + 6x) then f (x) = x + 6 Let g (x) = x / then g (x) = x -/ We can also write that when g (f) = (f) / then g (f) = (f) -/ Now we can use the chain rule to find h (x) h (x) =g (f) f (x) = (f) -/ f (x) = -/ x +6x ( x + 6 ) -/ = ( x+) x +6x = = x+ x +6x / x+ q x +6x Solution (using Leibniz notation) Let u = x + 6x then du =x+6 Let y = u=u / Thus, by the chain rule = du du then du = u -/ = u -/ ( x + 6 ) = -/ x +6x ( x + 6 ) x+ = q x +6x This is the same result as that obtained using function notation. The chain rule also gives us a method for differentiating comosite functions involving trig functions. cfl HEIOT-WATT UNIVESITY 6

.6. THE CHAIN ULE 5 Examles. Trig. function with a multile angle. Find h (x) when h (x) = sin (5x) Solution (using function notation) Let f (x) = 5x then f (x) = 5 Let g (f) = sin (f then g (f) = cos (f) Thus, by the chain rule h (x) = g (f) f (x) = cos (f) 5 = cos (5x) 5 = 5 cos (5x) Solution (using Leibniz notation) Lety=sinuwithu=5x When y = sin u then du = cos u When u = 5x then du =5 Thus by the chain rule = du du = 5 cos (5x) = cos u 5 = cos u 5 = cos u 5 = cos u 5 This is the same result as that obtained using function notation.. Trig. function raised to a ower. Find the derivative of cos x Solution (using function notation) emember that cos x can be rewritten as (cos x) Let f (x) = cos x then f (x) = - sin x Let g (f) = (f) then g (f) = (f) Now we can use the chain rule to find the derivative of cos x d cos x = d ( cos x) =g (f) f (x) = (f) f (x) = ( cos x) ( - sin x) = - sin x cos x Solution (using Leibniz notation) Let u = cos x then du = - sin x Let y = u then du =u cfl HEIOT-WATT UNIVESITY 6

5 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION Thus, by the chain rule = du du = ( cos x ) ( - sin x) =u ( - sin x) = - sin x cos x This is the same result as that obtained using function notation. Now try the questions in the following exercise. Exercise 5 6 min There is an on-line exercise at this oint which you might find helful. Differentiate the following. You can choose to use either function or Leibniz notation, whichever you find easier. Q5: a) h (x) = (x - 5) h (x) = (x + x) 7 c) h (x) = ( 4 - x ) 5 d) h (x) = 6 x +9 e) h (x) = x 5 + 5 f) h (x) = x +x Q6: a) y = sin (x) y = cos x c) y = cos (x - ) d) y = sin 5- x e) y = sin (x +4) f) y = cos 6x + ß 4 Q7: a) f (x) = sin x f (x) = cos 5 x c) f(x) = sin x d) f(x) = sin x cfl HEIOT-WATT UNIVESITY 6

.7. INTEGATION OF SIN X AND COS X 5 Q8: When f (x) = ( - cos x) calculate f ß and f 6 ß Q9: Find the equation of the tangent to the grah y = = at the oint where x ( x + ) Q: a) Find the coordinates of the stationary oints on the curve f (x) = sin x + sin x for 6 x 6 ß Make a sketch of the curve for» x»ß. Q: Show that the grah of y = 5x + sin x is never decreasing. ffl.7 Integration of sin x and cos x Learning Objective ffi Integrate sine and cosine functions We have alrea seen that fi fl d (sin x) = cos x d (cos x) = - sin x Since integration is the reverse rocess to differentiation it therefore follows that Z Z cos x = sin x + C sin x = - cos x + C Again, note that x must be measured in radians. Examles. Find (8 + sin x) Solution (8 + sin x) =8x cos x + C. Evaluate Solution ß/4 cos x + sin x cfl HEIOT-WATT UNIVESITY 6

54 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION Z ß/4 h i ß/4 cos x + sin x = sin x cos x ψ! = = =4 Now try the questions in the following exercise. Exercise 6 + ( ) 6 min There is an on-line exercise at this oint which you might find helful. Q: Find the following a) sin x 5 cos x c) - ß sin d d) 8- cos u du e) x + sin x f) (6 sin t - cos t) dt g) + 7 sin!! h) cos x d! - 5 sin x + ß Q: Evaluate the following integrals a) c) d) e) f) ß/6 ß/4 ß/ ß/ ß/ ß/ ß/ ß/ 4 cos x sin x (cos t - 5 sin t) dt (6-5sin u) du ( x + cos x) ( sin ff - 5 cos ff + ß) dff cfl HEIOT-WATT UNIVESITY 6

.7. INTEGATION OF SIN X AND COS X 55 Q4: Evaluate the following integrals a) c) d) ß sin x ß ß ß sin x sin x The grah shown here is for y = sin x Calculate the shaded area between the curve and the x-axis. Q5: y = sin x a) Find the coordinates of the oints of intersection at a and b Calculate the shaded area in the diagram. a b y = cos x Q6: A article starts from the origin at time, t = and moves in a straight line along the x-axis so that its seed at time t is given by the formula v (t) = 5 + sin t a) Calculate the formula for s (t), the distance of the article from the origin, at time t. (emember that v = ds dt ) How far is the article from the origin at t = ß? cfl HEIOT-WATT UNIVESITY 6

56 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION ffl.8 Integration of (ax + n Learning Objective ffi Integrate functions of the tye (ax + We have alrea seen that d (ax + n =an(ax + n and therefore similarily d (ax + n+ =a(n+) (ax + n Dividing both sides by a (n + ) then gives us ψ! d (ax+ n+ = (ax + n a (n +) fi fl Since integration is the reverse rocess to differentiation we can now write: (ax+ n = (ax+n + a (n +) +C, n6= See the following examles. Examles. Find (4x + ) 4 Solution Z (4x + ) 4 = (4x+)5 = (4x+)5 +C. Find (x+5) /4 Solution Z (x + 5) /4 (x + 5)7/4 = 7 + C 4 = 4 7 (x + 5)7/4 = 7 (x + 5)7/4 + C + C. Evaluate Solution (x - ) cfl HEIOT-WATT UNIVESITY 6

.8. INTEGATION OF (AX + B) N 57 Z (x - ) = Z (x - ) - " (x - ) - = (-) =-» x - =-» - 5 =- - = # Now try the questions in the following exercise. Exercise 7 There is an on-line exercise at this oint which you might find helful. Q7: Find the following 6 min a) (x + 5) 7 (t - ) 4 dt c) (4x - ) - d) (-x) -6 e) (4r+) / dr f) (x +6) -4/5 Q8: Integrate the following a) x+9 c) d) e) f) (x - ) x+4 q ( -4x) 5x - q (4x + 7) 5 Q9: Find the general solution of a) = ( -x)4 cfl HEIOT-WATT UNIVESITY 6

58 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION ds dt = (4t + ) - Q4: Evaluate the following integrals a) c) d) e) f) (x - ) ( -x) 4 +x (4x - ) 9 9-x (x - ) Q4: Find the articular solution of the following differential equations a) du dt = 5-tgiventhatu=when t = 5 dt = given that x = 5 when t = / (t + ) Q4: Calculate the shaded area in the following diagrams. a) cfl HEIOT-WATT UNIVESITY 6

.9. INTEGATION OF SIN(AX + B) AND COS(AX + B) 59 ffl.9 Integration of sin(ax + and cos(ax + Learning Objective ffi Integrate functions of the tye sin (ax + and cos (ax + We have alrea seen that d (sin (ax + ) = a cos (ax + d (cos (ax + ) = - a sin (ax + Again, since integration is the reverse rocess to differentiation we can now write: fi fl Z Z cos (ax + = sin(ax + + C a sin (ax + = - cos(ax + + C a Examles. Find cos (x + 5) Solution cos (x + 5) = sin (x+5) +C. Evaluate ß/4 sin x + ß Solution Zß/4 sin x + ß» = =- =- - cos x + ß ß/4 h cos x + ß h cos ß - cos ß =- ( --) i ß/4 i = Now try the questions in the following exercise. Exercise 8 There is an on-line exercise at this oint which you might find helful. Q4: Find the following 6 min a) cos(x - ) cfl HEIOT-WATT UNIVESITY 6

6 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION sin (x+5) c) 6 cos (x - ) d) sin 4x + ß e) cos x f) 5 sin x+ ß Q44: Evaluate the following a) c) d) ß/4 ß/ ß/ ß/4 ß/6 cos x sin x t cos dt cos x + ß 4 Q45: Calculate the area of each shaded region. Q46: a) Show that the curves y = sin x and y = cosx intersect at x = ß Hence calculate the shaded area in the following diagram. and x = 7ß cfl HEIOT-WATT UNIVESITY 6

.. SUMMAY 6 Q47: a) Use the double angle formula cos x = cos x - to show that cos x= cos x + Hence find cos x Q48: a) Use the double angle formula cos x = - sin x to exress sin x in terms of cos x Hence find sin x Q49: Use the double angle formula sin x = sin x cos x to find sin x cos x Q5: Use double angle formulae to evaluate the following a) sin x - cos x cos x c) 4sin 5x cos 5 x d) x - sin x ffl. Summary Learning Objective ffi.. ecall the main learning oints from this toic d (sin x) = cos x d (cos x) = - sin x d (x+a)n =n(x +a) n- ; (a ;n Q ) fi fl cfl HEIOT-WATT UNIVESITY 6

6 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION. 4. 5. 6. 7. d (ax + n =an(ax + n-, (a ;n Q ) The chain rule Function notation h (x) = g (f (x)) f (x) Leibniz notation = du du Z cos x = sin x + C Z sin x = - cos x + C (ax + n = Z Z (ax + n + a (n +) +C, (a 6= ;n6= ) cos (ax + = sin (ax + +C, (a 6= ) a sin (ax + =- cos (ax+ +C; (a 6= ) a. Extended information There are links on the web which give a variety of web sites related to this toic.. eview exercise eview exercise in further diff and int min There is an on-line exercise at this oint which you might find helful. Q5: a) Differentiate - cos x with resect to x Given y = 4 sin x, find Q5: Find f (x) when f (x) = (x - 5) -4 Q5: a) Find sin x Integrate -4 cos x with resect to x c) Evaluate 4 (x -) cfl HEIOT-WATT UNIVESITY 6

.. ADVANCED EVIEW EXECISE 6. Advanced review exercise Advanced review exercise in further diff and int There is an on-line exercise at this oint which you might find helful. Q54: 6 min a) If f (x) = cos x+, find f (x) 5x Given that f (x) = (5 - x) 4, find the value of f () c) Differentiate x / + 5 sin x with resect to x d) Find the derivative of 6 x + sin x e) Find given that y = - sin x Q55: a) Find (4x - x - sin x) Evaluate c) Evaluate ß/ sin x (x + ) d) Find + x and hence find the exact value of Q56: a) Show that (cos x + sin x) = + sin x 4 +x Hence evaluate ß/4 (cos x + sin x) Q57: Differentiate sin 5 x with resect to x. Hence find sin 4 x cos x Q58: By writing cos x as cos (x + x) show that cos x =4cos x - cos x Hence find cos x Q59: An artist has designed a "bow" shae which he finds can be modelled by the shaded area below. Calculate the area of this shae. cfl HEIOT-WATT UNIVESITY 6

64 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION (Higher Mathematics) Q6: Linktown Church is considering designs for a logo for their arish magazine. The "C" is art of a circle and the centre of the circle is the mid-oint of the vertical arm of the "L". Since the "L" is clearly smaller than the "C", the designer wishes to ensure that the total length of the arms of the "L" is as long as ossible. y A x The designer decides to call the oint where the "L" and "C" meet A and chooses to draw coordinate axes so that the A is in the first quadrant. With axes as shown, the equation of the circle is x + y = a) If A has coordinates ( x, y ), show that the total length T of the arms of the "L" is given by T=x+ - x Show that for a stationary value of T, x satisfies the equation x= - x c) By squaring both sides, solve this equation. Hence find the greatest length of the arms of the "L". cfl HEIOT-WATT UNIVESITY 6

.4. SET EVIEW EXECISE 65 (Higher Mathematics).4 Set review exercise Set review exercise in further diff and int There is an on-line exercise at this oint which you might find helful. Q6: min a) Differentiate 4 cos x with resect to x Given y = sin x, find Q6: Find f (x) when f (x) = (x + ) / Q6: a) Find 5 sin x Integrate - 4 cos x with resect to x c) Evaluate (x -) 4 cfl HEIOT-WATT UNIVESITY 6

66 TOPIC. FUTHE DIFFEENTIATION AND INTEGATION cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC Further differentiation and integration evision exercise (age 8) Q: a) f (x) = 6x - x f (x) = 4x - 5 c) f (x) = 5 x - x d) f (x) = 8x + x Q: a) x - 4 x4/ +C x - 4 5 x5 +C 5 c) 5 z - z+c Q: a) 9 c) Answers from age 4. Q4: x ß ß ß ß m T - Q5: c) - sin x cfl HEIOT-WATT UNIVESITY 6

4 ANSWES: TOPIC Exercise (age 4) Q6: a) f (x) = cos x f (x) = - sin x c) f (x) = - 5 cos x d) f (x) = 4 sin x e) f (x) = 5 cos x + sin x f) f (x) = - sin x + cos x g) f (x) = 6 x + 5 sin x h) f (x) = 7 cos x - sin x i) f (x) = cos x + 5 x j) f (x) = - Q7: + sin x x/ d) a) - sin x 5 cos t + sin t c) - u - cos u Q8: a) c) d) Q9: a) + c) d) - Q: f (x) = cos x = at turning oints and thus x = ß or ß When x = ß then f (x) = sin ß = and when x = ß then f (x) = sin ß =- Q: y = x + ß cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC 5 Q: y=- x+ ß +8 4 Q: = + cos x and since -» cos x» then 6 6 Thus Q4: > and so y = x + sin x is never decreasing. a) When x = the gradient is When x = ß the gradient is ß 8.9 ffi Q5: y=6x-ß Answers from age 44. Q6:. f (x)=(x+). f (x)=(x+). f (x)=(x-) 4. f (x)=(x-) Answers from age 44. Q7: a) f (x)=4(x+) f (x) =4(x +) f (x)=5(x+) 4 f (x) =5(x +) 4 c) f (x)=6(x+) 5 f (x) =6(x +) 5 d) =4 ( x-) e) =5 ( x-)4 f) =6 ( x-)5 Exercise (age 45) Q8: a) 8(x+6) 7 5(x-) 4 c) -(x+8) -4 cfl HEIOT-WATT UNIVESITY 6

6 ANSWES: TOPIC d) ( x-4)/ e) - ( x+) q 5 f) ( x-6) g) - ( x-) h) x+5 Q9: f () = - 4 and f () = Q: x + 7y - 7 = Q: a) cosx+8(x+9) 7 - ( x-5) + sin x c) 4 cos x + 4 ( x-) /4 d) 4 x 7-5 ( x+4) 4 Answers from age 47. Q: a) c) d) =6 ( x+) = ( x+) =(5x-)=( 5x-) 5 =6 ( x+5) =( x + 5 ) = ( 4x - ) =( 4x - ) 4 Answers from age 47. Q: a) c) d) e) =8 ( x+) = ( x + ) = ( 5x + ) =4 ( 5x - )7 =8 ( 7x + 5 ) cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC 7 Exercise 4 (age 48) Q4:... 4. 5. 6. 7. 8. = ( x+5)6 =- ( -4x) -/ = ( 6x + ) =- ( -x) -6 =- ( x + ) =- 8 ( 7x+) 5 = q ( 5-4x) = -5 ( 5x + 6 ) Exercise 5 (age 5) Q5: a) h (x) = 6 (x - 5) h (x) = 7 (x +)(x + x) 6 c) h (x) = - x (4 - x ) 4 6x d) h (x) = 6x +9-5x 4 e) h (x) = x 5 + f) h (x) = 5 - x x +x 4 Q6: a) c) d) = cos (x) =- sin x =-sin( x - ) 5- =- cos x cfl HEIOT-WATT UNIVESITY 6

8 ANSWES: TOPIC e) f) Q7: =xcos x +4 6x =-6sin + ß 4 a) f (x) = sin x cos x = sin x f (x) = - 5 sin x cos 4 x c) f (x) = - cos x sin 4 x d) f (x) = cos x sin x Q8: f (x) = 4 sin x ( - cos x) therefore f ß 6 = - and f ß = Q9: 4x+9y-7= Q: ß a) Maximum turning oints at, ß and, 7ß Minimum turning oints at 6,- ß and 4 6,- 4 Q: = 5 + cos x and since -» cos x» then 6 6 7 Thus > and so y = 5x + sin x is never decreasing. Exercise 6 (age 54) Q: a) - cos x + C cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC 9 5 sin x + C c) ß cos +C d) 8u - sinu+c e) x -cosx+c f) - 6 cos t - sin t + C g) 6! - 7 cos! +C h) sin x + 5 sin x + ß x+c Q: a) - c) -5 d) ß - ß e) 9 + f) + ß Q4: a) - c) d) 4 square units Q5: a) At the oints of intersection sin x = cos x sin x = (cos x 6= ) cos x tanx= x= ß 4 or 5ß 4 Thusa has coordinates Shaded area = square units Q6: a) s (t) = 5t - cost + ß s = 5ß + ß 4, 5ß and b has coordinates 4, cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC Exercise 7 (age 57) Q7: a) 6 (x + 5)8 +C 5 (t - )5 +C c) - 4 (4x - ) - +C d) 5 (-x) -5 +C e) 6 (4r + )/ +C f) 5 (x +6) /5 +C Q8: q a) (x +9) +C - 4 (x - ) +C c) x+4+c d) - q ( -4x) 5 +C q e) (5x - ) +C f) - +C 6 q(4x + 7) Q9: a) y = - (-x)5 +C y = - Q4: a) 4 4 (4t+) +C - 5 4 c) d) 9 e) 8 f) - Q4: cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC a) u=- q (5 -t) x = (t + ) / +4 Q4: a) 7 ß :65 Exercise 8 (age 59) Q4: a) sin (x - ) +C - cos (x + 5) +C c) sin (x - ) +C d) - 4 cos 4x + ß +C e) sin x +C f) - cos x+ß +C Q44: a) c) 4 6 = d) Q45: = 6 a) square unit. Q46: + 4 square units. a) At the oints of intersection sin x = cos x hence cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC sin x cos x tanx = = cos x cos x (cos x 6= ; at the oints of intersection) tan x = ) x = ß 6 or 7ß 6 ) x = ß square units or 7ß Q47: a) cos x = cos x- cos x + = cos x cos x + = cos x cos x = cos x + 4 sin x + x+c Q48: a) sin x= ( - cos x) x- sin x + C 4 Q49: - 4 cos x + C Q5: a) - sin x + C sin 6x + x+c c) - cos x + C 5 d) x - x+ sin 4x + C 8 eview exercise in further diff and int (age 6) Q5: a) sin x = 4 cos x cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC Q5: f (x)=-4(x-5) -5 Q5: a) - cosx+c -4sinx+C c) 4 Advanced review exercise in further diff and int (age 6) Q54: a) - sin x cos x - 6 5x = - sin x - 6 5x 4 c) + 5 cos x x/ d) +sinxcosx= + sin x x x e) - Q55: cos x - sin x a) x 4 - x/ +cosx+c c) 5 4 d) + x = (+x)/ + C and Q56: a) Q57: 4 +x =8 (cos x + sin x) = cos x + cos x sin x + sin x = cos x + sin x + cos x sin x ß 4 + = + sin x sin 5 x = 5sin 4 x cos x sin 4 x cos x = 5 sin5 x+c cfl HEIOT-WATT UNIVESITY 6

4 ANSWES: TOPIC Q58: cos x = cos (x + x) = cos x cos x - sin x sin x = cos x- cos x - sin x cos x sin x = cos x - cos x - sin x cos x = cos x-cosx- - cos x cos Z = cos x - cos x - cos x + cos x = 4 cos x - cos x cos x = Z (cos x + cos x) 4 Q59: Q6: 4 a) Note that Hence When then Since dt = sin x + 4 sin x + C square units = at a stationary value then x +y = y =-x y = -x T =x+y =x+ -x T = x + -x =x+ - x / dt =+ - x / ( - x) x =- - x - x - x = x = - x x = - x cfl HEIOT-WATT UNIVESITY 6

ANSWES: TOPIC 5 c) x = - x x =4 - x x =8-4x 5x =8 x =6 x = 4 (only ositive values are valid) When x = 4 then y = (from x +y = ) Thus the greatest length of the arms of the "L" is T = x + y = units. Set review exercise in further diff and int (age 65) Q6: This answer is only available on the course web site. Q6: This answer is only available on the course web site. Q6: This answer is only available on the course web site. cfl HEIOT-WATT UNIVESITY 6

6 ANSWES: TOPIC Logarithmic and exonential functions evision exercise (age 68) Q: a) a 7 emember to add the indices. b 6 emember to subtract the indices. c) a 6 For a ower of a ower - multily the indices. d) a 5 b 4 c abc b c a Ξ ab c = a b 5 c c ab =ab4 c Q: When x =, the function crosses the y-axis. That is, when y = The oint where it crosses the y-axis is (, ) When y =, the function crosses the x-axis. That is, when x -x+=) (x - )(x - ) = It crosses the x-axis at the two oints (, ) and (, ) Q: Using the oint (, ) in the equation gives =a +k) =+k) k= Using the oint (, ) and k = in the equation gives =a +) a =9) a= (remember that a is ositive) The equation of the function is f (x) = x + Q4: A) For the related exonential the formula will take the form y = x +k eflection in the line y=xshowsthat y = x + k will ass through the oint (, ) and this gives = +k) k= The equation of the exonential grah is y = x + and the logarithmic function is f (x) = log x+ B) In a similar manner to grah A, k can be found and is equal to - The equation is f (x) = log x- C) The related exonential asses through the oints (, -) and (, ) The oint (, -) when substituted into y = a x + k gives k = - Then using (, ) gives =a -) a= Thus the equation of the log grah is f (x) = log x- D) The related exonential grah, y=a x + k asses through the oints (, -) and (, 6) Using (, -) gives k = - and subsequently using (, 6) gives 6=a -) a= The equation of the logarithmic grah is f (x) = log x- cfl HEIOT-WATT UNIVESITY 6