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MATHEMATICS Guide to Trigonometry, Geometry & Calculus GRADE: 12 S.C. Nhlumayo 2 FOREWORD Thank you for acquiring this booklet on Trigonometry, calculus & Geometry. I just hope that you will find it useful. The booklet can be used in grade 11 and 12. The focus, obviously, is on trigonometry, calculus & Euclidean geometry. These topics are tested in paper 1 & 2 of the National senior certificate examination. One can see that we do need to pay much attention to these topics. It is because of this reason I have decided to write this booklet. Albert Einstein once said that the human mind is forever trying to master everything about something. We, mathematics teachers, will never stop trying hard to make mathematics accessible to all our learners. The idea behind this project accessible to all learners. is to make mathematics The main focus is on using theories learnt in class to answer typical exam questions. For any errors you find in this booklet, please communicate with me via email. [email protected] do Thank you!!!! S.C. Nhlumayo [email protected] 3 TRIGONOMETRY Trigonometry is the branch of mathematics which is based on the relationship between the angles and the sides of a triangle. The roots of the subject lie thousands of years in the past when early humans became interested in astronomy and used it to predict the seasons and planting time and for navigation and cartography. The Babylonians understood some of the basic principles, but it was the ancient Greeks who began to develop the concepts of modern trigonometry systematically. The Greek astronomer Ptolemy constructed the first table of sines in the second century AD. Trigonometry in its modern form is a highly evolved area of mathematics and is used in fields such as engineering, astronomy, architecture and surveying. Trigonometric equations represented graphically are extremely important in physics, chemistry, astronomy, meteorology and even in economics The tides of the sea, the appearance of Halley’s Comet, the vibrations of a molecule are all examples of periodic functions. I am now going to give a summary of the things that you need to know before you sit for grade 12 final examinations. The summary will be followed by a series of activities that you must complete as you prepare for the exam. S.C. Nhlumayo [email protected] 4 TRIGONOMETRIC IDENTITIES sin x sin 2 x cos 2 x 1 cos x You need to know the above identities because, usually, they are never included in the formula sheet. tan x sinA B sin A cos B cos A sin B cosA B cos A cos B sin A sin B sin 2A 2 sin A cos A cos 2A cos2 A sin2 A 1 2 sin2 A 2 cos2 A 1 THE THREE ELEMENTS( x ; y & r) IN SPACE(xy-plane) Remember the following definitions: If θ is the angle that the terminal ray(r) makes with the positive x-axis then sin y r cos x r tan y x The above identities will help you: Solve trigonometric equations. Simplify trigonometric expressions. Prove more complex identities. Sometimes these will be used together with reduction formulae S.C. Nhlumayo [email protected] 5 WORKING ON THE XY-PLANE USING THE THREE ELEMENTS. What are the steps? I explain the steps using an example. If 5 sin 3 0 900; 2700 with the aid of a diagram, Show that sin2 cos2 1 STEP1: Write the first part in the form Trig function = fraction 3 sin 5 This, together with the given restriction, will help you decide on the quadrant to choose. STEP2: Decide on the quadrant to use. The given restriction tells us to choose between quadrant 2 and quadrant 3 but in STEP1 we see that sin is positive and we know we have to go for the 2nd quadrant where sin is positive. You can now draw your diagram. STEP3: Consider the signs. ‘r is always positive In the 2nd quadrant y is positive In the 2nd quadrant x is negative. From trig definitions we know that sin y r y = 3 and r = 5 STEP4: Use the Pythagorean theorem to look for the 3rd element, in this case x. x 2 y2 r2 You can then answer the question. S.C. Nhlumayo [email protected] 6 EXERCISES: Use the information attempt the following exercises. from the previous pages and SCN1 1. 2. 3. 15 8 2 sin x 3 cos x 4 sin x cos x If tan x a prove b sin cos a b sin cos ab If tan that: P is the point in the first quadrant such that 13 cos 5 0 R is a point in the second quadrant such that POR =900 3.1 Determine: tan 3.2 k S.C. Nhlumayo 00 x 900 calculate [email protected] 7 SCN2 8 900 2700 , with the 17 aid of a diagram determine: Given: sin 2.1 tan 2.2 sin(900 ) 2.3 cos 2 SCN3 If 2 tan 3 0 3.1 sin( 900) 3.2 cos 2 SCN4 If sin 24 = p, determine 900 3600 the ff in 4.1 Cos 240 4.2 Sin 480 4.3 Cos 330 4.4 Sin 120 cos 120 – sin (-660) tan 2040 4.5 Cos 156o S.C. Nhlumayo determine terms of Sometimes you need to apply some clever techniques to solve these problems. [email protected] p 8 SCN5 Simplify 5.1 sin(180 x). tan(180 x). cos(x). sin(90 x) cos(x 360). cos(90 x). tan(x) 5.2 tan 425. tan(295) cos 115. cos(65) 5.3 tan 135. tan(105). tan 735 5.4 5 sin2 4 cos2 5.5 sin(180 x). cos(90 x) cos(180 x). cos(360 x) 6.1 If a cos b SCN6 and c sin d show that: a2c2 b 2c2 a2d2 6.2 6.3 Determine Write down the the maximum value of minimum value of sin x cos x 1 3 2 sin 2 6.4 f(x) 14 cos x 5 sin x. It is also given that f(x) R cos(x ) (00 ; 900) 6.4.1 Find the value of R and . 6.4.2 Write down the minimum S.C. Nhlumayo value of 14cosx -5sinx [email protected] 9 SCN7 If tan(A B) 7.1 7.2 7.2.1 Prove sin(A B) cos(A B) that: tan(A B) hence: If 1 tan α ; 3 0 α 90 Prove: tan 2 tan A tan B 1 tan A tan B and tan β 1 7 0 β 90 3 4 7.2.2 Evaluate: tan(2 ) 7.3 Three squares are arranged as shown below: 7.3.1 S.C. Nhlumayo Show that HINT: think in terms of the tangents of the angles!!!! [email protected] 10 SCN8 Prove the following identities 8.1 2 sin x sin 2x 2 sin x 4 3 cos x cos 2x 5 2 cos x 8.2 sin 4t sin 2t tan t cos 4t cos 2t 8.3 sin A sin 3A 2 sin A sin2 A cos2 A 8.4 sin 3x sin x sin x 2 2 cos 2x 8.5 3 sin(x 60) sin(x 30) cos x SCN9 9.1 If cos x 2 sin y cos y 0 Show that: x 2y 90 9.2 Determine the general solution: 9.2.1 1 4 sin2 A 5 sin A cos 2A 0 9.2.2 sin 2θ 4 cos θ 9.2.3 sin(3y 50) cos(y 10) 0 9.3 If cos(α β) a that a b cos α cos β 2 S.C. Nhlumayo x and y are acute 3 sin θ 2 3 0 and cos(α β) b prove [email protected] 11 SCN10 10.1 10.2 10.3 Prove that sin 6x cos 6x 2 sin 2x cos 2x Simplify: sin3x sin7x cos3x cos7x (HINT: 3x 5x - 2x and 7x 5x 2x ) Consider the identity: 1 cos 2 tan sin 2 10.3.1 For which values of θ is the identity defined? 10.3.2 Prove the identity. 10.3.3 Deduce that tan 150 2 3 determine cos 20o in terms of w. 10.4 If sin 50 w, 10.5 If sin14 0 y determine cos 38o in terms of y. SCN11 Determine the general solution 11.1 11.2 sin x cos 2x 1. 6 cos x 5 4 cos x ; cos x 0 11.3 cos 2x 1 3 cos x 11.4 sin 2 x cos 2x cos x 0 11.5 sin x 2 cos 2 x 1 S.C. Nhlumayo [email protected] 12 SCN12 12.1 a. b. 12.2 12.3 Given: sin sin 1 cos( ) cos( ) 2 Prove the identity Hence, simplify: 1 2 sin 700 0 2 sin 10 0 0 0 Prove that cos 20 cos 40 cos 80 1 8 2 2 If cos sin 1 then show that: cos4 sin4 1 2 cos6 sin6 1 3 SCN13 Use compound angle identities to verify reduction formulae 13.1 sin(1800 x) sin x 13.2 cos(1800 x) cos x 13.3 sin(900 x) cos x 13.4 tan(1800 x) tan x 13.5 cos(3600 x) cos x 13.6 cos(2700 x) sin x S.C. Nhlumayo [email protected] 13 SCN14 14.1 Given that P and Q are both acute, solve for P and Q if 1 sin P sin Q cos P cos Q and 2 1 sin P cos Q cos P sin Q 2 14.2 If cos 61o = p, express the following in terms of p. a. sin 209o 14.3 b. cos(-421o) c. cos 10. Prove: cos(A B) cos(A B) 2 cos A cos B Hence, evaluate cos 750 cos150 14.4 show that sin 3x 4 sin x cos 2 sin x sin 3x sin x 2 2 cos 2 x for which values of x is the above identity valid? hence, or otherwise prove the identity: 14.5 If sin θ = p 15.1 If sin 230 = p, write down the following in terms of p. and 2cos θ = 4p, find the value of p. SCN15 cos 1130 cos 230 sin 460 15.2 S.C. Nhlumayo Simplify: sin4 x 4 cos2 x cos4 x 4 sin2 x [email protected] 14 SOLUTION TO TRIANGLES In any triangle ABC a b c sin A sin B sin C 1 ab sin C 2 1 ac sin B 2 1 bc sin A 2 (sine rule) Area ABC (area rule) a2 b2 c2 2bc cos A b2 a2 c2 2ac cos B (cosine rule) c2 a2 b2 2ab cos C Types of triangles: Right angled triangle Oblique triangle (one angle equals 900) (no angle equals 900) Choosing which rule to use: Right angled triangle - Pythagoras theorem………..’if you know two sides and looking for the 3rd side. - SOH-CAH-TOA………..’if you know one angle and one side. Oblique triangle - Area rule……. For calculating the area of a triangle. - sine rule: (S-S-A); (A-A-S) - cosine rule: (S-S-S) ; (S-A-S) (S-A-S) All these rules can be used whether you are responding to a 2D or 3D problem. S.C. Nhlumayo [email protected] 15 In the figure below, CD is a vertical mast. The points B, C, and E are in the same horizontal plane. BD and ED are cables joining the top of the mast to pegs on the ground. DE =28.1 m and BC = 20.7 m. The angle of elevation of D from B is 43.6o. CBE = 63o and BDE = 35.7o SCN16 16.1 Calculate the length of BD 16.2 Calculate the length of BE. 16.3 Calculate the area of ∆BEC. S.C. Nhlumayo [email protected] 16 SCN17 In the diagram below, TQR is a horizontal plane on a sports field. PQ is a vertical flag pole. TQ = TR = y. QR = x units 17.1 Show that sin sin 2 17.2 Hence, prove that in ∆PQR 2 y cos PR cos 17.3 If TR = 2QR, prove that the area of ∆TQR = x 2 . cos 2 17.4 Calculate the area of ∆TQR if x = 25 units and θ = 30o S.C. Nhlumayo [email protected] 17 SCN18 The diagram below shows a rectangular block of wood. A plane cut is made through the vertices E, G and B, revealing a triangular plane EBG. 18.1 Calculate the lengths of EB, EG and BG. 18.2 Hence, determine the magnitude of EBG. 18.3 Determine the area of ∆EBG. SCN19 ABCD is a parallelogram with AC =7cm and Calculate the numerical values of a S.C. Nhlumayo BD =12cm and b [email protected] 18 SCN20 In the figure below, AC =e Show SCN21 that and BD = f e2 f 2 2(a2 b2) In the ff diagram, AB =2p, BC = 2q, DC =AC= q & AD = p Show that cos 1 4 SCN22 BD = x and AC = 6 Calculate the value of x. S.C. Nhlumayo [email protected] 19 SCN22 22.1 22.2 22.3 SCN23 Express MC in terms of h and Show If that BC h = 25mm and BC Area of ∆BCM Consider 3h cos sin = 43o calculate: ∆ABC 23.1 Use Sine Rule and show that: a b sin A sin B c sin C 23.2 If c2 a2 b 2 2ab cos C show that: 2ab 1 2 1 cos C c2 a b S.C. Nhlumayo [email protected] 20 SCN24 In the figure alongside, B, C and D are 3 points on the same horizontal plane and AB is a vertical pole of length p metres. The angle of elevation of A from C is θ. 24.1 Express CDB in terms of θ. 24.2 Express BC in terms of p and a trigonometric function of θ. 24.3 Hence, show that p 41 3 tan 24.4 Use the results in c to calculate the value of θ if the length of the pole is 12 m. S.C. Nhlumayo [email protected] 21 SCN25 25.1 Nomabham’bheshe is standing on level ground, at B, a distance of 19 metres away from the foot E of a tree TE. She measures the angle of elevation of the top of the tree at a height of 1.55 metres above the ground as 32˚. Calculate the height TE of the tree. Give your answer correct to 3 significant figures 25.2 When Amuh is standing 400m from the base of a mountain, the angle of elevation from the ground to the top of the mountain is θ. He then walks 500m straight back and measure the angle of elevation to be . a. b. c. Express h in terms of x and θ. Express h in terms of x and . If θ = 25o and = 20o , solve for x and then find the height of the mountain, h. S.C. Nhlumayo [email protected] 22 EUCLIDEAN GEOMETRY In Euclidean geometry, I concentrate a lot on applying theorems and postulates in solving riders. Bookwork (proving basic theorems) is not covered in this booklet. You can check proofs in your textbook. You are advised to know all the theorems so that you will be able to use them in proving riders. Although theorems won’t be proven in this booklet, we’ll look at the proof of the theorem of Pythagoras using similar triangles. LOGIC – The way of reasoning. In Euclidean geometry we use the “if-then” statement when we analyse and trying to find meaning from the theorems and postulates. This is the same reasoning we use in the real world. Look at this argument: All rugby players drink too much. This means that, if John is a rugby player then he drinks too much. This is how we work with theorems. Take the theorem, break it into substatements and then see which statements are bases and which one is the conclusion Look at this argument from Newton2 of motion: An object will remain at rest or continue moving at constant velocity unless acted upon by an unbalanced force. Bases: the object is at rest. The object is moving at constant velocity. Conclusion: forces acting on the object are balanced A theorem is just an argument which you need to break as shown above so that you will understand it. Trying to memorise a theorem is a total waste of time because memorizing is not understanding. S.C. Nhlumayo [email protected] 23 THE NINE BASIC THEOREMS ON THE CIRCLE A line segment drawn from the centre to the midpoint of a chord is perpendicular to that chord The angle at the centre of a circle is twice the angle at the circumference. (both angles subtended by the same arc) An angle subtended by a diameter is 900. S.C. Nhlumayo [email protected] 24 Opposite angles of a cyclic quadrilateral add up to 1800 The exterior angle of a cyclic quadrilateral equals the interior opposite angle. Angles in the same segment are equal S.C. Nhlumayo [email protected] 25 An angle between a tangent and a radius is 90o. Tangents from the same point outside the circle are equal in length. An angle between a tangent and a chord equals the angle subtended by that chord in the alternate segment. S.C. Nhlumayo [email protected] 26 THE THEOREM OF PYTHAGORAS There are a lot of ways of proving the pythagorean theorem but in this section we’ll use similar triangles to prove the theorem. The proof has got two parts. Part one: if you drop a perpendular from a vertex of right angle of a right angled triangle to the hypotenuse you will generate two more triangles which are similar to each other and to the original triangle. GIVEN: Right angled triangle ABC with AD BC R.T.P: ∆ABC///∆DBA///∆DAC PROOF: In ∆ABC and ∆DBA D1 = A………….each equals 90o B is common A1 = C………..third angle ∆ABC /// ∆DBA In ∆ABC and ∆DAC C is common A = D2 A2 = B,……third angle ABC ///DAC The two small triangles are both similar to the bigger triangle. We can conclude that ∆ABC///∆DBA///∆DAC S.C. Nhlumayo [email protected] 27 For part two, all you need to remember is that, if triangles are similar then their corresponding sides are proportional. (sides opposite equal angles) In ∆ABC and ∆DBA AB BC AB2 BD BC BD AB In ∆ABC and ∆DAC AC BC AC2 BC DC DC AC PART TWO 2 2 2 R.T.P: BC AB AC Proof: by substitution AB2 AC2 BC BD BC DC BC(BD DC) BC BC BC2 That is how prove the pythagorean theorem in grade 12. It’s the same theorem you proved in grade 9 using four congruent triangles. It looks like proving the theorem of Pythagoras is easy. The important part is that of proving the three triangles to be similar. (PART 1) S.C. Nhlumayo [email protected] 28 REMEMBER!!!!!!! If triangles are similar then the following equivalent. Triangles are equingular Corresponding sides are proportional two statements are THE PROPORTINALITY THEOREM A line parallel to a side of a triangle cuts the other two sides so as to divide them (internally or externally) in the same proportion. (line // one side of a∆) Since DE // BC then AD AE BD EC AB AC AD AE you can use any combination There is a special case of this theorem called the mid-point theorem. You did it in grade 10. However I’ll remind you. S.C. Nhlumayo [email protected] 29 THE MIDPOINT THEOREM The line segment joining the midpoints of the two sides of a triangle is parallel to the third side. Moreover, the line segment is half the length of the third side. Now, Let’s analyse the theorem and find meaning. There is a triangle. You join the midpoints of the two sides You will get two results: 1. The new line segment is parallel to the third side. 2. The new line segment is half the length of the third side YZ // WX YZ 1 WX 2 This was just a summary of the basic theorems that you really have to understand before you can attempt any euclidean geometry problem in grade 12. You are now going to attempt the following activities using the above theorems. S.C. Nhlumayo [email protected] 30 SCN26 A, B, C & D are points on the circle as shown below. AC is the diameter. Calculate the values of x and y SCN27 T is a point outside circle O. TA & TB are tangents to the circle. TPQ is a secant. X is the midpoint of PQ. Prove: OXBT is a cyclic quadrilateral BOT = BXT OT bisects A MA//QT S.C. Nhlumayo [email protected] 31 SCN28 PT is a tangent. Calculate the value of x, y and z SCN 29 Prove that PT2 = PA PB S.C. Nhlumayo [email protected] 32 SCN 30 Prove that: MKP = PKN SMK = KTN SCN 31 Prove that QS S.C. Nhlumayo 1 PS 2 [email protected] 33 SCN 32 Calculate: A1 D1 O1 B1 B2 A2 C1 D2 SCN 33 Determine : S.C. Nhlumayo C ADB BDC [email protected] 34 SCN 34 Prove that ACBD is a cyclic quadrilateral Can you conclude that AC//DB? Discuss. SCN 35 Determine AD BC S.C. Nhlumayo [email protected] 35 SCN 36 Calculate the value of x. SCN 37 Name the similar triangles in this figure. Prove: DE2 = AE . DC CE2 + DE2 = BC2 + BE2 + AD2 + AE2 S.C. Nhlumayo CE2 BE 2 DE EA [email protected] 36 SCN 38 Find 7 angles which are equal to Q1 Deduce that RB is a common tangent to circles BPT and BAQ. If BQ:QT = 7 : 4 and PB = 5 Find AP Below I have included a number of theorems and postulates that may help you. Don’t be scared. Just read through. You will get used to them. You are advised to attempt more geometry problems in the previous P3 question papers. Well, this is just for enrichment S.C. Nhlumayo [email protected] 37 Postulates and Theorems Converse of the AEA Theorem: If two lines are cut by a transversal congruent alternate exterior angles, then the lines are parallel. forming Interior Supplements Theorem: If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. Converse of the Interior Supplements Theorem: If two lines are cut by a transversal forming interior angles on the same side of the transversal that are supplementary, then the lines are parallel. Parallel Transitivity Theorem: If two lines in the same plane are parallel to a third line, then they are parallel to each other. Perpendicular to Parallel Theorem: If two lines in the same plane are perpendicular to a third line, then they are parallel to each other. SAA Congruence Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. Angle Bisector Theorem: Any point on the bisector of an angle equidistant from the sides of the angle. Perpendicular Bisector Theorem: If a point of a segment, then it is equally distant segment. is is on the perpendicular bisector from the endpoints of the Converse of the Perpendicular Bisector Theorem: If a point is equally distant from the end-points of a segment, then it is on the perpendicular bisector of the segment. Isosceles Triangle Theorem: If a angles are congruent. triangle is isosceles, then Converse of the Isosceles Triangle: Theorem If two angles of a congruent, then the triangle is isosceles. its base triangle are Converse of the Angle Bisector Theorem: If a point is equally distant the sides of an angle, then it is on the bisector of the angle. Perpendicular Bisector Concurrency Theorem: The bisectors of a triangle are concurrent. Angle Bisector Concurrency Theorem: The are concurrent. S.C. Nhlumayo from three perpendicular three angle bisectors of a triangle [email protected] 38 Triangle Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Quadrilateral Sum Theorem: The quadrilateral is 360°. sum of the measures of the Medians to the Congruent Sides Theorem: In an isosceles medians to the congruent sides are congruent. four angles of a triangle, the Angle Bisectors to the Congruent Sides Theorem: In an isosceles the angle bisectors to the congruent sides are congruent. Altitudes to the Congruent Sides Theorem: In an isosceles altitudes to the congruent sides are congruent. triangle, triangle, the Isosceles Triangle Vertex Angle Theorem: In an isosceles triangle, the altitude to the base, the median to the base, and the bisector of the vertex angle are all the same segment. Parallelogram Diagonal Lemma: A diagonal of a parallelogram divides parallelogram into two congruent triangles. Opposite Sides Theorem: The opposite the sides of a parallelogram are congruent. Opposite Angles Theorem: The opposite angles of a parallelogram are congruent. Converse of the Opposite Sides Theorem: If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Converse of the Opposite Angles Theorem: If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Opposite Sides Parallel and Congruent Theorem: If one pair of opposite of a quadrilateral are parallel and congruent, then the quadrilateral parallelogram. Rhombus Angles Theorem: Each diagonal of a angles. rhombus bisects sides is a two opposite Parallelogram Consecutive Angles Theorem: The consecutive angles of a parallelogram are supplementary. Four Congruent Sides Rhombus Theorem: If a quadrilateral has sides, then it is a rhombus. four congruent Four Congruent Angles Rectangle Theorem: If a quadrilateral has congruent angles, then it is a rectangle. S.C. Nhlumayo four [email protected] 39 Rectangle Diagonals Theorem: The diagonals of a rectangle are congruent. Converse of the Rectangle Diagonals Theorem: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Isosceles Trapezoid Theorem: The base angles of an congruent. isosceles Isosceles Trapezoid Diagonals Theorem: The diagonals of an trapezoid are congruent. trapezoid are isosceles Converse of the Rhombus Angles Theorem: If a diagonal of a parallelogram bisects two opposite angles, then the parallelogram is a rhombus. Double-Edged Straightedge Theorem: If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus. Tangent Theorem A tangent: to a circle drawn to the point of tangency. is perpendicular to the radius Converse of the Tangent Theorem: A line that is perpendicular to a radius at its endpoint on the circle is tangent to the circle. Perpendicular Bisector of a Chord Theorem: The perpendicular bisector of a chord passes through the center of the circle. Inscribed Angle Theorem: The measure of an angle equals half the measure of its intercepted arc. inscribed Inscribed Angles Intercepting Arcs Theorem: Inscribed angles the same or congruent arcs are congruent. in a circle that intercept Cyclic Quadrilateral Theorem: The opposite angles of a cyclic quadrilateral are supplementary. Parallel Secants Congruent Arcs Theorem: Parallel arcs on a circle. lines intercept congruent Parallelogram Inscribed in a Circle Theorem: If a parallelogram is inscribed within a circle, then the parallelogram is a rectangle. Tangent Segments Theorem: Tangent congruent. segments from a point to a circle are Intersecting Chords Theorem: The measure of an angle formed by two intersecting chords is half the sum of the measures of the two intercepted arcs. S.C. Nhlumayo [email protected] 40 Intersecting Secants Theorem: The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measure of the larger intercepted arc and the measure of the smaller intercepted arc. AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. SAS Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. SSS Similarity Theorem: If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar. two Corresponding Altitudes Theorem: If two triangles are similar, then corresponding altitudes are proportional to the corresponding sides. Corresponding Medians Theorem: If two triangles are similar, then corresponding medians are proportional to the corresponding sides. Corresponding Angle Bisectors Theorem: If two triangles are similar, then corresponding angle bisectors are proportional to the corresponding sides. Parallel/Proportionality Theorem: If a line passes through two sides of a triangle parallel to the third side, then it divides the two sides proportionally. Converse of the Parallel/Proportionality Theorem: If a line passes through two sides of a triangle dividing them proportionally, then it is parallel to the third side. Three Similar Right Triangles Theorem: If you drop an altitude from the vertex of a right angle to its hypotenuse, then it divides the right triangle into two right triangles that are similar to each other and to the original right triangle. Altitude to the Hypotenuse Theorem: The length of the altitude hypotenuse of a right triangle is the geometric mean between of the two segments on the hypotenuse. to the the length The Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. S.C. Nhlumayo [email protected] 41 Converse of the Pythagorean Theorem: If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right triangle. Hypotenuse Leg Theorem: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent. Square Diagonals Theorem: The diagonals of a perpendicular, and bisect each other. square are congruent, Converse of the Parallelogram Diagonals Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Converse of the Kite Diagonal Bisector Theorem: If only one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite. Triangle Mid-segment Theorem: A mid-segment of a triangle the third side and half the length of the third side. is parallel to Rectangle Midpoints Theorem: If the midpoints of the four sides of a rectangle are connected to form a quadrilateral, then the quadrilateral formed is a rhombus. Rhombus Midpoints Theorem: If the midpoints of the four sides of a rhombus are connected to form a quadrilateral, then the quadrilateral formed is a rectangle. Kite Midpoints Theorem: If the midpoints of the four sides of a kite are connected to form a quadrilateral, then the quadrilateral formed is a rectangle. Product of Interior Segments Theorem: If two chords intersect in a circle, then the product of the segment lengths on one chord is equal to the product of the segment lengths on the other chord. In the next section, we look at differential calculus. DIFFERENTIAL CALCULUS S.C. Nhlumayo [email protected] 42 In calculus we study the mathematics of motion and rates of change. Calculus was developed in the seventeenth century to meet the mathematical needs of Physicists. It enables Scientists to define slopes of curves and to calculate velocities and acceleration of moving bodies like cannon balls and planets. Calculus deals with two major problems, namely tangent and area problems. The study of the tangent problem is called Differentiation and the study of the area problem is called integration. Both these studies rest on a more fundamental concept known as a limit. 1. THE TANGENT AND THE DERIVATIVE. You already know how to calculate the average gradient of a curve using the NEWTON’S QUOTIENT: f ( x h) f ( x ) Where h is the horizontal distance between two h chosen points. Well, we are now going to look at how we can approximate the gradient of a curve at a certain point. To accomplish this, we are going to use the truth that a straight line has a constant gradient. You have already used that when calculating the average gradient of a curve. It is clear that a curve does not have a constant gradient since its steepness changes as you move along the curve. Consider the following diagram: S.C. Nhlumayo [email protected] 43 PQ is the secant ( a line segment joining two points on a curve) RPS is the tangent to the curve of f. (a line segment that touches the curve externally at only one point.) h is the horizontal distance between P and Q. If P is fixed and PQ is rotated clockwise then: a. The horizontal distance (h) between P and Q becomes negligible (h→0). b. PQ will eventually coincide with RS. This means it will become a tangent to the curve of f. c. As you rotate further, the line will get back to being a secant and then a tangent. S.C. Nhlumayo [email protected] 44 This means that, the gradient of RS is the limiting value as h becomes negligible (as h tends to zero.) Calculating the gradient of RS is the same as calculating the gradient of the curve of f at P. We can now sum up what we have just said. Determining the gradient of a curve includes using the limiting process to the NEWTON’S QUOTIENT. f ( x h) f ( x ) h h 0 This can be written mathematically as: f ' ( x ) lim This is called the derivative. You remember I said a curve has so many gradients. A set of all gradients of a curve is called the derivative. The derivative can be used to find the gradient of a curve at any point. FINDING THE DERIVATIVE OF A CURVE USING THE LIMITING PROCESS (i.e. from first principles). Example. Given: f ( x) 2 x 2 f ' ( x ) lim h 0 lim h0 lim h0 lim h0 lim h0 4x f (x h) f ( x) h 2 2 x h 2 x 2 h 2 2 x 2 xh h 2 2 x 2 h 2 2 x 4 xh 2 h 2 2 x 2 h h 4 x 2 h h t his results from cancelling S.C. Nhlumayo h with h and " forgeting 2h" [email protected] 45 You remember I said, in the limiting process h becomes negligible. I did not say h becomes zero. Most learners are tempted to replace h with zero, which is WRONG. One other mistake most learners do is omitting lim h0 just after writing the formula and continuing without writing it. By so doing, they lose marks. You must only omit it in the final step where you write the answer. SCN39. Determining the derivative from first principles (i.e. using the limiting process) 1. f ( x) x 2 2. f ( x) 3x 2 3. f ( x) x 2 2 4. f ( x) 2 3x 2 5. f ( x) 1 1 2 x 2 SCN40. Determine the derivative from first principles in each case 1. f ( x) 2 x 3 2. f ( x) x 3 3. f ( x) 4 2 x 3 4. f ( x ) 2 x 5. f ( x ) 3 x S.C. Nhlumayo [email protected] 46 SCN41. In each case, find the derivative of f(x) at the point where x = 1, using the definition of the derivative. 1. f ( x) 2 x 2 5 x 1 2 5 2 3. f ( x) x 4 x 3 2. f ( x) x 2 3x 1 4 5. f ( x) 3 x 2 5 4. f ( x) x 2 6 x 14 6. f ( x) 5 x 2 2 x 2 x 7. f ( x) 4 FINDING THE DERIVATIVE USING THE POWER RULE RULE: If y ax n then dy anx n1 (for any real number n) dx Examples : dy (2 3) x 31 Given: y 2x then dx 6x2 3 Most people fail calculus because they are not good in algebra. It is important that you keep horning your skills in algebra so that you will enjoy calculus. S.C. Nhlumayo [email protected] 47 A LOOK AT FRACTIONS AND RADICALS 1 x2 x 2 y 1. If y 2. If then x x x then then dy 1 43 x dx 3 1 2 dx 2 x 3 dy 1 12 x dx 2 1 y 3 x 3. If dy 1 3 OTHER RULES. d f ( x ) g ( x ) d f ( x ) d g ( x ) dx dx dx d kf ( x) k d f ( x) k a constant dx dx SCN42 Determine 3 dy in each case: dx 2 1. y 2 x 4 x 6 x 3 4 2 2. y x 3 x 2 x 1 6 x 3x 4 8x 2 7 2 3 2 4. y 4 x x 7 x 1 3. y SCN43 Determine the derivative using rules 1. f ( x) x 2 ( x 3) 2 2. y (3 x )( x 2) 2 3. y x 1 x 3 S.C. Nhlumayo 2 [email protected] 48 SCN44. Find the derivative using rules: 1. y 3 x x 2. y 3. y 2 2 1 3 x3 x x x6 3 x4 4. y 3 x 2 SCN45: Determine the derivative using rules x3 2 x 2 1. y x 1 x4 2. y 13 x 2 x 2 4x 3 x 3. y 5 x 1 x 4. y 2x 1 2 x x 5. y 2x 3 S.C. Nhlumayo [email protected] 49 SCN46: Determine 1. Dx 2 x 3 x 2 2. d 3 x 1 dx 3 x 3. Dx 1 4. Dx 5. d dx 2 2 x x 2 3 x x x x3 2 2 SCN47: Determine the derivative using rules 2x x2 1. y 3 2 3 2. yx x 2 y 8 2 3. 2 xy 2 x 7 x 6 2 4. x 3 y 2 x 18 3 3 1 6 x 5. y x TANGENTS TO CURVES At the beginning I said we can use the gradient of a tangent to a curve to find the gradient of that particular curve. We can reverse this. Given the equation of a curve and the point of contact, we can find the equation of a tangent. S.C. Nhlumayo [email protected] 50 SCN48: Determine the equation of a tangent to a curve in each case. 1. f ( x) x 3 4 x 2 11x 30 at x 1. 2. f ( x) 3 x 2 x 1 at x 2 3. y 4. y 6 x at x - 2 4 1 x at x 2 SCN49 2 1. Find the equation of the tangent to y x 4 x which is parallel to y 2 x 3 2 2. Find the equation of the tangent to y x 2 x which is parallel to y x 4 0 f ( x ) ax 3 bx 2 3. The gradient of the tangent to the curve is equal f(1) 5. to 13 Calculate where x the values = -1 and of a and b. 2 4. If y 2 x is a tangent to the curve of y x ax b at (2; 4), find the values of a and b. 5. Show that y x2 x 2 y x 1 is a tangent to at (1;2) 2 6. If y 4 x 4 is a tangent to y x ax b at x= 1, find a and b. S.C. Nhlumayo [email protected] 51 APPLICATIONS OF THE DERIVATIVE. Calculus gives you the tools you need to measure change both qualitatively and quantitatively. It has widespread applications in science and engineering and is used to solve complex and expansive problems for which algebra alone is insufficient. Differential calculus explores and analyses rates of change quantitatively and qualitatively. You are now going to use the derivative to solve many mathematical problems. We will start by looking at how we can use calculus in curve sketching. CURVE SKETCHING Initially I said tangents are used to calculate the gradients of curves. We all know that when a function is increasing, its gradient is positive. If the function is decreasing then its gradient is negative. What happens at stationary points? S.C. Nhlumayo [email protected] 52 Consider the following diagram: You can see that the tangents (dotted lines) to f and g are horizontal lines. The gradient tells you by how much y changes as x changes. With horizontal lines, as x changes y remains constant. From the gradient formula we can see that m y 0 0 . The x x gradient of a horizontal line is always zero. This tells us that at stationary points, the gradient is zero. We can therefore say that the derivative is zero. We are going to use this truth in finding the x-coordinates of the stationary points. S.C. Nhlumayo [email protected] 53 Example: Determine co-ordinates of the turning points of f ( x) x 3 4 x 2 x We have just said that at the turning points, the derivative is zero. f ' ( x ) 3 x 2 8 x 1 f ' ( x) 0 3x 2 8 x 1 0 then by u sin g the quadratic x 0.13 x 7.87 and formula you can then use these values of x to find the correspond ing values of y y -(0.13) 3 4(0.13) 2 (0.13 ) 0.06 y (7.87) 3 4(7.87) 2 (7.87) 247.57 Min(7.87; -247.57) max(0.13; -0.06) CONCAVITY AND POINTS OF INFLECTION Read this several times until you understand. If x a is a stationary point of a function f (x) then: f " (a) 0 means that the curve is concave up at x a and there is a minimum turning point there. f " (a) 0 Means that the curve is concave down and there is a maximum turning point there. The concavity of a graph at any value of x is determined by the sign of its second derivative at that value of x. S.C. Nhlumayo [email protected] 54 To find the point of inflection you need to use the second derivative to find x. And then use the original function to find the y-coordinate. WORKSHEETS The only recipe for sure success in mathematics is practice, practice, practice and more practice. SCN50 1. For which values of b is the graph of y x 4 bx 3 5 x 2 6 x 8 concave down when x 2 ? 2. For which values of a is the graph of y x 3 ax 2 9 x concave up when x 3? 3. Determine the x-coordinates of the stationary points of the graph f ( x) x 3 3x 2 25x 21 of and then use calculus methods to determine where the maximum and the minimum values occur. 3 2 4. Given y x dx 25x 21 and the point of inflection is (-1 ; 48). Determine the value of d. 3 2 5. The point of inflection of y ax 3x 36 x 37 occurs at x 6. For 1 . Determine the value of a. 2 which values of x is the graph defined by y x 3 3 x 2 2 concave up? S.C. Nhlumayo [email protected] 55 SCN51 3 2 Consider the function: f ( x ) 2 x 3x 36 x 37 1. Show that x 1 is a factor of f. 2. Determine the x-intercepts of f. 3. Determine the stationary points of f. 4. Determine the point of inflection of f. 5. Sketch the graph of f in the xy-plane below. y 84 77 70 63 56 49 42 35 28 21 14 7 x -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 -7 -14 -21 -28 -35 -42 6. Determine the turning points of f(x+3) 7. Determine the turning points of f(x) + 1. 8. Determine the equation of the tangent to f at x = 1. S.C. Nhlumayo [email protected] 56 SCN52 3 2 Consider the function: g ( x) x 4 x 11x 30 1. Determine the coordinates of the turning points of g. 2. Determine the point of inflection of g. 3. Calculate the x and y-intercepts of g. 4. Sketch the graph of g in the Cartesian plane below. 40 y 35 30 25 20 15 10 5 x -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 -5 -10 -15 5. Determine the equation of the tangent to the graph of g at x =1. 6. Determine the x-coordinate of the point at which the tangent in question 5 cuts the graph of g again. 7. Determine the value(s) of p for which g(x) = p will have only two roots. S.C. Nhlumayo [email protected] 57 SCN53 Given: f ( x) ( x 1)( x 1)( x 3) 1. Write down the x-intercepts of f. 2. Calculate the y-intercept of f. 3. Determine the turning points of f. 4. For which values of x is f concave down? 5. Write down the point of inflection of f. 6. In the Cartesian plane below, sketch the graph of f. y 4 3 2 1 x -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 -1 -2 -3 -4 7. Determine the values of x for which f ( x) 0 S.C. Nhlumayo [email protected] 58 SCN54 2 Given: h( x ) ( x 1) ( x 4) 1. Determine the x and y-intercepts of h. 2. Determine the stationary points of h. 3. Sketch the graph of h. y x -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 -4 -8 -12 -16 4. Determine the turning points of g if g ( x) h( x 3) 5. Determine the turning points of h( x) 2 6. Determine the point of inflection of h. 7. For which values of x is h' ( x) 0? S.C. Nhlumayo [email protected] 59 SCN55 3 Given: f ( x) x 12 x 16 1. Solve for x if f ( x) 0 2. Determine the coordinates of the turning points of f. 3. Determine the inflection point of f. 4. Sketch the graph of f in the xy-plane below. y 5 x -6 -4 -2 2 4 6 -5 -10 -15 -20 -25 -30 5. For which values of x is f ' ( x). f ( x) 0? S.C. Nhlumayo [email protected] 60 SCN56 3 2 Given: f ( x) 2 x 3 x 12 x 4 1. For which values of x is f concave down? 2. Write down the point of inflection of f. 3. Determine the coordinates of the turning points of f. 4. If x – 2 is a factor of f then calculate all xintercepts of f. 5. Calculate the y-intercept of f. 6. Sketch the graph of f in the xy-plane below. y 16 14 12 10 8 6 4 2 x -6 -4 -2 2 4 6 -2 -4 -6 -8 -10 -12 7. For which values of x is f ' ( x) 0 S.C. Nhlumayo [email protected] 61 GRAPH INTEPRETATION You are now able to use calculus methods to sketch the cubic function. The next thing you must practise is using calculus to interpret graphs. We are going to look at the graph of the cubic function and the graph of the derivative. I’m now starting to enjoy Calculus. I’m now ready to learn more S.C. Nhlumayo [email protected] 62 SCN57 (GIVEN THE DEFINING EQUATION AND THE GRAPH) Consider the graph of f ( x ) (5 x )(x 1) . A and B are the 2 stationary points. A and C are the x-intercepts of f. y B f x A C 1. For which values of x is f concave up? 2. Write down the coordinates of A and C. 3. Determine the coordinates of B. 4. Determine the equation of the tangent to f at x = 2. 5. For which values of x is f ' ( x) 0 ? 6. For which values of k will f(x) = -k have only one root? S.C. Nhlumayo [email protected] 63 SCN58 (GIVEN THE DEFINING EQUATION AND THE GRAPH) 3 2 The graph below shows the graph of f ( x ) x 4 x 11x 30 . A and B are the turning points. B y x O f A 1. Calculate the x-intercepts of f. 2. Determine the coordinates of A and B. 3. For which values of x is f decreasing? 4. Determine the gradient of f at x =-2. 5. Determine the point of inflection of f 6. Determine the turning points of f(x-1). S.C. Nhlumayo [email protected] 64 SCN59 (GIVEN THE DEFINING EQUATION AND THE GRAPH) Given the graph of g ( x) x 3 4 x 2 4 x 6 . A is the turning point of g. y x g A 1. Determine the coordinates of A. 2. Determine the point of inflection. 3. For which values of x is g increasing? 4. Determine the value of x where the gradient of g is -8. 5. Determine the equation of the tangent to the graph of g at x = 4. S.C. Nhlumayo [email protected] 65 SCN60. (Given x-cuts and one point on the graph) 3 2 Given the graph of f ( x ) ax bx cx d . (1 ; 16) is the point on the graph of f. the graph of f cuts the x-axis at x = -3, x = -1 and x = 2. y f (1 ; 16) x -3 -1 2 1. Determine the values of a, b , c and d. 2. Determine the coordinates of the turning points of f. 3. Determine the equation of the tangent to f at x = 1. 4. Determine the turning points of f(x)-3. 5. For which values of x is f decreasing? 6. For which values of x is f concave up? S.C. Nhlumayo [email protected] 66 SCN61 Sketched above is the graph of f ( x) 1 3 x ax 2 bx 10 . It is 2 further given that the graph of f cuts the x-axis at x =-2 , x = 2 and x =5. 1. Show that a 5 and b 2 2 2. Write down the y-cut of f. 3. Determine the turning points of f. 4. Determine the equation of the tangent to f at x = -1. 5. Determine the point of inflection of f. 6. For which values of x is f ( x ). f ' ( x ) 0 ? S.C. Nhlumayo [email protected] 67 SCN62 Sketched below is the graph of h( x ) x ax bx c . 3 2 y (2 ; 0) (-3 ; 0) x h A 1. Show that a =4, b = -3 and c = -18 2. Determine the coordinates of A, the turning point of h. 3. Determine the values of x for which h is decreasing. 4. Determine the point of inflection of h. 5. Determine the equation of the tangent to h at x = -1. 3 2 6. For which values of k will x 4 x 3x 18 k have three distinct roots? S.C. Nhlumayo [email protected] 68 SCN63 Consider the graph of f ( x ) x 3 ax 2 bx 18 . A is the turning point of f. y A f (2 ; 0) x (-3 ; 0) 1. Determine the values of a and b. 2. Calculate the x-coordinate of A. 3. For which values of x is f concave down? 4. Write down the point of inflection of f. 5. For which values of x is the gradient of f equals to 8? 6. Hence, write down the equation of the tangent to f at this point. S.C. Nhlumayo [email protected] 69 SCN64 3 2 Sketched below is the graph of f ( x) x 4 x ax b y A f (-2 ; 0) x (3 ; 0) 1. Show that a = -3 and b = 18. 2. Determine the coordinates of A, the turning point of f. 3. For which values of x is f ' ( x) 0? 4. For which values of x is f ( x). f ' ( x) 0 ? 5. Determine the point of inflection of f. S.C. Nhlumayo [email protected] 70 SCN65 3 2 Sketched below is the graph of g ( x ) x ax bx 18 . D is the turning point of g. y (3 ; 0) (-2 ; 0) x g D 1. Write down the y-intercept of g. 2. Determine the values of a and b. 3. For which values of x is g decreasing for x 0 ? 4. Determine the coordinates of D. 5. Determine the gradient of the tangent to g at x = 2. 6. For which values of x is g ' ( x) 0 ? S.C. Nhlumayo [email protected] 71 SOMETIMES YOU WILL BE GIVEN TURNING POINTS ONLY SCN66 3 2 Sketched below is the graph g ( x) x ax bx . A and B are the turning points of g. y A(-1 ; 3.5) x g B(2 ; -10) 1. Show that a 3 2 and b 6 2. Determine the x-intercepts of g. 3. Determine the turning points of g ( x 2) 4. Determine the equation of the tangent to g at x = -2. 5. Determine the x-coordinate of the point of inflection of g. 6. For which values of x is g concave up? 3 7. For which values of s will x 3 2 x 6 x s 0 have ONE 2 real root? S.C. Nhlumayo [email protected] 72 SCN67 3 2 Sketched below is the graph of f ( x) x ax bx 30 . A and B are the turning points of f. y B(11/3 ; 14.8) x f A(-1 ; -36) 1. Determine the values of a and b. 2. For which values of x is f ' ( x) f ( x) 0 ? 3. Determine the value(s) of k such that 16 x y k 0 is a tangent to the graph of f. 4. Determine the x-coordinates of the turning points of f ( x) 4 . 5. For which values of x is f(x) =0? 6. Write down the equation of h if h( x) f ( x) 2 S.C. Nhlumayo [email protected] 73 GIVEN A TURNING POINT AND AN INFLECTION POINT SCN68 3 2 The diagram below shows the graph of f ( x) 2 x ax bx 37 . A and C are the turning points of f. B is the point of inflection of the graph of f. y A(-2 ; 81) B(0.5 ; 18.5) f x C 1. Show that a 3 and b -36 2. For which values of x is f concave up? 3. Determine the coordinates of C. 4. Determine the x-coordinate of the point where the tangent to f at C cuts the graph of f. S.C. Nhlumayo [email protected] 74 SCN69 3 2 Sketched below is the graph of g ( x) ax 11x bx . B and C are stationary points of g. A is the point of inflection of g. y C g x A(11/6 : -2.65) B(3 : -9) 1. Determine the values of a and b. 2. Determine the x-intercepts of g. 3. Determine the coordinates of C. 4. Find the equation of the tangent to the graph of g at the point (2 ; -4). 5. Calculate the co-ordinates of the point where this tangent cuts the curve again S.C. Nhlumayo [email protected] 75 THE GRAPH OF THE DERIVATIVE. SCN70 2 Sketched below is the graph of f ' ( x ) ax bx c . The x-cuts are given as shown in the diagram. f is a cubic function. y f'(x) x -2 5 1. Write down the x-coordinates of the stationary points of f. 2. State whether each stationary point is a local minimum or a local maximum. Support your answer. 3. Determine the x-coordinate of the point of inflection of f. 4. For which values of x is f decreasing? 5. Draw a sketch graph of f. S.C. Nhlumayo [email protected] 76 SCN71 Consider the graph of f ' where f is a cubic function. y f'(x) x -2 5 1. For which values of x is f increasing? 2. For which values of x is f decreasing? 3. Write down the x-coordinate of the point of inflection. 4. Write down the x-coordinates of the turning points of f and state whether each stationary point is a local maximum or a local minimum. 5. Sketch the graph of f. S.C. Nhlumayo [email protected] 77 SCN72 Sketched below is the graph of g ' ( x) 1 x 52 4 where g is 2 a cubic function y C A x B g' D 1. Determine the x-coordinates of A, B and C. 2. Write down the x-coordinates of the stationary point of g. 3. For which values of x is g decreasing? 4. For which values of x is g increasing? 5. For which values of x is g concave up? 6. For which values of x is g concave down? 7. Write down the x-coordinate of the point of inflection of g. S.C. Nhlumayo [email protected] 78 OPTIMISATION Apply the derivative of a function to solve optimization problems Apply the derivative of a function to determine the rate of change of physical quantities. OPTIMIZATION: Looking for the best possible outcome. Seeking the best possible outcome could mean one of the following: Maximisation (seeking the largest possible outcome) or Minimisation (seeking the smallest possible outcome) Some physical quantities which can be maximized or minimized: Lengths and distances Areas and surface areas Volumes Velocity Acceleration Steps in Optimisation Draw a diagram, if necessary Set up a function for the object/ quantity to be optimised. Take note of the restrictions that could be connected to the practical situation. (e.g. it would be wrong to find something like -3 seconds) Use the first derivative to determine the critical numbers. Use the second derivative to determine whether maximums or minimums occur at the critical numbers. S.C. Nhlumayo [email protected] 79 SCN73. (NOV 2010: DIAGRAM MODIFIED) The object below is made of a cylinder with a hemisphere at each end. The radius of the cylinder is r and its height is h. The volume of the object is 1. Show that h . 6 1 4r 2 6r 3 2. Hence, show that the outer surface area of the object is given by S 4 r 2 3 3r 3. Calculate the minimum outer surface area of this object. V r 2 h (Cylinder) A 2rh (curved surface area of a cylinder) 4 V r 3 (Sphere) 3 TSA 4r 2 (Sphere) S.C. Nhlumayo [email protected] 80 SCN74 Ayabonga wants to make a bird-cage. She has 180 cm of wire and bends it to form a rectangular prism with an extra band around its middle. 1. Show that L 45 3 x 2. Show that the equation for the volume 3 enclosed by the frame is V 3x 45x 2 3. Calculate the dimensions of the bird-cage that will give the maximum volume. S.C. Nhlumayo [email protected] 81 SCN75 A cylinder fits exactly into one half of a spherical container as shown. The radius of the sphere is 2 3 cm. The height of the cylinder is x. 1. Show that r 2 12 x 2 2. Calculate the value of x that will maximize the volume of the cylinder. 3. Calculate the maximum volume 2 Volume of a Cylinder r h S.C. Nhlumayo [email protected] 82 SCN76 Gubevu wants to make an open box (i.e. a box has no lid) with a square base and a volume of 2.5 m3. 1. Determine the height of the box in terms of x. 2. He wants to paint the inner and the outer surface area of the box, including the base. Show that the area to be 2 painted is given by the equation: A 2 x 20 x 3. Hence, determine the value of x that will give the minimum area to be painted. S.C. Nhlumayo [email protected] 83 SCN77 A cylinder fits in a cone as shown in the diagram below. The length from the rim of the cylinder to the center of the base is 2 3 . The radius of the cylinder is r and its height is x. 2 3 2 1. Show that r 12 x 2 2. Write down the equation of the volume of the cylinder in terms of x. 3. Calculate the value of x that will maximize the volume of the cylinder. 4. Calculate the radius of the cylinder if it is of maximum volume. S.C. Nhlumayo [email protected] 84 SCN78. An object is made of a cylinder and a hemisphere. The radius of the cylinder is x mm. the total length of the object is to be (600 –x) mm as shown, 1. Show that the volume of the object is V 600x 2 4 3 x 3 2. Find the value of x at which maximum volume occurs. SCN79 1 4 3 The equation d (t ) t 1 2 t 4 shows the depth of water in 2 metres in a water tank. The time, t, is the number of hours that have passed since 8 a.m. 1. What is the depth of water at 10:30 a.m? 2. At what rate is the depth of water changing at 11:00 a.m? 3. At what time will the inflow of water be the same as the outflow? S.C. Nhlumayo [email protected] 85 SCN80 The relationship between the number of employees working for your uncle and the profit he makes is given by the formula P( x) 2 x 3 600 x 10000 1. Determine the number of employees he needs for the business to make maximum profit. 2. Calculate maximum profit. SCN81 It is given that, the number of bacteria at any moment be represented by the formula can f (t ) 5t 2 50t 1000 where f(t) is the number of bacteria (in millions) present and t is the number of hours after the beginning of the experiment. 1. Calculate the rate of change of the number of bacteria after 2 hours. 2. After how many hours does the population of bacteria start decreasing? 3. How long will it take for all the bacteria to die. S.C. Nhlumayo [email protected] 86 CALCULUS OF MOTION. In calculus of motion you need to know that the motion of an object is defined by its position relative to a fixed point, its velocity and acceleration. The term used for position is displacement. Velocity is the rate of change of displacement with respect to time. Acceleration is the rate of change of velocity with respect to time. If s f (t ) is the equation of motion, which gives position at time t Then v And a ds gives the velocity at time t. dt dv dt is the acceleration at any time t. ____________________________________________________________ S.C. Nhlumayo [email protected] 87 SCN82 The height that a stone is above ground level at any time, t seconds after being catapulted up is given by S (t ) 25t 5t 2 1. Determine the velocity of the stone at the instant that it is catapulted. 2. Determine the maximum height that the stone reaches. 3. What is the acceleration of the stone while it is in the air? SCN83. The stone is thrown vertically upwards. Its height, s metres at any given time(sec) is given by S (t ) 20t 5t 2 1. Find what height the stone is after (i) 1sec (ii) 2 sec (iii) 3sec 2. At what time will the stone again hit the ground? 3. What is the greatest height reached by the stone? 4. Find an expression for the acceleration of the stone. S.C. Nhlumayo [email protected] 88 SCN84 The distance in metres covered by two objects in t seconds is given by the f (t ) t 3 2t 2 3t 4 functions and g (t ) 3t 2 5t 2 . 1. After how many seconds is the velocity of the two objects the same? 2. How far has each object travelled after 2 seconds? 3. What is the acceleration of an object that has a velocity given by r (t ) 6t 5 ? SCN85 Let the displacement (in metres) of an object be represented by the function S (t ) 2t 2 7t 4 where t is the time in seconds. 1. Determine the velocity of the object at t = 3 seconds. 2. Determine the acceleration of the object at any time. S.C. Nhlumayo [email protected] 89 This marks the end of my worksheets on Trigonometry, Geometry and Calculus. Unfortunately I could not provide solutions to my worksheets. I trust teachers to help learners in that case. If you get stuck, you may still communicate with me via email: [email protected] or call me: 082 291 7325/ 072 922 5920. S.C. Nhlumayo [email protected]