Core Maths C2. Revision Notes


 Cody Ferguson
 2 years ago
 Views:
Transcription
1 Core Maths C Revision Notes November 0
2 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations... 6 Trigonometr... 6 Radians... 6 Connection between radians and degrees... 6 Arc length, area of a sector and area of a segment... 6 Trig functions... 6 Basic results... 6 Eact values for 0 o, 45 o and 60 o... 7 Graphs of trig functions... 7 Graphs of = sin n, = sin( ), = sin( + n) etc Sine & Cosine rules and area of triangle... 8 Identities... 8 Trigonometric equations... 9 Coordinate Geometr... 0 Mid point... 0 Circle... 0 Centre at the origin... 0 General equation... 0 Equation of tangent... 4 Sequences and series... Geometric series... Finite geometric series... Infinite geometric series... Proof of the formula for the sum of a geometric series... Binomial series for positive integral inde... 4 Pascal s triangle... 4 Factorials... 4 n Binomial coefficients or n C r or... 4 r 5 Eponentials and logarithms... 5 Graphs of eponentials and logarithms... 5 Rules of logarithms... 5 Changing the base of a logarithm... 6 Equations of the form a = b... 7 C 4/04/0 SDB
3 6 Differentiation... 7 Increasing and decreasing functions... 7 Stationar points and local maima and minima (turning points) Using second derivative... 8 Using gradients before and after... 9 Maimum and minimum problems Integration... Definite integrals... Area under curve... Numerical integration: the trapezium rule... Inde... C 4/04/0 SDB
4 Algebra Polnomials: +,,,. A polnomial is an epression of the form a n n + a n n +... a + a + a 0 where all the powers of the variable,, are positive integers or 0. +,, of polnomials are eas, must be done b long division. Factorising General eamples of factorising: ab + 6ac = a(b + c ) = ( )( ) 6 = ( 6) 6 0 = ( + )( 5) Standard results: = ( )( + ), difference of two squares ( + ) = + +, ( ) = + Long division See eamples in book Remainder theorem If 67 is divided b 6 the quotient is 04 and the remainder is. This can be written as 67 = In the same wa, if a polnomial P() = a 0 + a + a +... a n n is divided b (c + d) to give a quotient, Q() with a remainder r, then r will be a constant (since the divisor is of degree one) and we can write P() = (c + d) Q() + r If we now choose the value of which makes (c + d) = 0 = d / c then we have d P( c ) = 0 Q() + r d P( c ) = r. C 4/04/0 SDB
5 Theorem: If we substitute = d / c in the polnomial we obtain the remainder that we would have after dividing the polnomial b (c + d). Eample: The remainder when P() = a  4 is divided b (  ) is 7. Find a. Solution: Choose the value of which makes (  ) = 0, i.e. = /. Then the remainder is P( ) = 4 ( )  ( ) + a ( ) a ( )  4 = a ( ) = 7 a = = 7 You ma be given a polnomial with two unknown letters, a and b. You will also be given two pieces of information to let ou form two simultaneous equations in a and b. Factor theorem Theorem: If, in the remainder theorem, r = 0 then (c + d) is a factor of P() d P( c ) = 0 (c + d) is a factor of P(). Eample: A quadratic equation has solutions (roots) = / and =. Find the quadratic equation in the form a + b + c = 0 Solution: The equation has roots = ½ and = it must have factors ( + ) and ( ) b the factor theorem the equation is ( + )( ) = 0 5 = 0. Eample: Show that ( ) is a factor of P() = and hence factorise the epression completel. Solution: Choose the value of which makes ( ) = 0, i.e. = remainder = P() = = = 0 ( ) is a factor b the factor theorem. We have a cubic and so we can see that the other factor must be a quadratic of the form (6 + a ) and we can write = ( ) (6 + a ) Multipling out we see that the 6 The term is and +6 terms are correct. 4 C 4/04/0 SDB
6 Multipling out the term comes from must come to + a which a = 7. We must now check the term which is 9. Multipling out with a = 7 the term is ( 7) + ( ) 6 = 7 = 9, which works! = ( )(6 7 ) which is now factorised completel. = ( )( )( + ) Choosing a suitable factor To choose a suitable factor we look at the coefficient of the highest power of and the constant (the term without an ). Eample: Factorise Solution: is the coefficient of and has factors of and. 6 is the constant and 6 has factors of,, and 6 so the possible linear factors of are ( ± ), ( ± ), ( ± ), ( ± 6) ( ± ), ( ± ), ( ± ), ( ± 6) But ( ± ) = ( ± ) and ( ± 6) = ( ± ), so the are not new factors. We now test the possible factors using the factor theorem until we find one that works. Test ( ), put = giving Test ( + ), put = giving ( ) + ( ) ( ) Test ( ), put = giving = = 0 and since the result is zero ( ) is a factor. We now divide in, as in the previous eample, to give = ( )( + 5 ) = ( )( )( + ). C 4/04/0 SDB 5
7 Cubic equations Factorise using the factor theorem then solve. N.B. The quadratic factor might not factorise in which case ou will need to use the formula for this part. Eample: Solve the equation + = 0. Solution: Possible factors are ( ± ) and ( ± ). Put = we have + = 0 ( ) is not a factor Putting = we have + = = 0 ( ) is a factor + = ( )( + ) = 0 = or + = 0 this will not factorise so we use the formula = or = ± ( ) 4 = 0.68 or. Trigonometr Radians A radian is the angle subtended at the centre of a circle b an arc of length equal to the radius. Connection between radians and degrees 80 o = π c Degrees Radians π / 6 π / 4 π / π / π / π / 4 5π / 6 π π / π Arc length, area of a sector and area of a segment Arc length s = rθ and area of sector A = ½r θ. Area of segment = area sector area of triangle = ½r θ ½r sinθ. Trig functions Basic results sin A tan A= ; cos A sin( A) = sin A; cos( A) = cos A; tan( A) = tan A; 6 C 4/04/0 SDB
8 Eact values for 0 o, 45 o and 60 o From the equilateral triangle of side we can read off sin 60 o = / sin 0 0 = ½ cos 60 o = ½ cos 0 o = / 0 o tan 60 o = tan 0 o = / 60 o and from the isosceles right angled triangle with sides,, we can read off sin 45 0 = / cos 45 o = / tan 45 o = 45 o Graphs of trig functions = sin = cos = tan Graphs of = sin n, = sin( ), = sin( + n) etc. You should know the shapes of these graphs = sin is like = sin but repeats itself times between 0 o and 60 o = sin( ) = sin and = tan( ) = tan are the graphs of = sin and = tan reflected in the ais. = cos( ) = cos is just the graph of = cos. = sin( + 0) is the graph of = sin translated through 0. 0 C 4/04/0 SDB 7
9 Sine & Cosine rules and area of triangle a b c = = : be careful the sine rule alwas gives ou two answers for each sin A sin B sinc angle so if possible do not use the sine rule to find the largest angle as it might be obtuse; or ou might want both answers if it is possible to draw two different triangles from the information given. a = b + c bc cosa etc. Unique answers here! Area of a triangle = ab sinc = bcsin A = acsin B. Identities sin A tan A = cos A Eample: Solve sin = 4 cos. Solution: First divide both sides b cos sin = 4 tan = 4 cos = 5. o, or. o. tan = 4 / sin A + cos A = Eample: Given that cos A = 5 / and that 70 o < A < 60 o, find sin A and tan A. Solution: We know that sin A + cos A = sin A = cos A sin A = ( 5 / ) sin A = ± /. But 70 o < A < 60 o sin A is negative sin A = /. sin A Also tan A = cos A tan A = 5 = / 5 =.4. 8 C 4/04/0 SDB
10 Eample: Solve sin + sin cos = Solution: Rewriting cos in terms of sin will make life easier so using sin + cos = cos = sin sin + sin cos = sin + sin ( sin ) = sin + sin = 0 ( sin )(sin + ) = 0 sin = / or = 4.8 o, 8.9 o, or 70 o. N.B. If asked to give answers in radians, ou are allowed to work in degrees as above and then convert to radians b multipling b π / 80 So answers in radians would be = 4.80 π / 80 = 0.70, or π / 80 =.4, or 70 π / 80 = π /. Trigonometric equations Eample: Solve sin ( π / 4 ) = 0.5 for 0 c π c, giving our answers in radians in terms of π. Solution: First we know that sin 60 o = 0.5, and 60 o = π / radians π / 4 = π / or π π / = π / = 7π / or π  /. Eample: Solve sin = 0.47 for 0 o 60 o, giving our answers to the nearest degree. Solution: First put X = and find all solutions of sin X = 0.47 for 0 o X 70 o X = 8., or = 5.9 or = 88., or = 5.9 i.e. X = 8., 5.9, 88., 5.9 = 4 o, 76 o, 94 o, 56 o to the nearest degree. There are several eamples in the book. C 4/04/0 SDB 9
11 Coordinate Geometr Mid point The mid point of the line joining P (a, b ) and Q (a, b ) is ( ( a a ), ( b b )) Circle + +. Centre at the origin Take an point, P, on a circle centre the origin and radius 5. Suppose that P has coordinates (, ) P (, ) Using Pthagoras Theorem we have + = 5 + = 5 r which is the equation of the circle. and in general the equation of a circle centre (0, 0) and radius r is + = r. General equation In the circle shown the centre is C, (a, b), and the radius is r. CQ = a and PQ = b and, using Pthagoras r P (, ) b CQ + PQ = r ( a) + ( b) = r, C (a, b) a Q which is the general equation of a circle. Eample: Find the centre and radius of the circle whose equation is = 0. Solution: First complete the square in both and to give = = 5 ( ) + ( + ) = 5 which is the equation of a circle with centre (, ) and radius 5. 0 C 4/04/0 SDB
12 Eample: Find the equation of the circle on the line joining A, (, 5), and B, (8, 7), as diameter. Solution: The centre is the mid point of AB is ( 8), (5 7) ) ( and the radius is ½ AB = ½ (8 ) + ( 7 5) = 6. 5 equation is ( 5.5) + ( + ) = = (5½, ) Equation of tangent Eample: Find the equation of the tangent to the circle = 64 which passes through the point of the circle ( 6, 4). Solution: First complete the square in and in to give ( + ) + ( ) = 69. Net find the gradient of the radius from the centre (, ) to the point ( 6, 4) which is / 5 gradient of the tangent at that point is 5 /, since the tangent is perpendicular to the radius and product of gradients of perpendicular lines is equation of the tangent is 4 = 5 ( + 6) 5 = 98. Eample: Find the intersection of the line = + 4 with the circle + = 5. Solution: Put = + 4 in + = 5 to give + ( + 4) = = = 0 (5 + )( + ) = 0 =. or = 0.4 or line intersects circle at (., 0.4) and (, ) If the two points of intersection are the same point then the line is a tangent. Note. You should know that the angle in a semicircle is a right angle and that the perpendicular from the centre to a chord bisects the chord (cuts it eactl in half). C 4/04/0 SDB
13 4 Sequences and series Geometric series Finite geometric series A geometric series is a series in which each term is a constant amount times the previous term: this constant amount is called the common ratio. The common ratio can be or, and positive or negative. Eamples:, 6, 8, 54, 6, 486,..... with common ratio, 40, 0, 0, 5, ½, ¼,..... with common ratio ½, ½,, 8,, 8, 5,.... with common ratio 4. Generall a geometric series can be written as S n = a + ar + ar + ar + ar up to n terms where a is the first term and r is the common ratio. The nth term is u n = ar n. The sum of the first n terms of the above geometric series is n r S n = a r a r n = r. Eample: Find the n th term and the sum of the first terms of the geometric series whose rd term is and whose 6 th term is 6. Solution: 6 = r 6 = r r = 8 r = Now = r = r = ( ) = ½ n th term, n = ar n = ½ ( ) n and the sum of the first terms is S = ( ) = S = 4 ½ Infinite geometric series When the common ratio is between and + the series converges to a limit. S n = a + ar + ar + ar + ar up to n terms n r and S n = a. r Since r <, r n 0 as n and so a S n S = r C 4/04/0 SDB
14 Eample: Show that the following geometric series converges to a limit and find its sum to infinit. S = ¾ + Solution: Firstl the common ratio is / 6 = ¾ which lies between  and + therefore the sum converges to a limit. a 6 The sum to infinit S = = r S = 64 4 Proof of the formula for the sum of a geometric series You must know this proof. S n = a + ar + ar + ar + ar n + ar n, multipl through b r r S n = ar + ar + ar + ar n + ar n + ar n subtract S n r S n = a ar n ( r) S n = a ar n = a( r n ) n n r r S n = a = a. r r Notice that if < r < + then r n 0 and S n S = a r. C 4/04/0 SDB
15 Binomial series for positive integral inde Pascal s triangle When using Pascal s triangle we think of the top row as row 0. row 0 row row row row 4 row 5 row To epand (a + b) 6 we first write out all the terms of degree 6 in order of decreasing powers of a to give a 6 + a 5 b + a 4 b + a b + a b 4 + ab 5 + b 6 and then fill in the coefficients using row 6 of the triangle to give a 6 + 6a 5 b + 5a 4 b + 0a b + 5a b 4 + 6ab 5 + b 6 = a 6 + 6a 5 b + 5a 4 b + 0a b + 5a b 4 + 6ab 5 + b 6 Factorials Factorial n, written as n! = n (n ) (n ).... So 5! = 5 4 = 0 Binomial coefficients or n n C r or r If we think of row 6 starting with the 0 th term we use the following notation 0 th term st term nd term rd term 4 th term 5 th term 6 th term C 0 6 C 6 C 6 C 6 C 4 6 C 5 6 C where the binomial coefficients n C r or n are defined b r 4 C 4/04/0 SDB
16 n n n! C r = = r ( n r)! r! This is particularl useful for calculating the numbers further down in Pascal s triangle e.g. The fourth term in row 5 is 5 5 5! 5! 5 4 C 4 = = = = 5 7 = 65 4 (5 4)!4!!4! 4 You ma find an n C r button on our calculator. Eample: Find the coefficient of in the epansion of ( ) 5. Solution: The term in is 5 C ( ) = 0 ( 8) = 80 so the coefficient of is Eponentials and logarithms Graphs of eponentials and logarithms = is an eponential function and its inverse is the logarithm function = log. Remember that the graph of an inverse function is the reflection of the original graph in = = = log Rules of logarithms log a = = a log a = log a + log a log a ( ) = log a log a log a n = n log a log a = 0 log a a = Note: log 0 is often written as lg C 4/04/0 SDB 5
17 Eample: Find log 8. Solution: Write log 8 = 8 = = 4 log 8 = 4. To solve log equations we can either use the rules of logarithms to end with log a = log a = or log a = = a Eample: Solve log a 40 log a = log a 5 Solution: log a 40 log a = log a 5 log a 40 log a 5 = log a log a (40 5) = log a log a 8 = log a = 8 =. Eample: Solve log 5 = + ½ log 5. Solution: log 5 = + ½ log 5 log 5 ½ log 5 =.5 log 5 = log 5 = = 5 = 5. Changing the base of a logarithm log a b = log log c c b a Eample: Find log 4 9. log Solution: log 9 = = =. 4 log A particular case logb b log a b = log a 4. 0 b = This gives a source of eam questions. log a b Eample: Solve log 4 6 log 4 = Solution: log 4 6 = log 4 (log 4 ) log 4 6 = 0 (log 4 )( log 4 + ) = 0 log 4 = or = 4 or 4 = 64 or / 6. 6 C 4/04/0 SDB
18 Equations of the form a = b Eample: Solve 5 = Solution: Take logs of both sides log 0 5 = log 0 log 0 5 = log 0 log0.9 = = =.59. log Differentiation Increasing and decreasing functions is an increasing function if its gradient is positive, is an increasing function if its gradient is negative, Eample: d > 0; d d < 0 d For what values of is = + 7 an increasing function. Solution: = + 7 d = d For an increasing function we want values of for which > 0 Find solutions of = 0 ( + )( ) = 0 =  / or so graph of meets ais at  / and and is above ais for <  / or > So = + 7 is an increasing function for < / or >. C 4/04/0 SDB 7
19 Stationar points and local maima and minima (turning points). An point where the gradient is zero is called a stationar point. minimum maimum Turning points Stationar points of inflection Local maima and minima are called turning points. The gradient at a local maimum or minimum is 0. Therefore to find ma and min firstl differentiate and find the values of which give gradient, d, equal to zero: d secondl find second derivative d and substitute value of found above d second derivative positive minimum, and second derivative negative maimum: N.B. If d = 0, it does not help! In this case ou will need to find the gradient d just before and just after the value of. Be careful: ou might have a stationar point of inflection thirdl substitute to find the value of and give both coordinates in our answer. Using second derivative Eample: Find the local maima and minima of the curve with equation = Solution: = First find d d = At maima and minima the gradient = d d = = = 0 ( + 4) = 0 8 C 4/04/0 SDB
20 ( + 4)( ) = 0 = 4, 0 or. d Second find d = +4 6 When = 4, When = 0, When =, d = = 80, positive min at = 4 d d = 6, negative, ma at = 0 d d = = 0, positive, min at =. d Third find values: when = 4, 0 or = 5, 7 or 0 Maimum at (0, 7) and Minimum at ( 4, 5) and (, 0). d N.B. If = 0, it does not help! You can have an of ma, min or stationar d point of inflection. Using gradients before and after Eample: Find the stationar points of = Solution: = d = 4 + = 0 for stationar points d ( + ) = 0 ( ) = 0 = 0 or. d d = which is (positive) when = 0 minimum at (0, 7) and which is 0 when =, so we must look at gradients before and after. = 0.9. d d = stationar point of inflection at (, ) N.B. We could have ma, min or stationar point of inflection when the second derivative is zero, so we must look at gradients before and after. C 4/04/0 SDB 9
21 Maimum and minimum problems Eample: A manufacturer of cans for baked beans wishes to use as little metal as possible in the manufacture of these cans. The cans must have a volume of 500 cm : how should he design the cans? Solution: h cm We need to find the radius and height needed to make cans of volume 500 cm using the minimum possible amount of metal. Suppose that the radius is cm and that the height is h cm. The area of top and bottom together is π cm and the area of the curved surface is πh cm the total surface area A = π + πh cm. (I) cm We have a problem here: A is a function not onl of, but also of h. But we know that the volume is 500 cm and that the volume can also be written as π h cm π h = 500 h = 500 π and so (I) can be written A = π + π 500 π A = π da d = π = 4π 000 = 4π. For stationar values of A, the area, da d = 0 4π = π = 000 = = = π = 4.0 to S.F. h = 500 = 860. π We do not know whether this value gives a maimum or a minimum value of A or a stationar point of inflection d A 000 so we must find = 4 π = 4 π +. d Clearl this is positive when = 4.0 and thus this gives a minimum of A minimum area of metal is 49 cm when the radius is 4.0 cm and the height is 8.60 cm. 0 C 4/04/0 SDB
22 7 Integration Definite integrals When limits of integration are given. Eample: Find d Solution: d = [ 4 + ] no need for +C as it cancels out = [ 4 + ] [ 4 + ] put top limit in first = [] [ ] =. Area under curve The integral is the area between the curve and the ais, but areas above the ais are positive and areas below the ais are negative. Eample: Find the area between the ais, = 0, = and = 4. Solution: 0 4 d 8 = = [ 8] ] [0 0] 0 6 = which is negative since the area is below the ais required area is Eample: Find the area between the ais, =, = 4 and =. Solution: First sketch the curve to see which bits are above (positive) and which bits are below (negative). = = ( ) A meets ais at 0 and 4 A so graph is as shown. Area A, between and, is above ais: area A, between and 4, is below ais 5 so we must find these areas separatel. C 4/04/0 SDB
23 A = d = = [4.5] [ ] = / 6. and 4 5 d = = [ ] [4.5] = 6 and so area A (areas are positive) = + 5 / 6 so total area = A + A = / + 5 / 6 = 5 / 6. Note that 4 4 = [ ] [ ] = 6 d which is A A = / 5 / 6 = /. 4 Numerical integration: the trapezium rule Man functions can not be anti differentiated and the trapezium rule is a wa of estimating the area under the curve. Divide the area under = f () into n strips, each of width h. = f () Join the top of each strip with a straight line to form a trapezium. n Then the area under the curve sum of the areas of the trapezia 0 h h h a b b f ) d h ( + ) + h ( + ) + h ( + ) h ( + ) a b ( 0 n f ) d h ( ) a b ( 0 n f ) d h ( + + ( )) a ( 0 n n area under curve ½ width of each strip ( ends + middles ). n n C 4/04/0 SDB
24 Inde Area of triangle, 8 Binomial series, 4 binomial coefficients, 4 n C r or n, 4 r Circle Centre at origin, 0 General equation, 0 Tangent equation, Cosine rule, 8 Cubic equations, 6 Differentiation, 7 Equations a = b, 7 Eponential, 5 Factor theorem, 4 Factorials, 4 Factorising eamples, Functions decreasing, 7 increasing, 7 Integrals area under curve, definite, Logarithm, 5 change of base, 6 rules of logs, 5 Mid point, 0 Pascal s triangle, 4 Polnomials, long division, Radians arc length, 6 area of sector, 6 area of segment, 6 connection between radians and degrees, 6 Remainder theorem, Series Geometric, finite, Geometric, infinite, Geometric, proof of sum, Sine rule, 8 Stationar points gradients before and after, 9 maima and minima, 8 maimum and minimum problems, 0 second derivative, 8 Trapezium rule, Trig equations, 9 Trig functions basic results, 6 graphs, 7 identities, 8 Turning points, 8 C 4/04/0 SDB
Core Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationSenior Secondary Australian Curriculum
Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationSolve 9 x = x = 3 3 2x = 3 x = 3 2
www.mathsbo.org.uk CORE Summar Notes Indices Rules of Indices a b = a + b 0 = a n = n m a b = a b + Solve 0 + = 0 ( a ) b = ab = a n = ( n ) m = n m Solve 9 = 7 = = = Let = () 0 + = 0 0 + = 0 ( )( ) =
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationCALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES
6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationQuestion: What is the quadratic formula? Question: What is the discriminant? Answer: Answer:
Question: What is the quadratic fmula? Question: What is the discriminant? Question: How do you determine if a quadratic equation has no real roots? The discriminant is negative ie Question: How do you
More informationFunctions and Graphs
PSf Functions and Graphs Paper 1 Section B 1. The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form. 2 2. hsn.uk.net Page 1 Questions
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationPROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
More informationAQA Level 2 Certificate FURTHER MATHEMATICS
AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationx 2 would be a solution to d y
CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationPolynomials Past Papers Unit 2 Outcome 1
PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : NLYTICL METHODS FOR ENGINEERS Unit code: /0/0 QCF Level: Credit value: OUTCOME TUTORIL LGEBRIC METHODS Unit content Be able to analyse and model engineering situations and solve problems using algebraic
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationSection C Non Linear Graphs
1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,
More informationpp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 11: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 12: Algebraic Expressions
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and
More informationPURE MATHEMATICS AM 27
AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationMathematics. Total marks 100. Section I Pages marks Attempt Questions 1 10 Allow about 15 minutes for this section
04 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Boardapproved calculators ma be
More informationMathematics Paper 1 (NonCalculator)
H National Qualifications CFE Higher Mathematics  Specimen Paper A Duration hour and 0 minutes Mathematics Paper (NonCalculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
More informationExaminers Report. Summer Pearson Edexcel GCE in Core Mathematics 2 (6664/01)
Eaminers Report Summer 014 Pearson Edecel GCE in Core Mathematics (6664/01) Edecel and BTEC Qualifications Edecel and BTEC qualifications are awarded by Pearson, the UK s largest awarding body. We provide
More information1.6 Powers of 10 and standard form Write a number in standard form. Calculate with numbers in standard form.
Unit/section title 1 Number Unit objectives (Edexcel Scheme of Work Unit 1: Powers, decimals, HCF and LCM, positive and negative, roots, rounding, reciprocals, standard form, indices and surds) 1.1 Number
More informationAble Enrichment Centre  Prep Level Curriculum
Able Enrichment Centre  Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationKnow the following (or better yet, learn a couple and be able to derive the rest, quickly, from your knowledge of the trigonometric functions):
OCR Core Module Revision Sheet The C exam is hour 0 minutes long. You are allowed a graphics calculator. Before you go into the exam make sureyou are fully aware of the contents of theformula booklet you
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationNATIONAL QUALIFICATIONS
H Mathematics Higher Paper 1 Practice Paper A Time allowed 1 hour 0 minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions 1 0 (40 marks) Instructions
More informationNew HigherProposed OrderCombined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC
New HigherProposed OrderCombined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiationbut not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationEQ: How can regression models be used to display and analyze the data in our everyday lives?
School of the Future Math Department Comprehensive Curriculum Plan: ALGEBRA II (Holt Algebra 2 textbook as reference guide) 20162017 Instructor: Diane Thole EU for the year: How do mathematical models
More informationIf (a)(b) 5 0, then a 5 0 or b 5 0.
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be realvalued functions in a single variable. A function is realvalued if the input and output are real numbers
More informationPractice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths.
Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths. Your formal test will be of a similar standard. Read the description of each assessment standard carefully to
More informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
More informationUNIVERSITY OF WISCONSIN SYSTEM
Name UNIVERSITY OF WISCONSIN SYSTEM MATHEMATICS PRACTICE EXAM Check us out at our website: http://www.testing.wisc.edu/center.html GENERAL INSTRUCTIONS: You will have 90 minutes to complete the mathematics
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationTriangle Definition of sin and cos
Triangle Definition of sin and cos Then Consider the triangle ABC below. Let A be called. A HYP (hpotenuse) ADJ (side adjacent to the angle ) B C OPP (side opposite to the angle ) sin OPP HYP BC AB ADJ
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More information(x) = lim. x 0 x. (2.1)
Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
More informationPrep for Calculus. Curriculum
Prep for Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular
More informationHigher Education Math Placement
Higher Education Math Placement 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry (arith050) Subtraction with borrowing (arith006) Multiplication with carry
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More informationThe Mathematics Diagnostic Test
The Mathematics iagnostic Test Mock Test and Further Information 010 In welcome week, students will be asked to sit a short test in order to determine the appropriate lecture course, tutorial group, whether
More informationSAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?
SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More information3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?
1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, coordinate geometry (which connects algebra
More informationClever Keeping Maths Simple
Clever Keeping Maths Simple Grade Learner s Book J Aird L du Toit I Harrison C van Dun J van Dun Clever Keeping Maths Simple Grade Learner s Book J Aird, L du Toit, I Harrison, C van Dun and J van Dun,
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationKS4 Curriculum Plan Maths FOUNDATION TIER Year 9 Autumn Term 1 Unit 1: Number
KS4 Curriculum Plan Maths FOUNDATION TIER Year 9 Autumn Term 1 Unit 1: Number 1.1 Calculations 1.2 Decimal Numbers 1.3 Place Value Use priority of operations with positive and negative numbers. Simplify
More informationWe d like to explore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( x)
Inverse Trigonometric Functions: We d like to eplore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( ) sin. Recall the graph: MATH 30 Lecture 0 of 0 Well, we can
More informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationUnit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2
Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More information29 Wyner PreCalculus Fall 2016
9 Wyner PreCalculus Fall 016 CHAPTER THREE: TRIGONOMETRIC EQUATIONS Review November 8 Test November 17 Trigonometric equations can be solved graphically or algebraically. Solving algebraically involves
More informationHigher Mathematics Homework A
Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(3,1) and B(5,3) 2. Find the equation of the tangent to the circle
More informationMaclaurin and Taylor Series
Maclaurin and Talor Series 6.5 Introduction In this block we eamine how functions ma be epressed in terms of power series. This is an etremel useful wa of epressing a function since (as we shall see) we
More informationGCSE Maths Linear Higher Tier Grade Descriptors
GSE Maths Linear Higher Tier escriptors Fractions /* Find one quantity as a fraction of another Solve problems involving fractions dd and subtract fractions dd and subtract mixed numbers Multiply and divide
More informationMath Analysis. A2. Explore the "menu" button on the graphing calculator in order to locate and use functions.
St. Mary's College High School Math Analysis Introducing the Graphing Calculator A. Using the Calculator for Order of Operations B. Using the Calculator for simplifying fractions, decimal notation and
More informationINTEGRATION FINDING AREAS
INTEGRTIN FINDING RES Created b T. Madas Question 1 (**) = 4 + 10 3 The figure above shows the curve with equation = 4 + 10, R. Find the area of the region, bounded b the curve the coordinate aes and the
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More information2009 Chicago Area AllStar Math Team Tryouts Solutions
1. 2009 Chicago Area AllStar Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More information