1627 simultaneous equations and intersections
TRANSCRIPT
Demo DiscDemo Disc
““Teach A Level Maths”Teach A Level Maths”Vol. 1: AS Core ModulesVol. 1: AS Core Modules
© Christine Crisp
Explanation of Clip-art images
An important result, example or summary that students might want to note.
It would be a good idea for students to check they can use their calculators correctly to get the result shown.
An exercise for students to do without help.
38: The Graph of tan43: Quadratic Trig
Equations
29: The Binomial Expansion33: Geometric series – Sum to Infinity
25: Definite Integration
13: Stationary Points
11: The Rule for Differentiation
9: Linear and Quadratic Inequalities
8: Simultaneous Equations and Intersections
6: Roots, Surds and Discriminant
The slides that follow are samples from the 51 presentations that make up the work for the
AS core modules C1 and C2.
18: Circle Problems 46: Indices and
Laws of Logarithms
26: Definite Integration and Areas
6: Roots, Surds and Discriminant
Demo version note: Students have already met the discriminant in solving quadratic equations. On the following slide the calculation is shown and the link is made with the graph of the quadratic function.
For the equation . . . 0742 xx. . . the discriminant
acb 42 12
There are no real roots as the function is never equal to zero
2816
The Discriminant of a Quadratic Function
If we try to solve , we get0742 xx
2124
x
The square of any real number is positive so there are no real solutions to 12
742 xxy0
Roots, Surds and Discriminant
8: Simultaneous Equations and Intersections
Demo version note: The following slide shows an example of solving a linear and a quadratic equation simultaneously. The discriminant ( met in presentation 6 ) is revised and the solution to the equations is interpreted graphically.
14 xy
32 xy
e.g. 214 xy )(232 xy )(1
Eliminate y:
1432 xx
The discriminant,
0)4)(1(444 22 acb0442 xx
0)2)(2( xx(twice)2 x
The quadratic equation has equal roots.
The line is a tangent to the curve.
72 yx
0442 xxSolving
Simultaneous Equations and Intersections
9: Linear and Quadratic Inequalities
Demo version note: Students are shown how to solve quadratic inequalities using earlier work on sketching the quadratic function. The following slide shows one of the two types of solutions that arise.The notepad icon indicates that this is an important example that students may want to copy.
542 xxy 542 xxy
Solution:e.g.2 Find the values of x that satisfy 0542 xx
0542 xx 0)1)(5( xx
5 x or 1x
1 xThere are 2 sets of values of x
Find the zeros of where )(xf 54)( 2 xxxf
542 xx is greater than orequal to 0 above the x-
axis
5xorThese represent 2 separate
intervals and CANNOT be combined
Linear and Quadratic Inequalities
11: The Rule for Differentiation
Demo version note: In this presentation, the rule for differentiation of a polynomial is developed by pattern spotting, working initially with the familiar quadratic function. A later presentation outlines the theory of differentiation.
),( 42Tangent at
2xy (2, 4)x
The Gradient at a point on a CurveDefinition: The gradient at a point on a curve
equals the gradient of the tangent at that point.e.g.
3
12
The gradient of the tangent at (2, 4) is 43
12 mSo, the gradient of the curve at (2, 4)
is 4
The Rule for Differentiation
13: Stationary Points
Demo version note: Stationary points are defined and the students practice solving equations to find them, using cubic functions, before going on to use the 2nd derivative to determine the nature of the points. The work is extended to other functions in a later presentation.
xxxy 93 23
0dxdy
The stationary points of a curve are the points where the gradient is zero
A local maximum
A local minimum
x
x
The word local is usually omitted and the points called maximum and minimum points.
e.g.
Stationary Points
18: Circle Problems
Demo version note:
The specifications require students to know 3 properties of circles. Students are reminded of each and the worked examples, using them to solve problems, emphasise the need to draw diagrams.
e.g.2 The centre of a circle is at the point C (-1, 2). The radius is 3. Find the length of the tangents from the point P ( 3, 0).
xC (-1, 2)
Solution:2
122
12 )()( yyxxd P (3,0)x
Method: Sketch!
• Find CP and use Pythagoras’ theorem for triangle CPA
A
222 ACPCAP 11920 AP
tangent
tangent • Use 1 tangent and join the radius.
The required length is AP.
22 )20())1(3( CP
Circle Problems
20416 CP
20
11
3
25: Definite Integration
Demo version note:
The next slide shows a typical summary. The clip-art notepad indicates to students that they may want to take a note.
SUMMARY
Find the indefinite integral but omit C
Draw square brackets and hang the limits on the end
Replace x with • the top
limit• the bottom limit
Subtract and evaluate
The method for evaluating the definite integral is:
Definite Integration
26: Definite Integration and AreasDemo version note:
The presentations are frequently broken up with short exercises. The next slide shows the solution to part of a harder exercise on finding areas. The students had been asked to find the points of intersection of the line and curve, sketch the graph and find the enclosed area.
2xy24 xy
,02 yx
Area of the triangle
2x 1xorSubstitute in : 2xy
31 yx
Area under the curve
1
2
31
2
2
344
xxdxx 9
3321
(b) ; 2xy24 xy
022 xx0)1)(2( xx
Shaded area = area under curve – area of triangle
29
29
Definite Integration and Areas
242 xx
29: The Binomial Expansion
Demo version note:
The following short exercise on Pascal’s triangle appears near the start of the development of the Binomial Expansion. Answers or full solutions are given to all exercises.
ExerciseFind the coefficients in the expansion of 6)( ba
Solution: We need 7 rows
1 2 1
1 3 3 1
1 1
1
1 4 6 4 11 5 10 110 5
1 6 15 120 15 6Coefficients
The Binomial Expansion
33: Geometric series – Sum to Infinity
Demo version note:
The students are shown an example to illustrate the general idea of a sum to infinity. A more formal discussion follows with worked examples and exercises.
Suppose we have a 2 metre length of string . . .
. . . which we cut in half
We leave one half alone and cut the 2nd in half again
m 1 m 1
m 1 m 21
. . . and again cut the last piece in half
m 1 m 21
m 41 m 4
1
m 21
Geometric series – Sum to Infinity
38: The Graph of tan
Demo version note:
The next slide shows part way through the development of the graph of using and .
tanysiny cosy
siny
y
cosy
yx
The graphs of and for are sin cos 3600
x x
190sin
090cos
This line, where is not defined is called an asymptote.
tan
tany
Dividing by zero gives infinity so is not defined when .
tan 90
The Graph of tan
43: Quadratic Trig Equations
Demo version note:
By the time students meet quadratic trig equations they have practised using both degrees and radians.
e.g. 3 Solve the equation for the interval , giving exact answers. 20
02cos3cos2 2
21cos 2cos or
0232 2 ccFactorising:
0)2)(12( cc 221 cc or
The graph of . . .cosy
Solution: Let . Then,cosc
shows that always lies between -1 and +1 so, has no solutions for .
cos
2cos
0 2
1
y
cosy-1
Quadratic Trig Equations
cosy-1
0 2
1
y
Principal Solution: 3
60
Solving for .21cos 20
50y
3
35
Ans: 3
5,3
Quadratic Trig Equations
46: Indices and Laws of Logarithms
Demo version note:
The approach to solving the equationstarted with a = 10 and b an integer power of 10. The word logarithm has been introduced and here the students are shown how to use their calculators to solve when x is not an integer.The calculator icon indicates that students should check the calculation.
ba x
A logarithm is just an index.To solve an equation where the index is unknown, we can use logarithms.e.g. Solve the equation giving the
answer correct to 3 significant figures.410 x
x is the logarithm of 4 with a base of 10 4log410 10 xxWe
write
In general if bx 10 then bx 10log
log
index
( 3 s.f. )6020
Indices and Laws of Logarithms
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