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DEPARTMENT OF MATHEMATICS
AS level Mathematics
Core mathematics 1 – C1
2015-2016
Name: _____________________________________
Page
C1 workbook contents
Indices and Surds ______________________________________________
Simultaneous equations ______________________________________
Quadratics ______________________________________________________
Inequalities ______________________________________________________
Graphs __________________________________________________________
Arithmetic series ______________________________________________
Coordinate Geometry _________________________________________
Differentiation _______________________________________________
Integration ______________________________________________________
C1WB: Indices & Surds
page
Indices and Surds Numeracy - Notes BAT apply the laws of indices to calculations
BAT simplify and rationalise surds
BAT solve equations using the rules for indices and surds
𝒂𝟎 = 𝟏
𝒂−𝒎 =𝟏
𝒂𝒎
𝒂𝟏 𝒏 = 𝒂𝒏
𝒂𝒎 × 𝒂𝒏 = 𝒂𝒎+𝒏
𝒂𝒎 ÷ 𝒂𝒏 = 𝒂𝒎−𝒏
𝒂𝒎 𝒏 = 𝒂𝒎×𝒏
𝒂𝒎 𝒏 = 𝒂𝟏 𝒏 𝒎
C1WB: Indices & Surds
page
WB1 Solve
a) 𝒙𝟏 𝟐 = 𝟗 b) 2𝒙−𝟏 𝟐 = 𝟑𝟐
c) 3𝑥1 2 = 48 d) 𝑥3 4 = 27 e) 7𝑥−1 2 =1
7 f) 𝑥−2 3 =
4
9
WB2 Rearrange to single powers of 2, 3 or 5
a) 𝟖 × 𝟐𝒙 b) 𝟏𝟔𝟐𝒙
c) 𝟖𝒙 d) 𝟒𝟑−𝒙 e) 𝟑𝟒 × 𝟑𝒙 f) 𝟐 × 𝟐𝒙 g) 𝟗 × 𝟑𝟑 h) 𝟑𝟐 ÷ 𝟖 𝟒
C1WB: Indices & Surds
page
WB3
a) Solve 27𝑥 = 9
b) Solve 43−𝑥 = 8𝑥
c) Solve 7𝑥+1 3 = 492𝑥
d) Solve 5𝑥 2 = 12512
C1WB: Indices & Surds
page
WB4 𝒂 = 𝟐𝒙 𝒃 = 𝟑𝒙 Write in terms of a and b
a) 𝟐𝒙+𝟏 + 𝟑𝒙−𝟏b) 𝟒𝒙 + 𝟐𝟕𝒙+𝟏
c) 𝟐𝒙+𝟐 d) 𝟖𝒙 e) 𝟒 × 𝟖𝒙 f) 𝟐𝟐𝒙 + 𝟑𝒙+𝟐g) 𝟐𝟕𝒙 − 𝟗𝒙+𝟏 h) 𝟑 × 𝟗𝟐𝒙+𝟏
C1WB: Indices & Surds
page
Indices & Surds Algebra - Notes BAT manipulate expressions using rules of indices
C1WB: Indices & Surds
page
WB5
Write as a single power a) 𝑥 b) 1
𝑥 c)
3
𝑥
Rearrange d) 9𝑥1 2 e) 9𝑥 1 2 f) 3𝑥−1 2
WB6
Rearrange and simplify a) 𝟔𝒙𝟐 ×𝟐
𝟑 𝒙 b)
𝟒𝒙𝟐+𝟏𝟎𝒙+𝟔
𝟐𝒙
C1WB: Indices & Surds
page
Calculations with Surds - Notes BAT simplify and rationalise surds
BAT solve equations using the rules for indices and surds
C1WB: Indices & Surds
page
WB7
Simplify a) 12 b) √49
16
c) 1200 d) 50 + 18 e) 11 × 11 f) √4
9 × 18
C1WB: Indices & Surds
page
WB8
Find length BC and express your answer as
an exact simplified value
WB9
B is (11 , 6) and C is ( 1.5, 10)
Show that length BC is 5 17
2
4
6
C
B
A
A C
B
C1WB: Indices & Surds
page
WB10
work out the area of the rectangle
give an exact simplified answer
WB11
a special case: ‘making an integer (2 + 3)(2 - 3)
2 − 3
4 + 3
C1WB: Indices & Surds
page
WB12
rationalise each surd a) 𝟐
𝟓 b)
𝟒
𝟐+ 𝟑
WB13
Simplify 𝟒 − 𝟑
𝟑×
𝟔 + 𝟑
𝟒
C1WB: Simultaneous equations
page
Simultaneous equations - Notes BAT know and use three methods for solving simultaneous equations
BAT solve simultaneous equations from index problems
C1WB: Simultaneous equations
page
WB1 Solve these simultaneous equations
a) 𝒚 = 𝟐𝒙 + 𝟖 And 𝒚 = 𝟕𝒙 − 𝟐
b) 𝒚 = 𝒙𝟐 − 𝟐𝒙 And 𝒚 = 𝟑𝒙 − 𝟔
C1WB: Simultaneous equations
page
WB2 Solve these simultaneous equations
a) 𝒚 = 𝒙𝟐 + 𝟑𝒙 − 𝟖 And 𝒚 − 𝟐𝒙 = 𝟒 b) 𝒚 = 𝟒 + 𝒙 − 𝒙𝟐 And 𝒚 = 𝟕 − 𝟑𝒙
C1WB: Simultaneous equations
page
WB3
Solve the simultaneous equations below algebraically
𝒚 = 𝒙𝟐 − 𝟐𝒙 + 𝟑 And 𝒚 + 𝟐𝒙 = 𝟕
C1WB: Simultaneous equations
page
WB4
Solve the simultaneous equations below algebraically
𝒙𝟐 − 𝟒𝒚 = 𝟗 And 𝒚 − 𝟐𝒙 = −𝟒
WB5
Solve the simultaneous equations below algebraically
𝒙𝟐 + 𝟑𝒚 − 𝟐𝟕 = 𝟎 And 𝒙 + 𝒚 = 𝟗
C1WB: Simultaneous equations
page
WB6
Solve the simultaneous equations below algebraically
𝒙𝟐 + 𝒚𝟐 = 𝟐𝟔 And 𝒚 = 𝟑𝒙 + 𝟐
WB7
Solve the simultaneous equations below algebraically
𝒙𝟐 + 𝒚𝟐 = 𝟐𝟗 And 𝟑𝒙 + 𝒚 = 𝟏𝟏
C1WB: Simultaneous equations
page
WB8
Solve the simultaneous equations below algebraically
a) 𝟏𝟔𝒙
𝟖𝒚=
𝟏
𝟒 And 𝟒𝒙𝟐𝒚 = 𝟏𝟔
b) 𝟒𝒂 √𝟐𝒃 And 𝟖𝒃
𝟐𝒂= 𝟒(√𝟐𝒂𝟑
)
c) 𝟑𝒄
𝟑𝒅= 𝟐𝟕 And 𝟗𝒄 𝟑𝒅 =
𝟏
𝟑
C1WB: Quadratics
page
Quadratics - Notes BAT convert between completed square and normal form
BAT rearrange and solve quadratics using completed square form
C1WB: Quadratics
page
WB1
Rearrange into completed square form 𝑥2 + 12𝑥 + 10
WB2
Rearrange into completed square form 𝑥2 + 7𝑥 + 15
C1WB: Quadratics
page
WB3
Rearrange into completed square form 𝑥2 + 12𝑥 + 10 and solve
WB4
Rearrange into completed square form 𝑥2 + 14𝑥 + 4 and solve
C1WB: Quadratics
page
WB5
Rearrange into completed square form 4𝑥2 + 24𝑥 + 12
WB6
Rearrange into completed square form 3𝑥2 + 12𝑥 + 10
C1WB: Quadratics
page
WB7
Rearrange into completed square form 3𝑥2 + 12𝑥 + 10 and solve
C1WB: Quadratics
page
Quadratics - Notes BAT manipulate quadratic expressions and solve using the quadratic
formula
BAT rearrange and solve disguised quadratics
C1WB: Quadratics
page
WB8
Solve 3𝑥2 − 𝑥 − 1 = 0 using the quadratic formula
WB9
Solve 2𝑥2 − 7𝑥 + 4 = 0 using the quadratic formula
C1WB: Quadratics
page
WB10
Solve 2𝑥 + 1 =21
𝑥
WB11
Solve 𝑦4 + 3𝑦2 − 28 = 0
C1WB: Quadratics
page
WB12
Solve 𝑥 + 9√𝑥 + 14 = 0
C1WB: Quadratics
page
Quadratic graphs and discriminant - Notes BAT know how to use the discriminant to solve problems and
understand properties of quadratics
BAT Sketch quadratic graphs showing intersections and max/min point
C1WB: Quadratics
page
WB12
𝑓(𝑥) = 2𝑥2 + 12𝑥 + 𝑐 Given that f(x)=0 has equal roots, find the value of c and hence solve f(x)=0
WB13
𝑓(𝑥) = 𝑥3 − 𝑘𝑥 + 16, where k is a constant
a) Find the set of values of k for which the equation 𝑓(𝑥) = 0 has no real solutions
b) Express 𝑓(𝑥) in the form (𝑥 − 𝑝)2 + 𝑞
c) find the minimum value of 𝑓(𝑥) and the value of x for which this occurs
C1WB: Quadratics
page
WB14
The equation 8𝑥2 − 4𝑥 − (𝑘 + 3) = 0, where k is a constant has no real roots
Find the set of possible values of k
WB15
Sketch 𝒇(𝒙) = 𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟖
C1WB: Quadratics
page
WB16
Sketch 𝒇(𝒙) = −𝒙𝟐 + 𝟏𝟎𝒙 −16
C1WB: Inequalities
page
Inequalities - Notes BAT Solve quadratic and linear inequalities
BAT solve inequalities problems in context
C1WB: Inequalities
page
WB1
Solve 𝒙𝟐 + 𝟕𝒙 − 𝟏𝟖 > 𝟎
WB2
Solve 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟐 ≤ 𝟎
C1WB: Inequalities
page
WB3
i) Solve 5x – 2 > 3x + 7
ii) Solve 𝒙𝟐 − 𝟕𝒙 − 𝟏𝟖 < 𝟎
iii) Solve to find when both inequalities hold true
WB4
The specification for a new rectangular car park states that the length L is to be 18 m more
than the breadth and the perimeter of the car park is to be greater than 68 m
The area of the car park is to be less than or equal to 360 m2
Form two inequalities and solve them to determine the set of possible values of L
C1WB: Inequalities
page
C1WB: Inequalities
page
- Notes BAT
C1WB: Inequalities
page
WB
WB
C1WB: Inequalities
page
- Notes BAT
C1WB: Inequalities
page
WB
WB
C1WB: Graphs
page
Transformations - Notes BAT know and use the six types of transformations to graphs
C1WB: Graphs
page
Transformations - Notes (i) Shifts
(ii) Stretches
iii) Reflections
+A
(x, y) (x + A, y)
)( Axf is a shift
in the x direction
+A
(x, y)
(x , y + A)
Axf )( is a shift
in the y direction
× A1
(x, y)
(A1 x , y)
×A
(x, y)
(x, Ay)
)(Axf is a stretch by scale
factor A
1 in the x direction
)(xAf is a stretch by scale
factor A in the y direction
(x, – y)
(x, y)
(x, y) (-x, y)
)( xf is a reflection of
the graph in the y axis
)(xf is a reflection of
the graph in the x axis
C1WB: Graphs
page
WB1
Draw a sketch graph of 𝒚 = (𝒙 − 𝟑)𝟐 + 𝟐
WB2
Draw a sketch graph of 𝒇(𝒙 + 𝟒) − 𝟐
C1WB: Graphs
page
WB3
Draw a sketch graph of 𝑓(3𝑥)
WB4
Draw a sketch graph of 2𝑓(𝑥)
C1WB: Graphs
page
WB5
Draw a sketch graph of 𝑓(−𝑥)
C1WB: Graphs
page
WB6
Describe these transformations
a) −𝑓(𝑥) b) 𝑓(𝑥) + 4
c) 𝑓(2𝑥) d) 3𝑓(𝑥) + 1
Extension : if 𝒇(𝒙) = 𝟐𝒙𝟑 Work out the equations of the transformed graphs
C1WB: Graphs
page
Graphs - Notes BAT explore cubic and reciprocal graphs
BAT explore graph properties – asymptotes and limits
C1WB: Graphs
page
WB7
Sketch y = (x – 3)(x + 2)(x – 5)
C1WB: Graphs
page
WB8
Sketch the graph of
a) 𝑓(𝑥) = 1 +2
𝑥−3 b) 𝑓(𝑥) = 4 −
1
𝑥
C1WB: Arithmetic series
page
Series - Notes BAT Explore arithmetic series and derive (new) formulas for the nth term
and sum of terms
BAT practice solving series problems of all types up to ‘build and solve
simultaneous equations’
C1WB: Arithmetic series
page
WB1 a) The third term of an arithmetic sequence is 11 and the seventh term is 23. Find the
first term and the common difference
b) An arithmetic series has first term 6 and common difference 2 ½ . Find the least value
of n for which the nth term exceeds 1000
c) Find the number of terms in the arithmetic series 13 + 17 + 21 … + 93
C1WB: Arithmetic series
page
WB2
The 5th term of an arithmetic sequence is 24 and the 9th term is 4
a) Find the first term and the common difference
b) The last term of the sequence is -36. How many terms are in this sequence
C1WB: Arithmetic series
page
WB3
The first term of an arithmetic sequence is 3, the fourth term is -9. What is the sum of the
first 24 terms?
WB4
The first term of an arithmetic sequence is 2, the sum of the first 10 terms is 335. Find the
common difference
C1WB: Arithmetic series
page
WB5
An arithmetic sequence for building each step of a spiral has first two terms 7.5 cm and 9 cm
What will be (i) the length of the 40th line of the spiral
(ii) the total length of the spiral after 40 steps?
WB6
Sim eqn
An arithmetic sequence is used for modelling population growth of a Squirrel colony starting
at three thousand in the year 2000. The 2nd and 5th numbers in the sequence are 14 and 23
showing the increase in population those years. Find: (i) the first increase in population (ii)
the 16th increase (iii) the population after 16 years?
C1WB: Arithmetic series
page
WB7
Sim eqn
The first three terms of an arithmetic sequence are (4x – 5), 3x and (x + 13) respectively
a) Find the value of x b) Find the 23rd term
WB8
sim eqn
The sum of an arithmetic sequence to n terms is 450
The 2nd and 4th terms are 40 and 36. Find the possible values of n
C1WB: Arithmetic series
page
Sigma notation - Notes BAT use sigma notation and solve series problems
C1WB: Arithmetic series
page
WB9
Evaluate ∑ (𝑟2 + 1)85
WB10
Evaluate ∑ (7𝑟 − 3)461
C1WB: Arithmetic series
page
WB11
Evaluate ∑ (3𝑟 + 5)221
WB12
Show that ∑ (3𝑟 + 4)𝑛1 = 3∑ 𝑟𝑛
1 + 4𝑛
C1WB: Arithmetic series
page
Recurrence relations - Notes BAT solve problems involving recurrence relations
C1WB: Arithmetic series
page
WB13
The sequence of positive numbers 𝑢1, 𝑢2, 𝑢3 , … is given by 𝑢𝑛+1 = (𝑢𝑛 − 6)2 , where
𝑢1 = 9
a) Find 𝑢2, 𝑢3 and𝑢4
b) Write down the value of 𝑢20
WB14 The nth term of a sequence is 𝑢𝑛, the sequence is defined by 𝑢𝑛+1 = 𝑝𝑢𝑛 + 𝑞, where
𝑝 & 𝑞 are constants
The first three terms of the sequence are
Find 𝑢1 = 2, 𝑢2 = 5 and 𝑢3 = 14
a) Show that 𝑞 = −1 and find the value of 𝑝
b) Find the value of 𝑢4
C1WB: Linear Geometry
page
Lines - Notes BAT explore gradients of parallel and perpendicular line
BAT rearrange and find equations of lines
C1WB: Linear Geometry
page
WB1 For each of these equations,
i) rearrange it into the form y = mx + c ii) give the gradient
iii) give the intercept on the y-axis
a) 2𝑥 + 𝑦 − 10 = 0 b) 5𝑥 − 2𝑦 + 6 = 0
WB2
Give the General equation of the perpendicular line to 2𝑥 + 𝑦 − 8 = 0
that goes through (4, 9)
C1WB: Linear Geometry
page
WB3
Give the General equation of the perpendicular line to 𝑥 + 5𝑦 − 6 = 0
that goes through (3
5, 7)
WB4
Two points A(1,2) and B(-3,6) are joined to make the line AB.
Find the equation of the perpendicular bisector of AB
C1WB: Linear Geometry
page
Points, Lines, Gradients - Notes BAT find distances between points
BAT explore equations of lines, know the general equation of a line
BAT use a new formula to find equations of lines
C1WB: Linear Geometry
page
WB5
Find the line that joins these points (-2, 8) and (3,-7)
C1WB: Linear Geometry
page
WB6
Find the equations of the lines that join these points
(-6, 1) (2, 5) (-3, -5)
C1WB: Linear Geometry
page
WB7
Find the line that joins points (4, 9) and (8, 12)
in the form ax + by + c = 0
WB8
Find the line that joins points (-2, 8) and (3,-7)
in the form ax + by + c = 0
C1WB: Linear Geometry
page
WB9
Find the general equation of each line through (3, 7) and is perpendicular to y = 2x + 8
WB10
Find the equation of the line that goes through
(3, 7) and is perpendicular to y = 2x + 8
C1WB: Linear Geometry
page
WB11
The line l1 has gradient -3 goes through (-2, 3)
Line l2 is perpendicular to l1 and goes through (-2, 3)
Find the equations of lines l1 and L2
WB12
A line has equation 6𝑥 + 3𝑦 = 4 Find the equation of the line parallel to this which
passes through the point (5, 5)
C1WB: Linear Geometry
page
WB13
Find, in the form 𝑦 = 𝑚𝑥 + 𝑐 the equation of the line through (3, 11) which is parallel to
𝑦 = 3𝑥 + 13
C1WB: Linear Geometry
page
More Problems - Notes solve linear geometry problems
C1WB: Linear Geometry
page
WB14
Line l1 joins points A (3, 6) and B (6, 4)
a) What is the equation of the perpendicular line through midpoint of AB ?
b) Show this line goes through (3, 11/4)
WB15
L1 has equation 2x + y - 6 = 0 and goes through points A(0, p) and B(q, 0)
a) Find the values of p and q
b) What is the equation of the perpendicular line from point
C(4, 5) to line L1 ?
c) What is the area of triangle OAB?
C1WB: Linear Geometry
page
WB16
Line L1 goes through points A(-3, 2) and B(3, -1)
a) Find distance AB
b) Find the equation of L1 in the form ax + by + c = 0
Perpendicular Line L2 has equation 2x – y + 3 = 0 and crosses L1 at point D.
c) Find coordinates of point D
Line L2 crosses the y-axis at point Q
d) Find the area of triangle AQB
C1WB: Linear Geometry
page
WB17
The points A(-6, -5), B(2, -3) and C(4, -28) are the vertices of triangle ABC. Point D is the midpoint of the line
joining A to B
a) Show that CD is perpendicular to AB
b) Find the equation of the line passing through A and B in the form ax + by + c = 0, where a, b and c are
integers
WB18
The straight line L1 ha equation 4y +x = 0
The straight line L2 has equation y = 5x - 4
a) The lines L1 and L2 intersect a the point A. Calculate, as exact fractions the coordinates of A
b) Find an equation of the line though A which is perpendicular to L1. Give your answer in the form ax + by = c
C1WB: Linear Geometry
page
WB19 The points A and B have coordinates (5, -1) and (10, 4)
AB is a chord of a circle with centre C
a) Find the gradient of AB
The midpoint of AB is point M
b) Find an equation for the line through C and M
Given that the x-coordinate of point C is 6,
b) Find the y coordinate of C
c) Show that the radius of the circle is 17
WB20 The points A(3, 7) B(22, 7) and C(p, q) form the vertices of a triangle. Point D(9, 2) is the midpoint of AC
a) Fins the values of p and q
The line L, which passes through D and is perpendicular to AC, intersects AB at E
b) Find an equation for line L in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0
c) Find the exact x-coordinate of E
C1WB: Differentiation
page
The Gradient function - Notes BAT explore differentiation and the gradient function of curves
BAT differentiate polynomials and find the gradient of curves
C1WB: Differentiation
page
WB1
Find the gradient function of a) 𝑦 = 4𝑥7
b) 𝑦 = 9𝑥3 c) 𝑦 =3
4𝑥−8 d) 𝑦 = √𝑥 e) 𝑦 =
1
2𝑥2
WB2
Find the gradient function of a) a) 𝑦 = 2𝑥2+ 5𝑥 − 3
b) 𝑦 = 2𝑥5 + 6𝑥8 c) 𝑦 = 3𝑥2 − 7𝑥 + 6
d) 𝑦 = 4𝑥3 − 3𝑥2 + 8𝑥 − 10 e) 𝑦 =3
5𝑥5 +
1
2𝑥2
C1WB: Differentiation
page
WB3
Determine the points on the curve 𝑦 = 𝑥3 + 5𝑥 + 4
Where the gradient is equal to 17
WB4
a) Find the coordinate of the point on the curve 𝑦 = 4𝑥2 − 10𝑥 + 6
Where the gradient is -2
b) Sketch this graph and point on a diagram
C1WB: Differentiation
page
WB5
Sketch the graph of 𝑦 = 𝑥2 − 4𝑥 − 21 showing the minimum point and the places where
the graph intersects the axes
WB6
a) Given that 𝑦 = 5𝑥3 + 6 +5
𝑥2 find
𝑑𝑦
𝑑𝑥 in its simplest form
b) Given that 𝑦 = 9𝑥3 − 8√𝑥 +9𝑥2+4
𝑥 find
𝑑𝑦
𝑑𝑥 in its simplest form
C1WB: Differentiation
page
Stationary points - Notes BAT determine the coordinates and nature of stationary points
BAT find and use the second derivative
C1WB: Differentiation
page
WB7
Find the coordinates of the points on each of these curves at which the
Gradient is zero
a) y = 𝑥2 − 2𝑥 − 3 b) y = 8𝑥 − 2𝑥2
Now sketch the graphs of each curve
C1WB: Differentiation
page
WB8
Find the coordinates of the points on each of these curves and determine their nature
a) y = 𝑥2 − 8𝑥 + 14 b) y = 𝑥3 + 3𝑥2 + 1
Now sketch the graphs of each curve
C1WB: Differentiation
page
WB9
Differentiate 4𝑥2 −8
𝑥 and hence find the x-coordinate of the curve y = 4𝑥2 −
8
𝑥
WB10
Find the coordinates of the stationary point(s) on the
curve y =1
3𝑥3 − 2𝑥2 + 4𝑥 + 1 and determine their nature
C1WB: Differentiation
page
WB
WB
C1WB: Differentiation
page
WB
WB
C1WB: Differentiation
page
Tangents and normal - Notes BAT
C1WB: Differentiation
page
WB
WB
C1WB: Integration
page
- Notes BAT Integrate functions using the reverse process to differentiation
BAT Integrate functions in context
BAT find the value of + C
C1WB: Integration
page
WB1
Find a) ∫ 3𝑥2 dx
b) ∫ 7𝑥6 + 8 +4
𝑥2 c) ∫(𝑥3 + 3𝑥 + 2) dx
d) ∫ 𝑥6 + 𝑥8 dx e) ∫(𝑥 + 3)(𝑥 − 2) dx
C1WB: Integration
page
WB2
Find a) ∫2
√𝑥3 dx
b) ∫ √𝑥 dx c) ∫ 2𝑥 −1
𝑥3 dx
d) ∫1
𝑥2 dx e) ∫
𝑥3−3𝑥
𝑥 dx
C1WB: Integration
page
WB3
The gradient of a curve at the point (x, y) on the curve is given by 𝑑𝑦
𝑑𝑥= 3𝑥2 − 4𝑥
Given that the point (1, 2) lies on the curve, determine the equation of the curve
WB4
A curve passes through the point (1, 5) and 𝑑𝑦
𝑑𝑥= 16𝑥7 − 6𝑥
Find its equation
C1WB: Integration
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WB5
The gradient of a curve at the point (x, y) on the curve is given by 𝑑𝑦
𝑑𝑥= 𝑥2(2𝑥 + 1)
The curve passes through the point (1, 5) Find the equation of the curve
WB6
a) Given that 𝑓′′(𝑥) = 2 −2
√𝑥3 and f ′(1) = 0 Find f ′(x)
b) Given further that 𝑓(1) = 8 find f(x)
C1WB: Integration
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WB7
The curve C has equation 𝑦 = 𝑓(𝑥) where 𝑓′(𝑥) = 2𝑥 − 6√𝑥 +8
𝑥2
Given that the point P (4, -14) lies on C
a) Find 𝑓(𝑥) and simplify your answer b) Find an equation of the normal to C at point P
WB8
A curve has equation 𝑦 = 12𝑥2 − 15𝑥 − 2𝑥3
The curve crosses the x-axis at the origin, O, and the point A (2, 2) lies on the curve
a) Find the gradient of the curve at point A
b) Hence, find the equation of the normal to the curve at point A giving your answer in
the form 𝑥 + 𝑝𝑦 + 𝑞 = 0
C1WB: Integration
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WB9 The curve C with equation 𝑦 = 𝑓(𝑥) passes through the point (2, 9)
Given that 𝑓′(𝑥) = 5𝑥 +8
𝑥2
a) Find 𝑓(𝑥) b) Verify that 𝑓(−2) = 12 c) Find an equation for the tangent at C at the point (-2, 12) giving your answer in the form
𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0
WB10 Given that
𝑑𝑦
𝑑𝑥 =
9𝑥+12𝑥5 2⁄
√𝑥
a) Write 𝑑𝑦
𝑑𝑥 in the form 9𝑥𝑝 + 12𝑥𝑞
b) Given that y = 90 when x = 1, find y in terms of x, simplifying the coefficient of each term
