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Inhaltsverzeichnis

Frontmatter

1. Proportionality and Linear Laws

Abstract
When you have completed this chapter you should be able to
1.
draw a straight-line graph to represent data
 
2.
find the gradient and intercept of a straight-line graph and hence write down the line’s equation
 
3.
draw a straight-line graph from its equation
 
4.
interpret the gradient as a rate of change
 
5.
recognize whether variables are directly or inversely proportional and find the appropriate law between the variables
 
6.
find and use a straight line of best fit as a model for making predictions
 
John Berry, Patrick Wainwright

2. Quadratics and Polynomials

Abstract
When you have completed this chapter you should be able to
1.
identify a polynomial and write down its degree
 
2.
sketch the graph of a quadratic
 
3.
factorize a quadratic
 
4.
find the roots of a quadratic either from a sketch graph or by factorizing or by using ‘the formula’
 
5.
sketch the graph of a cubic polynomial
 
6.
understand the difference between a local maximum, a local minimum and a point of inflexion
 
7.
evaluate polynomials by nested multiplication
 
8.
find the roots of a polynomial either from a sketch graph or by the interval bisection method
 
John Berry, Patrick Wainwright

3. Logarithms and Exponential Functions

Abstract
When you have finished working through this chapter you should be able to
1.
manipulate positive, negative and fractional indices in algebraic expressions
 
2.
recognize and manipulate surds
 
3.
understand the components of a function and functional notation
 
4.
explain what is meant by an inverse function
 
5.
define exponential functions to base 10 and base e and be able to sketch and recognize their graphs
 
6.
understand the physical significance of the number e
 
7.
define the logarithm functions to base 10 and base e and know their algebraic rules
 
8.
find the half-life and decay constant of radioactive substances
 
9.
find power laws and exponential laws between variables using appropriate logarithm graphs, log-log and log-linear graph paper
 
John Berry, Patrick Wainwright

4. Trigonometric Functions

Abstract
When you have completed this chapter you should be able to
1.
state the convention for measuring angles
 
2.
define sine and cosine as projections of a unit vector
 
3.
sketch the graphs of sine, cosine and tangent
 
4.
find the primary and secondary solutions to sine and cosine equations
 
5.
state the period and the equations of the asymptotes of the tangent function
 
6.
obtain the general solution to a trigonometric equation
 
7.
obtain solutions within a specified range to a trigonometric equation
 
8.
solve more complex trigonometric equations, including squares, multiple angles and phase angles
 
9.
define sec, cosec and cot in terms of cos, sin and tan
 
10.
sketch graphs of the form a sin(bx + c), a cos(bx + c) and a tan(bx + c) and determine their amplitude and period
 
11.
convert from degrees to radians, and from radians to degrees
 
12.
sketch graphs and solve equations using radians for angles
 
13.
use trigonometric functions to model mechanical vibrations and alternating currents
 
John Berry, Patrick Wainwright

5. Trigonometry

Abstract
When you have finished studying this chapter you should be able to
1.
use the appropriate trigonometric identity to simplify expressions involving the trigonometric functions sine, cosine and tangent
 
2.
quote and use the double angle formulas for sin(2θ) and cos(2θ)
 
3.
simplify the expression a sin x + b cos x and solve equations of the form a sin x + b cos x = c
 
4.
define and use the inverse trigonometric functions
 
5.
quote and use the sine rule and the cosine rule for finding angles and side lengths in oblique triangles
 
John Berry, Patrick Wainwright

6. Further Algebraic Skills

Abstract
When you have completed this chapter you should be able to
1.
define a sequence
 
2.
state the limit of a sequence when it exists
 
3.
use formulae to compute the sums of powers of positive integers
 
4.
define arithmetic progressions and sum their terms
 
5.
define geometric progressions and sum their terms
 
6.
state the condition for a geometric progression to have a sum to infinity and compute such a sum
 
7.
carry out polynomial divisions, computing quotients and remainders
 
8.
use the factor theorem to test a polynomial for factors
 
9.
state when a quadratic polynomial is irreducible over the real numbers
 
10.
identify proper and improper partial fractions
 
11.
define the three types of reduction for partial fractions
 
12.
reduce an algebraic fraction to partial fractions
 
13.
expand (a + b) n where n is a positive integer using Pascal’s triangle and the Binomial Theorem
 
14.
expand (1 + x) n using the Binomial Theorem and state when the expansion is valid
 
John Berry, Patrick Wainwright

7. Differentiation

Abstract
When you have completed this chapter you should be able to
1.
distinguish between average and instantaneous rates of change
 
2.
find the derivative of a function by a limiting process
 
3.
use rules to differentiate polynomial, power, trigonometric, exponential and logarithmic functions
 
4.
use rules to differentiate composite functions, products of functions and quotients of functions
 
5.
find and identify the nature of stationary points
 
6.
use differentiation as a tool in curve sketching
 
John Berry, Patrick Wainwright

8. Integration

Abstract
When you have completed this chapter you should be able to
1.
approximate areas by summing the areas of rectangles
 
2.
approximate areas by summing the areas of trapezia
 
3.
express a definite integral as a limit of a sum
 
4.
relate definite integrals to areas and state the convention for the signs attached to areas
 
5.
state the relationship between integration and differentiation
 
6.
integrate polynomial, exponential, power and trigonometric functions
 
7.
apply substitutions to evaluate integrals
 
8.
integrate by parts
 
9.
use partial fractions to help with integration
 
10.
calculate volumes of solids of revolution
 
11.
locate the centroid of a body
 
John Berry, Patrick Wainwright

9. Differential Equations

Abstract
When you have finished this chapter you should be able to
1.
identify a first order differential equation
 
2.
sketch the direction field for a first order differential equation
 
3.
given initial conditions sketch a particular solution curve on the direction field plot
 
4.
obtain the general solution to a first order differential equation by separating the variables
 
5.
use initial conditions to obtain a particular solution from the general solution
 
6.
set up differential equations
 
7.
use simple numerical methods to obtain approximate solutions to differential equations
 
John Berry, Patrick Wainwright

10. Numerical Analysis

Abstract
After studying this chapter you should be able to
1.
estimate the error interval associated with arithmetic calculations and in the evaluation of functions
 
2.
solve equations using iterative methods
 
3.
understand the importance of convergence of a numerical method
 
4.
state and use the Newton-Raphson method
 
5.
state and use Simpson’s rule for numerical integration
 
John Berry, Patrick Wainwright

11. Vectors

Abstract
When you have completed your study of this chapter you should be able to
1.
recognize vector and scalar quantities
 
2.
find the sum of two vectors geometrically
 
3.
write two dimensional vectors in component form and add vectors algebraically
 
4.
apply vector methods to simple problems involving forces, displacements and velocities
 
5.
write the equation of a straight line in vector form
 
6.
find the velocity and acceleration vectors from the position vector and vice versa using calculus
 
John Berry, Patrick Wainwright

12. Matrices

Abstract
When you have finished this chapter you should be able to
1.
add and subtract matrices
 
2.
multiply matrices
 
3.
state that in general ABBA
 
4.
write down the 2 × 2 and 3 × 3 identity matrices
 
5.
rewrite simultaneous equations in matrix form
 
6.
calculate the determinant of a 2 × 2 and 3 × 3 matrix
 
7.
calculate the inverse of a 2 × 2 and 3 × 3 matrix when the inverse exists
 
8.
solve simultaneous equations by inverse matrix methods
 
9.
solve simultaneous equations by elimination
 
10.
understand why simultaneous equations do not always have a unique solution
 
11.
find when simultaneous equations have no solutions
 
12.
find when simultaneous equations have infinitely many solutions, and find such solutions
 
John Berry, Patrick Wainwright

13. Mathematical Modelling

Abstract
When you have finished studying this chapter you should be able to
1.
distinguish between theoretical and empirical modelling
 
2.
know the difference between a model and modelling
 
3.
improve your problem solving skills
 
John Berry, Patrick Wainwright

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