Welcome to Engineering Mathematics

1. Welcome to Engineering Mathematics • We will cover 7 topics this semester • 1. Algebraic Manipulation • 2. Complex numbers • 3. Cartesian coordinates and curves • 4. Functions • 5. Differentiation of a function of one variable • 6. Differentiation of a function of two variables • 7. Integration of a function of one variable It is essential that you read the HELM workbooks. These are found on WebCT. Go to http://webct.nottingham.ac.uk – You should be registered for Engineering Mathematics 1 (EM1). You will find lecture summaries, lecture slides and the HELM workbooks here.

2. Introduction • We will cover 5 topics today • 1. Real Numbers and Inequalities • 2. Exponential Functions, the number ‘e’ • 3. Exponential Growth and Decay • 4. Partial Fractions • 5. The Binomial Theorem

3. Real Numbers, Powers & Inequalities Integer– An integer is a whole number. It can be positive, negative or zero. Rational Number – A rational number is any number that can be expressed as a fraction having the form p/q, where p and q are integers. Irrational Number – An irrational number is any number that is not a rational number or an integer. Examples include π and √2. Infinity – The symbol ∞is used to indicate infinity. Infinity can not be used in algebra as a normal number. Claims such as ∞/ ∞ = 1 and ∞ - ∞ = 0 are false.

4. Real Numbers, Powers & Inequalities Powers of numbers take the form ax. The power ‘x’ is called the exponent or index of the expression. Rules for exponents The concept of an identity needs to be distinguished from an equation. ax.ay = ax+y a0 = 1 a-x = 1/ax (ax)y = axy ax.bx = (a.b)x a1/2 = √a This is an equation It is only true for certain values of ‘x’. This is an identity The solution to x2 = 2 is x = ±√2 It is true for all values of ‘x’ and ‘y’.

5. Real Numbers, Powers & Inequalities The symbols for inequalities have the following meanings There is also the modulus or absolute value of the variable, for example < ‘is less than’ > ‘is greater than’ ≤ ‘is less than or equal to’ ≥ ‘is greater than or equal to’ i.e. |-3| = 3 |3| = 3 The variable ‘x’ satisfies the inequalities 2 < x ≤ 4 and |x| ≤ 3. Express x as a single equation 2 < x ≤ 3 Do not forget that the range of x can also extend to infinity i.e. 0 < x ≤ ∞

6. Exponential Functions Exponential Functions are always of the form A standard base can be chosen at your convenience For example, at one time base 10 was used to simplify the arithmetic used in large calculations Where a is called the base for the exponential function. Graphs of exponential functions with different bases are shown below. The base that is now in most general use is given the letter ‘e’. We will call this the natural exponential function and it has the value 2.71828… The natural exponential function is written ex. It is often written exp(x) in programming and sometimes on calculators. This base is used most frequently because it has special properties and these will be discussed later.

7. Logarithmic Functions The Natural Logarithm is a special logarithm that is commonly used and is the inverse of the natural exponent. It is written as Logarithms are the inverse of Exponents. Logarithms help solve the following problem which is spoken as ‘log to the base e’ This is the solution to the following problem Where x and a are known but y is not. The solution to this problem is written as Rules for Logarithms Loga(ax) ≡ x Loga(x.y) ≡Logax + Logay Loga(x/y) ≡Logax - Logay Loga(xp) ≡p.Loga(x) Loga(1) ≡ 0 And spoken as ‘log to the base a of x’ For all x log10(1000) = y What is y? 103 = 1000 Hence y = 3 For x > 0 Loga(x) → ∞ as x → ∞ Loga(x) → -∞ as x → 0

8. Logarithmic Functions Obtain y in terms of x when Equate the exponential functions on both sides of the equation Using the logarithmic rules we can deduce that And Hence

9. Exponential Growth and Decay Exponential growth and decay can be modeled using the following function If c < 0 then y is said to have exponential decay Where ‘A’ and ‘c’ are constants. If c > 0 then y is said to have exponential growth If |c| is used then exponential decay is expressed as We can calculate the doubling period (T)of y. Consider a moment of time (t). At sometime later (t + T) y will have doubled in value i.e. Then y halves after every time period T [= (1/c).ln(2)]. This is period is known as the half-life of the function. Which reduces to hence Thus, y (=A.ec.t) doubles its value after every time interval of T.

10. Exponential Growth and Decay In a radioactive element the number of radioactive nuclei (x) present at time ‘t’ is given by Where k is a constant and x0 is the number of radioactive nuclei present at time t = 0 Find the time for 90% of the nuclei to decay. Where T is the time taken for the number of radioactive nuclei to decay by 90%

11. Partial Fractions The following expressions are examples of polynomials A polynomial is more generally expressed as the sum of a number of the following terms A partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator of a rational function. The outcome of a partial fraction expansion expresses that function as a sum of fractions. Where ‘n’ is a positive integer and ‘a’ is any number. The degree of a polynomial is equal to the highest value of n. A rational function is defined as We shall use the symbol ‘≡’ to indicate an identity i.e. Where P(x) and Q(x) are polynomials Remember that an identity holds for all values of x. It is the denominator that determines which form the constituent partial fractions will take. There are some general cases that work for a wide variety of partial fractions, these are shown on the following slide.

12. Rules for Partial Fractions A simple factor. IfQ(x) = ax + b then Where ‘L’ is a constant The numerator P(x) is only used to determine the constants L, M and N A repeated factor. IfQ(x) = (ax + b)n then Examples will be shown on the following slides Where ‘Ln’ are constants An irreducible quadratic. IfQ(x) = px2 + qx +r and q2 < 4pr then Where M and N are constants.

13. Partial Fractions Examples in partial fractions Express Thus The denominator is two simple factors. Hence we can write the solution as A and B are unknown constants and we can find A and B in the following way Hence

14. Partial Fractions Examples A simpler method is to substitute obvious values of x to simplify the equation, i.e. in partial fractions Express If x = 1 then The denominator has repeated simple factors. Hence we can write the solution as Hence Therefore Where A, B and C are unknown constants. Find A, B and C. If x = -1/2 then Therefore We could multiply out this equation and find values for A, B and C. However this is a long winded method. If x = 0 then Therefore

15. Partial Fractions Examples To find the constants we can put x equal to zero and immediately see that in partial fractions Express Unfortunately there are no other values of x that we can easily choose. The denominator is the product of a simple factor and an irreducible quadratic. Hence we can write the solution as However, by substituting A into the previous equation we can determine that Thus Once again we can determine that This condition can only be true if and Hence

16. The Binomial Theorem The binomial theorem is an important formula that gives the expansion of powers of a sum. For example, consider expressions of the form This is known as Pascal’s Triangle Where n is a positive integer. The coefficients occurring in these polynomials are known as the binomial coefficients. They can be determined for any value of n by writing out the triangle. However, this is a long and cumbersome process. We will look at a better way of calculating the coefficients. For small values of n we can expand (1 + x)n and obtain You can see that the coefficients are symmetrical around the centre of each equation and that the equations form a triangle. There are n+1 terms in each equation.

17. The Binomial Theorem Consider the following scenario For example, there are 3 ways to arrange one ‘x’ and two ‘1’s. There are also 3 ways to arrange two ‘x’s and one ‘1’. Multiply out the brackets but do not simplify There is only 1 way to arrange three ‘x’s or three ‘1’s. When we simplify this equation we get Then You can see that each term is determined by multiplying 3 elements together. Hence the value of each binomial coefficient is equal to the number of ways of rearranging the n elements (n = 3 in this case).

18. The Binomial Theorem In order to calculate the binomial coefficients we need to know how to calculate the number of ways of arranging r distinct objects within n distinct objects. This is known as a combination. The general solution is presented as Where ‘n’ and ‘r’ have been defined previously and ‘C’ stands for combination. For example, if we had 4 marbles and 2 of them were blue how many different positions could we put the marbles in? Sometimes the following notation is used In this case we can calculate the number of ways of arranging 2 marbles from 4.

19. The Binomial Theorem Consider the following expression The binomial theorem yields the following general result This expression can be expanded using the following identity For example Hence the powers of a sum can be expanded for any integer value of n. The general form is expressed as Or more generally However, currently we are restricted on the types of sum we are able to expand. We may increase the range of sums we may expand using the following method.

20. Binomial Theorem Example Use the Binomial Theorem to expand the following expression Recall that Thus Hence

21. Binomial Theorem Example Use the Binomial Theorem to expand the following expression

22. Conclusion • You have learned about • Real numbers, powers and inequalities • Exponential functions and the number ‘e’ • Exponential growth and decay • Partial fractions • The Binomial theorem • Essential Reading • HELM Workbook 2.3: 1 – 1 and Inverse Functions. • HELM Workbook 2.7: Common Engineering Functions • HELM Workbook 3.6: Partial Fractions • HELM Workbook 6.1: Exponential Functions • HELM Workbook 6.4: Logarithmic Functions