Lecture 1 Overview (MA2730,2812,2815) Lecture 1

Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Contents of the teaching and assessment blocks MA2730: Analysis I

Lecture slides for MA2730 Analysis I Analysis — taming inﬁnity Maclaurin and Taylor series. Simon Shaw Sequences. people.brunel.ac.uk/~icsrsss Improper Integrals. [email protected] Series.

College of Engineering, Design and Physical Sciences Convergence. bicom & Materials and Manufacturing Research Institute Brunel University LATEX 2ε assignment in December. Question(s) in January class test. Question(s) in end of year exam. September 21, 2015 Web Page: http://people.brunel.ac.uk/~icsrsss/teaching/ma2730

Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Lecture 1 Lecture 1 MA2730: topics for Lecture 1 Functions — Level 1 revision

What is a function? How do we deﬁne a function? What ingredients do we need? A function is a triple: Lecture 1 A rule, for example f(x)= x2 + sin(√x). A domain, for example. . . Revision of functions and factorials (from Level 1) D = [0, ) R. ∞ ⊆ Maclaurin polynomials — some intuition A range, for example. . . Taylor polynomials R = [ 1, ) R − ∞ ⊆ Deﬁnitions and formality We then write f : D R with rule f(x)= x2 + sin(√x). → Examples and exercises Often the range is replaced with the co-domain. The co-domain is The introductory material covered in this lecture can be found in usually R. Usually the domain is taken to be the largest subset of The Handbook in Chapter 1 and Section 1.1. See, for example, R that is possible. equation (1.2) with a = 0. So we usually work with just the rule, but bear in mind that this is just a lazy convenience! Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Lecture 1 Lecture 1 What is a factorial? Motivation: plot of sin(x) and a polynomial

We’re going to use these a lot. x3 x5 x7 3! = 3 2 1 = 6 x + × × − 3! 5! − 7! 5! = 5 4 3 2 1 = 120 1.5 × × × × sin(x) 1! = 1 polynomial 0! = 1 — just accept it! 1

In general, 0.5

n(n 1)(n 2) (3)(2)(1) if n > 1 0 if n N 0 then n!= − − · · · ∈ ∪ { } 1 if n = 0. −0.5

We will also need the mean value theorem and Rolle’s theorem, −1 but these can wait. −1.5 Although it would be commendably pro-active of you to revise −10 −5 0 5 10 them sooner rather than later Shaw , bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Lecture 1 Lecture 1 Motivation: plot of sin(x) and a special polynomial Most functions are (nearly) polynomials x3 x5 x7 x9 x11 x13 x15 x17 x19 x21 x + + + + + − 3! 5! − 7! 9! − 11! 13! − 15! 17! − 19! 21! So what?

1.5 sin(x) Question: How does your calculator compute sin(1)? polynomial

1 All computers can do is add and subtract...... in binary 0.5 Answer: they use approximations — like polynomials

0 Maclaurin and Taylor Series −0.5 In this ﬁrst part of Analysis I we are going to study Maclaurin and −1 Taylor series.

−1.5 −10 −5 0 5 10 This is very important material; the results will pervade much of your subsequent mathematical studies.

Lecture 1 Lecture 1 Who are they? Very Important Results. . .

Colin Maclaurin (1698 — 1746) Brook Taylor (1685 — 1731) The Maclaurin series expansion for sin(x) x3 x5 x7 ∞ ( 1)nx2n+1 sin(x)= x + + = − − 3! 5! − 7! · · · (2n + 1)! n=0 X Similarly, for cos(x) and exp(x):

x2 x4 x6 ∞ ( 1)nx2n cos(x) = 1 + + = − , − 2! 4! − 6! · · · (2n)! n=0 We just met the Maclaurin series expansion for sin(x): X x2 x3 x4 ∞ xn ex = 1+ x + + + + = . 3 5 7 n 2n+1 2! 3! 4! · · · n! x x x ∞ ( 1) x n=0 sin(x)= x + + = − X − 3! 5! − 7! · · · (2n + 1)! n=0 X There are of profound importance — remember them. . . forever!

Lecture 1 Lecture 1 Let’s develop some intuition. . . Digging deeper. . .

The Maclaurin series expansion for sin(x) Calculating the Maclaurin series expansion for sin(x) x3 x5 x7 ∞ ( 1)nx2n+1 We just found a = 0, so we now need a , a , a . . . such that, sin(x)= x + + = − 0 1 2 3 − 3! 5! − 7! · · · (2n + 1)! n=0 2 3 4 X sin(x)= a1x + a2x + a3x + a4x + · · · How can we calculate this? How could you calculate a1? HINT: differentiate. . . Well, start by assuming that coefficients a0, a1, a2,... exist such that, d 2 3 4 Apply to both sides: sin(x)= a0 + a1x + a2x + a3x + a4x + dx · · · 2 3 How could you calculate a0? cos(x)= a + 2a x + 3a x + 4a x + 1 2 3 4 · · · Now, how can we find a1? Take x = 0 again. We get a1 = 1. How about taking x = 0? We get a0 = 0.

Lecture 1 Lecture 1 Stop and look back Countdown

Calculating the Maclaurin series expansion for sin(x) Derive the Maclaurin series expansion for sin(x) We started by assuming, x3 x5 x7 ∞ ( 1)nx2n+1 sin(x)= x + + = − 2 3 4 − 3! 5! − 7! · · · (2n + 1)! sin(x) = a + a x + a x + a x + a x + n=0 0 1 2 3 4 · · · X d 2 gave: cos(x) = a + 2a x + 3a x2 + 4a x3 + Start with: sin(x)= a0 + a1x + a2x + dx 1 2 3 4 · · · · · ·

Taking x = 0 gave a0 = 0.

Diﬀerentiating and taking x = 0 gave a1 = 1.

How could you ﬁnd a2? Diﬀerentiate and take x = 0. . .

How could you ﬁnd a3? Diﬀerentiate and take x = 0. . .

Oﬀ you go then. 5! seconds. . . Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Lecture 1 Lecture 1 Here it is. Child’s play... The Maclaurin expansion of sin(x)

Start with the assumption that,

2 3 4 sin(x)= a0 + a1x + a2x + a3x + a4x + · · · We have just discovered how to derive Take x = 0 to get a0 = 0 and differentiate, x3 x5 x7 ∞ ( 1)nx2n+1 2 3 sin(x)= x + + = − cos(x)= a1 + 2a2x + 3a3x + 4a4x + − 3! 5! − 7! · · · (2n + 1)! · · · nX=0 Take x = 0 to get a1 = 1 and differentiate, This is the Maclaurin expansion of sin(x). sin(x) = 2a + 3 2a x + 4 3a x2 + − 2 × 3 × 4 · · · The same ‘x = 0 and differentiate’ technique can be applied for

Take x = 0 to get a2 = 0 and diﬀerentiate, ‘any’ function on the left hand side...

cos(x) = 3 2a + 4 3 2a x + − × 3 × × 4 · · · Take x = 0 to get a = 1/3! ..., 3 − Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Lecture 1 Lecture 1 Exercise Solution: boardwork.

Examples

x2 x3 x4 ∞ xn Show that ex =1+ x + + + + = 2! 3! 4! · · · n! n=0 X Start by assuming ex = a + a x + a x2 + a x3 + and then use 0 1 2 3 · · · the ‘x = 0 and diﬀerentiate’ technique.

You have cosec4(π/4) minutes.

Lecture 1 Lecture 1 Polynomial approximation An application

Remember that earlier we saw that chopping oﬀ 3 5 So sin(x) x x + x near to x = 0. A computer/calculator ≈ − 3! 5! x3 x5 x7 ∞ ( 1)nx2n+1 could use this in an algorithm. sin(x)= x + + = − − 3! 5! − 7! · · · (2n + 1)! Computers cannot look up the value of sin(x). n=0 X Computers can only add and subtract in binary. Multiplication is at the x5 term gave a good approximation to sin(x) near to x = 0. like repeated addition and division is like repeated subtraction.

1.5 3 5 sin(x) x x polynomial And that is how they can do arithmetic: + . x + looks like: 1 − ×÷ − 3! 5! 0.5 They cannot do anything more complicated than this. x3 x5 Key Point: if x is close to zero 0 But they can calculate x 3! + 5! . 2 3 4 − then x , x , x ,... all become −0.5 And that gives sin(x) accurately near to x = 0, for example. smaller and more negligible as −1 But what if you want sin(47◦)?

−1.5 x 0. −10 −5 0 5 10 →

Lecture 1 Lecture 1 What about ‘expanding about x = a’? Taylor expansions

But what if you want sin(47◦)? Let’s see how this works in general when we expand any function

Well, notice that 47◦ is close to π/4 radians. we like about any a we choose. Write: We must always use radians in calculus: f(x)= a + a (x a)+ a (x a)2 + a (x a)3 + a (x a)4 + 0 1 − 2 − 3 − 4 − · · · 47◦ = a radians where a = 47π/180. Take x = a to get a = f(a)= f(a)/0!. Differentiate. . . Instead of writing 0 2 3 f ′(x)= a1 + 2a2(x a) + 3a3(x a) + 4a4(x a) + sin(x)= a + a x + a x2 + a x3 + a x4 + − − − · · · 0 1 2 3 4 · · · Take x = a to get a1 = f ′(a)= f ′(a)/1!. Differentiate. . . let’s instead try 2 f ′′(x) = 2a2 + 3!a3(x a) + 4 3a4(x a) + 2 3 4 − × − · · · sin(x)= a0+a1(x π/4)+a2(x π/4) +a3(x π/4) +a4(x π/4) + − − − − · · · Take x = a to get a2 = f ′′(a)/2= f ′′(a)/2!. Differentiate. . .

f ′′′(x) = 3!a + 4!a (x a)+ Now if x is close to π/4 we can expect the higher powers of 3 4 − · · · (x π/4) to become negligible as x π/4. Take x = a to get a = f (a)/3!. Can you see the pattern? − → 3 ′′′ Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Lecture 1 Lecture 1 Taylor expansions Maclaurin’s and Taylor’s theorems

By writing By writing

2 3 4 2 3 4 f(x)= a0 + a1(x a)+ a2(x a) + a3(x a) + a4(x a) + f(x)= a + a (x a)+ a (x a) + a (x a) + a (x a) + − − − − · · · 0 1 − 2 − 3 − 4 − · · · and using the ‘take x = a and differentiate’ technique we found and using the ‘take x = a and differentiate’ technique we found (n) that, that an = f /n!. f(a) a = The Taylor and Maclaurin expansion (series) 0 0! The inifintely differentiable function f : R R has the formal f ′(a) → a1 = expansion about a R given by: 1! ∈ f ′′(a) a = 2 f ′′(a) 3 f ′′′(a) 2 f(x)= f(a) + (x a)f ′(a) + (x a) + (x a) + 2! − − 2! − 3! · · · f (a) a = ′′′ 3 3! ...to infinity. If a = 0 this is called the Maclaurin expansion or Can you see the pattern? series, otherwise it is called the Taylor expansion or series. Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1

Lecture 1 Lecture 1 Maclaurin’s and Taylor’s theorems Exercise

The inifintely differentiable function f : R R has the formal → expansion about a R given by: ∈ ∞ f (n)(a) f(x)= (x a)n . Examples − n! n=0 X Expand sin(x) about x = π/4 up to third powers. Use it to approximate sin(47◦) = sin(47π/180) and determine the error. Some questions arise. . . What does ‘summing to ’ actually mean? Try this before the next lecture. We’ll pick it up again then. ∞ Can we differentiate a function infinitely many times? How can we tame infinity?

That’s what analysis is all about.

Lecture 1 Lecture 1 Summary Summary. We deduced the general form:

The Taylor and Maclaurin expansion or series We have met the Maclaurin series expansions: The inifintely differentiable function f : R R has the formal → expansion about a R given by: x3 x5 x7 ∞ ( 1)nx2n+1 ∈ sin(x) = x + + = − (n) − 3! 5! − 7! · · · (2n + 1)! ∞ n f (a) n=0 f(x)= (x a) . X − n! x2 x4 x6 ∞ ( 1)nx2n n=0 cos(x) = 1 + + = − , X − 2! 4! − 6! · · · (2n)! n=0 If a = 0 this is called the Maclaurin expansion or Maclaurin series, X otherwise it is called the Taylor expansion or Taylor series. x2 x3 x4 ∞ xn ex = 1+ x + + + + = . 2! 3! 4! · · · n! And we used Taylor’s series to approximate a function about x = 0. n=0 6 X This idea is useful. See also the first example in Section 1.1 in The Can you see how these might be related? Handbook where it is shown that, 1+ x + x2 + x3 = 40 + 34(x 3) + 10(x 3)2 + (x 3)3. − − − Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1

End of Lecture

Computational and αpplie∂ Mathematics

We’re all born ignorant but one must work hard to remain stupid Benjamin Franklin

This lecture was based on Section 1.1, in Chapter 1 of The Handbook. Homework: attempt Questions 1 and 2 on Exercise Sheet 1a.

Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16