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ACS123 Functions

ACS123 Functions. Dr Viktor Fedun Automatic Control and Systems Engineering, C09 Based on lectures by Dr Anthony Rossiter. Why is mathematics important?. Why do engineers need to be good at mathematics? Is it sufficient to memorise key results?

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ACS123 Functions

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  1. ACS123Functions Dr Viktor Fedun Automatic Control and Systems Engineering, C09 Based on lectures by Dr Anthony Rossiter

  2. Why is mathematics important? Why do engineers need to be good at mathematics? Is it sufficient to memorise key results? Just because a learning technique worked at school, does that make it the best method now? What mathematics do I need to be good at?

  3. Mathematics is a tool-kit A good engineer: • knows which is the best tool to use? • Is proficient in using the tool? • Can adapt the tool to a new use. It is not good enough to memorise key results as the most important skill is abstraction. You must put your effort into understanding.

  4. Module assessment • 3 in class tests in weeks 4, 7, 11, 13 These will be similar to exam questions. • An exam in May or June If you want feedback on an answer you have done, ask in a tutorial. 20 credit module, similar pattern in semester 2.

  5. Module organisation I will teach the first semester Lectures and tutorials Of the 5 timetabled hours, 2-3 will be used for lectures (these times may vary each week).

  6. MOLE Please use the discussions board to ask questions. Then everyone can see the question and answer. I will not respond to email queries unless of a personal or private nature.

  7. Resources Learning is only effective where students engage in self-discovery. • What you hear, you will usually forget. • You only really understand something when you use it. • We will provide ample materials, but YOU will only learn if you use these properly. [5-6 hours per week] Lecturers are here to guide – NOT TO TEACH! We will answer queries and be as helpful as possible, but only you can do the work.

  8. Be a function Stand up. • Use your arms to illustrate y=x. • What about y=-x? • Can you do y=x2, or even x3. • What about sine(x) – you may need a partner. Now do cosine(x). • Can you y=mod(x)? Or even y=sqrt(x2)? • Can you think of any more?

  9. Common functions • sine, cosine, tangent (and their inverses) • logarithm, exponential • sinh and cosh • straightline, quadratic, general polynomial • combinations of above as products, composites and fractions. You should be familiar with shapes of common functions and be able to sketch quickly.

  10. Example 1 (Page 136, Kuldeep and Singh, Example 3) [Mechanics] The displacement, φ(t), of a particle at time t is given by: φ(t)= 2t3 +t2 - 10t + 10 • Evaluate φ(2), φ(3), φ(5). • Find simplified expressions for: (i) φ(t2) (ii) φ(t + 1)

  11. Example 1 Solution Solution: • We have • φ(2) = (2 x 23) + 22 – (10 x 2) + 10 = 10 • φ(3) = (2 x 33) + 32 – (10 x 3) + 10 = 43 • φ(5) = (2 x 53) + 52 – (10 x 5) + 10 = 235

  12. Example 1 Solution (b) (i) For φ(t2) we replace the t with t2 in φ(t)= 2t3 +t2 - 10t + 10: φ(t2) = 2(t2)3 +(t2)2 – 10(t2) + 10 = 2t6 +t4 – 10t2 + 10 (ii) For φ(t + 1) we replace t with t+1 in φ(t)= 2t3 +t2 - 10t + 10: φ(t+1)= 2(t+1)3 +(t+1)2 – 10(t+1) + 10:

  13. What is a function? • A rule which translates an input, usually to a single output. • What are the functions for: • Double the input • Shift the input by 3 • Cube the input and subtract 1. • Write down in words the functions for

  14. What variables can a function have? What is the difference between the functions f(x), g(w), h(y) and k(x) A function describes a relationship, the variable names are unimportant. Engineers typically use variable names that relate to the topic: W for weight, h for height, L for length, etc.

  15. What is a function argument? The part that appears in the brackets; • For y=f(x), x is the argument. • For z=g(w), w is the argument. Thus argument is another word for the input to the function. Independent and dependent variables: what do you think these are? Use common sense.

  16. Composition of functions What do the following statements mean?

  17. Evaluate the following Find y when x=pi/2. Find w when z=1.

  18. Function products Evaluate A given that: A = y2h with x=2 and z=3 Write down a detailed function expression to express A.

  19. Example 2 (Page 152 Kuldeep Singh, Example 16) [Reliability Engineering] The failure density function, f(t), for a component is given by: f(t) = 1/8 where 0 < t < 8 years. Find F(t), R(t) and h(t) where these are defined as: F(t) = tf(t) (Failure Distribution function) R(t) = 1-F(t) (Reliability function) h(t) = f(t) / R(t) (hazard Rate function) and 0 < t < 8 years.

  20. Example 2 (Page 152 Example 16) Solution We have: F(t) = tf(t) = t(1/8) = t/8. R(t) = 1-F(t) = 1- t/8 h(t) = f(t) / R(t) = (1/8)/(1-t/8) = 1/(8-t)

  21. Graphs and sketching By first producing a suitable table, sketch the graphs of the following functions in the domain -3 to 3. Domain is the values allowed to the argument or independent variable. Range is the values the output (dependent variable) can take. What is the range of these?

  22. Example 3 (Page 110 Example 7) [Fluid Mechanics] The streamlines of fluid flow are given by: y = x2 + c where c is constant. Sketch the streamlines for c = 0, -1 ,1, -2, 2, -3 and 3.

  23. Example 3 (Page 110 Example 7) Solution The graphs of y = x2 + c for c = 0, -1 ,1, -2, 2, -3 and 3 are: (c=0) y = x2 (c=-1) y = x2 - 1 (c=1) y = x2 + 1 (c=-2) y = x2 -2 (c=2) y = x2 +2 (c=-3) y = x2 – 3 (c=3) y = x2 + 3

  24. y c = 3 c =2 c =1 c =0 c =-1 3 c =-2 2 c =-3 1 1 -2 -1 2 0 x -1 -2 -3 Notice how the graph of y=x2 + c varies as c changes. The c is where the curve cuts the y axis.

  25. Inverse function mean that all we’ve done is made a switch in emphasis

  26. Inverse function mean that all we’ve done is made a switch in emphasis 7 – 4 = 3 3 + 4 = 7

  27. Inverse function mean that all we’ve done is made a switch in emphasis 7 – 4 = 3 3 + 4 = 7 Both of this statements say the same thing, but with a change in emphasis

  28. Inverse function mean that all we’ve done is made a switch in emphasis 7 – 4 = 3 3 + 4 = 7 Both of this statements say the same thing, but with a change in emphasis

  29. Inverse function mean that all we’ve done is made a switch in emphasis 7 – 4 = 3 3 + 4 = 7 Both of this statements say the same thing, but with a change in emphasis -1

  30. Inverse function example y=2x-7; y=f(x)=2x-7

  31. Inverse function example y=2x-7; y=f(x)=2x-7 Identity function

  32. Inverse function example y=2x-7; y=f(x)=2x-7 Identity function and If inverse function

  33. Inverse function example y=2x-7; y=f(x)=2x-7 Identity function and If inverse function Composition of functions

  34. Inverse function -1 f f -1 A function f and its inverse f . Because f maps 1 to 4, the inverse f  maps 4 back to 1. -1

  35. One to one function For every value of x, there is a distinct value of y and for every value of y there is a distinct value of x. Which of the following is one to one? Draw the graph and it should be obvious.

  36. Inverse function What about? Proof

  37. Inverse functionexample

  38. Inverse functionexample Sometimes the inverse of a function cannot be expressed by a formula with a finite number of terms. For example, if f is the function -1 then f is one-to-one, and therefore possesses an inverse function f . The formula for this inverse has an infinite number of terms:

  39. Inverse functionexamples link

  40. Many-to-one and one-to-many Give some examples of many-to-one and one-to-many functions. The logic goes from independent variable to dependent variable.

  41. Notation Get into groups and decide three example functions with the following properties [3 for each item]. • Continuous • Discontinuous • Periodic (Why are these important?) • Odd • Even

  42. Odd Even

  43. Summary Independent variable (domain) Dependent variable (range) Function Many-to-one (one-to-one,…) Odd, even, periodic Inverse function Continuous/discontinuous Composite function Straight lines

  44. Exponential functions On some rough paper, do a sketch of the following functions. In what sense are the functions equivalent? With a suitable rescaling of x, they are all the same shape. Functions of this form are called exponentials.

  45. Exponential properties If you double the value of the independent variable, you square the value of the dependent variable. There is a constant ratio which depends solely on the difference of the argument: For all x!

  46. Exponential properties

  47. Exponential properties Exponentiation is not commutative 4 5 4 + 5 = 5 + 4 4 * 5 = 5 * 4 but 4 = 5 256 = 625

  48. Exponential properties Exponentiation is not commutative 4 5 4 + 5 = 5 + 4 4 * 5 = 5 * 4 but 4 = 5 256 = 625 Exponentiation is not associative (2 + 3) + 4 = 2 + (3 + 4) (2 * 3) * 4 = 2 * (3 * 4) 4 4 ( ) 3 3 ( ) 2 = 2.417.851.639.229.258.349.412.353 2 = 4096 but

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