CIVE1620- Engineering Mathematics 1.1 Lecturer: Dr D Borman Differentiation of a function From 1 st principles General techniques (trigonometric, logarithmic, exponential etc.) Chain Rule (Function of a Function) Lecture 2

Graph sketching y = f(x) = x x y y = x 1)Find where graph Crosses y-axis: (when x=0) 4) Look at the gradient of the line/curve -we’’ll come back to this 3) What happens as 2) Find where graph Crosses x-axis: (when y=0) Crosses y-axis:when x=0 y = 32 Crosses x-axis:when y=0 x = 0 – =

y =f(x) =a +bx y = mx + c Reminder: Equation of a straight line y =2x + 1 e.g. y =x + 1 y =-2x + 1 y x Gradient = up. across

Gradient of a Straight Line f(x) = y = a + bx Gradient = up. across

Gradient of a Curve Gradient = up. across : As

y = f(x) = x 2 Therefore, the ’derivative’ (gradient at x) is Principle of Differentiation Consider the gradient at a general point x (its value not yet specified). If It is then clear that as :

Derivatives of Some Basic Functions

Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 1) A B C D

Multiple choice Choose A,B,C or D for each of these: Differentiate the following wrt x : 2) A B C D

Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 3) A B C D

Multiple choice Choose A,B,C or D for each of these: Differentiate the following wrt x : 4) B D A C txxf51)t/sin(t)( 

Multiple choice Choose A,B,C or D for each of these: Differentiate the following wrt to t: 5) A B C D

Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 6) A B C D

Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 6) A B C D

Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 7) A B C D

Differentiate the following

1) 2) 3) 4) 5) 6) 7)

Differentiate the following 1) 2) 3) 4) 5) 6) 7)

Differentiating Composite functions =>Chain Rule (Function of a function) If you can differentiate sin(2x) you are already applying the CHAIN RULE Consider the relationship where => much simpler form to differentiate But we want where Chain rule

Differentiating Composite functions =>Chain Rule Also called Differentiating (Function of a function) Example with more than one function (3 parts in this case, but can be more) (just differentiate each part and multiply by the result) 1) Differentiate the exponential e f(x) part 2) Differentiate the sin(f(x)) part 3) Differentiate the x 2 part

Differentiate the following 1) 2) 3) 4) 5)

Differentiate the following 1) 2) 3) 4) 5)

What is the gradient of this graph at the point x=1? 5

Extension, 5 gradient at x=1, was found to be 5 A line is perpendicular to the curve at the point x=1, a)What’s the equation of this line? b)What possibilities do you have for finding where the line crosses the curve? (what problems might you have doing this?) (work in small groups 2-4 people)

Extension, 5 gradient at x=1, was found to be 5 a)y=mx +c and when x=1, y=2 b)How would you go about finding where the line crosses the curve?

There are lots of mathematical packages e.g. MATLAB that will allow us to solve an equation like this numerically (very simple to do – just type in equation and press go!). i.e. will find the values of x that satisfy the equation. Not possible to solve analytically (i.e. “mathematically!”) 3x^2+sin(x-1)+e^(x^2)-e^(2x-1)-2+1/x^2; -1/5x+11/5 graph-plotter.cours-de-math.eu

Simpler example that can be solved analytically. Where do these lines intersect? y = x 2 +2x+3 y = -x/2+3 y = x 2 +2x+3 y = -x/2+3 -x/2 + 3 = x 2 +2x+3 -x = 2x 2 + 4x 2x 2 + 5x = 0 x(2x + 5) = 0 So x= 0 or x = -5/2 = -2.5

-Differentiation -1 st principles -Standard functions sin, cos, exp, ln, etc - Chain Rule e.g. sin(2x) ****VLE –all slides and NOTES will be on VLE ****Maths lab tasks - log in and complete Week 1 task by Tuesday**** You can retrieve a password by entering your university

Wobbly millennium bridge- understanding & solving Finite element analysis and other modelling techniques require understanding of Series, Limits and partial differentiation. ©Albert Lee 2010, sourced from Available under Creative commons license

CFD modelling on new Arsenal stadium. Evaluating effects of wind on temperature and other climate factors inside the stadium

Arup Consulting engineers, designers, planners and project managers Eden Project – Arup's CFD flow analysis contributed to the design. They also helped develop software to model moisture concentrations in the Humid Tropics - to create a precise atmosphere for maximising plant growth. Computational fluid dynamic (CFD) image of the NASA Hyper - X at the Mach 7 test condition with the engine operating. NASA Modeling Air flow around building ©Kenneth Allen 2007, sourced from Available under creative commons license