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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
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Graph sketching y = f(x) = 32 + 1.8x x y y = 32 + 1.8x 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 – 32 1.8 = -17.78
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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
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Gradient of a Straight Line f(x) = y = a + bx Gradient = up. across
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Gradient of a Curve Gradient = up. across : As
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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 :
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Derivatives of Some Basic Functions
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Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 1) A B C D
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Multiple choice Choose A,B,C or D for each of these: Differentiate the following wrt x : 2) A B C D
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Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 3) A B C D
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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)(
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Multiple choice Choose A,B,C or D for each of these: Differentiate the following wrt to t: 5) A B C D
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Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 6) A B C D
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Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 6) A B C D
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Multiple choice Choose A,B,C or D for each of these: Differentiate the following: 7) A B C D
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Differentiate the following
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1) 2) 3) 4) 5) 6) 7)
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Differentiate the following 1) 2) 3) 4) 5) 6) 7)
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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
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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
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Differentiate the following 1) 2) 3) 4) 5)
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Differentiate the following 1) 2) 3) 4) 5)
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What is the gradient of this graph at the point x=1? 5
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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)
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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?
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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
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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
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-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 e-mail http://graph-plotter.cours-de-math.eu
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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 http://en.wikipedia.org/wiki/File:WBB.jpghttp://en.wikipedia.org/wiki/File:WBB.jpg Available under Creative commons license
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CFD modelling on new Arsenal stadium. Evaluating effects of wind on temperature and other climate factors inside the stadium
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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 http://www.geograph.org.uk/photo/462742http://www.geograph.org.uk/photo/462742 Available under creative commons license
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