CIVE1620- Engineering Mathematics 1.1 Lecturer: Dr Duncan Borman Differentiation –complex functions - Product Rule - Quotient Rule 2 nd Derivatives Lecture 3

Differentiation from first principles Gradient = up. across : As

Differentiating Composite functions =>Chain Rule (Function of a function) where => much simpler form to differentiate Chain rule Differentiate What about differentiating functions like ? Called function of a function – each function sat inside the next

Differentiating Composite functions =>Chain Rule Chain rule What about differentiating functions like ? Called function of a function – each function sat inside the next

Product Rule If our function is of the form: y = uv, (where the variables u and v themselves have relationships with x) What is this saying?? Differentiate the 2 nd part (v) and multiply by 1 st part (u) + Differentiate the 1 st part (u) and multiply by 2 nd part (v) => PRODUCT RULE Try these

2nd Differivative - differentiating twice wrt x

Quotient Rule If our function is of the form: => QUOTIENT RULE Try these

Results of Mathlab week 1 task Mathlab: Week 2 Task Available now – due next Tuesday 21 people who’ve not attempted the week 1 Task – please make attempt today! Fastest time for 100% - 21 mins

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Multiple choice Choose A,B,C or D for each of these: 2 nd Differential – differentiating twice wrt x 1) A B C D

Multiple choice Choose A,B,C or D for each of these: 2 nd Differential – differentiating twice wrt x 2) A B C D

Multiple choice Choose A,B,C or D for each of these: 2 nd Differential – differentiating twice wrt x 3) A B C D

Multiple choice Choose A,B,C or D for each of these: 2 nd Differential – differentiating twice wrt x 4) A B C D

A differential equation is… A) an equation that involves a derivative B) the highest order equation in a system of equations C) the smallest term in an equation D) an equation that gives an approximate solution when solved

A numerical solution to an equation is… A) an approximate solution that involves finding a solution at discrete points B) an exact solution C) the same as an analytical solution D) a solution that is a whole number

In these lectures so far the level of the content is… A) too hard B) about right C) too easy

My first impressions of Mathlab are : A) it’s brilliant- it’s getting in the way of my social life B) Useful, but it’s got a few an annoying bits C) not that impressed, but I guess it’s good to practice this stuff D) I hate it- please don’t make me use this

Question? A) B) C)

APPLICATIONS OF DIFFERENTATIONRate of change problems Instantaneous rate of change of a variable y = f(x) with respect to x. The instantaneous rate of change at point x = x 0 isevaluated at that point. Example 1 An object is falling such that its distance s metres from its initial position is related to time t seconds by the expression s = 4t 2. Find the instantaneous velocity at time t = 3 seconds. Hence, at t = 3, the instantaneous velocity is 8 x 3 =24 m/s. Velocity, v is, by definition, the rate of change of distance with respect to time, and so we differentiate to find:

APPLICATIONS OF DIFFERENTATIONRate of change problems Instantaneous rate of change of a variable y = f(x) with respect to x. The instantaneous rate of change at point x = x 0 isevaluated at that point. Examples - Velocity is rate of change of distance with respect to time Acceleration is rate of change of velocity wrt to time Rate of change of Area of a circle with respect to the radius (indicates the rate at which the area will change as the radius increases/decreases) Rate of change of Volume of a sphere with respect to the radius

Example 2 A spherical balloon is being inflated. Estimate the instantaneous rate of change of the volume of the balloon with respect to its radius, at a radius of 1 metre. and at a radius of r = 1 metre, this gives an instantaneous rate of change of volume with respect to radius of metres 3 /metre. Denoting the radius by r meters and the volume by V metres 3, it is known that these are related by Differentiating gives:

Rate of change question An object is falling such that its distance s metres from its initial position is related to time t seconds by the expression s = 4.9t 2. a) Find the instantaneous velocity at time t = 5 seconds. Acceleration is the RATE of CHANGE of velocity b) What would the acceleration of the object be at a) t=5 seconds?, b) t=20s?, c) t=3days? c) How would I write a general expression for acceleration in terms of s (using derivatives)?

Rate of change question An object is falling such that its distance s metres from its initial position is related to time t seconds by the expression s = 4.9t 2. a) Find the instantaneous velocity at time t = 5 seconds. Acceleration is the RATE of CHANGE of velocity b) What would the acceleration of the object be at a) t=5 seconds?, b) t=20s?, c) t=3days? c) How would I write a general expression for acceleration in terms of s (using derivatives)?

Problem A rectangular water tank is being filled at the constant rate of 20,000 cm 3 / second. The base of the tank has dimensions w =100cm and L = 200 cm. What is the rate of change of the height of water in the tank (wrt to time)? (express the answer in cm/second). Solution The volume V of water in the tank is given by: V = w*L*H Find the rate of change of the height H of water Rate of change of the volume = cm 3 /s. In V = w*L*H, only V and H are functions of time – so differentiate both sides wrt t = W*L* W and L can be considered constants W*L So ©emijrp 2006, sourced from Available under creative commons license

Finding the Maximum capacity - Example A plumbing company are manufacturing new linings for water tanks for large buildings, using only sheets of material of the sizes (1500mm x 2550mm) using the net shown below, what are the dimensions which would secure the highest capacity? 50mm There are 3 variables (b, h, l) and we can obtain 3 equations as follows: -From the restrictions (constraints) to the length and width we have: A function for the capacity can then be made by resolving these equations in terms of one single variable (in this case b) l h b And for the volume of the tank:

Differentiating to find the maximum capacity Can be solved to find b = 440mm or 1327mm Therefore 440mm is maximum Substitute b = 440cm to find h = 810mm and l = 960mm Using the sheet of size (1500mm x 2550mm) the maximum capacity of the tank would be 960mm x 440mm x 810mm l h b Differentiate again to find the nature of the turning point

Finding the Maximum capacity - Example To find maximum capacity... Can be solved to find b = 56mm or 175mm Therefore 56mm is maximum Substitute b = 56cm to find h= 810mm and l = 144mm With a sheet of size (1500mm x 2550mm) the maximum capacity of the tank would be 960mm x 440mm x 810mm A company who make orange juice cartons are manufacturing a new product, using only sheets of material of size (210mm x 297mm) using the net shown below, what are the dimensions which would secure the highest capacity (volume) carton?

Finding the Maximum capacity - Example A plumbing company are manufacturing new linings for water tanks for large buildings, using only sheets of material of the sizes (1500mm x 2550mm) using the net shown below, what are the dimensions which would secure the highest capacity. 50mm To find maximum capacity... Can be solved to find b = 440mm or 1327mm Therefore 440mm is maximum Substitute b = 56cm to find l = 1620mm and h = 960mm With a sheet of size (1500mm x 2550mm) the maximum capacity of the tank would be: 440mm x 1620mm x 960mm l h b

e.g 2 nd derivatives If ƒ is a differentiable function f′(x) is its derivative. The derivative of f′(x) (if it has one) is written f′′(x) and is called the second derivative of ƒ. Similarly, the derivative of a second derivative, if it exists, is written f′′′(x) and is called the third derivative of ƒ. These repeated derivatives are called higher-order derivatives. Higher derivatives

Functions that are NOT Differentiable Which is an equation of a straight line for each of negative and positive. ? But what happens exactly at Is the derivative/gradient 1 or -1? We can’t say, and for this reason we say thatis not differentiable at

Finally, How do I differentiate this?.i.e find Implicit differentiation

Finally, How do I differentiate this?.i.e find Implicit differentiation

Differentiation techniques you need to know and be confident with: Standard functions (sin, cos, ln, e x, x n...) Chain Rule (or function of a function) Product rule Quotient Rule 2nd derivatives (and 3 rd, 4 th derivative etc) Implicit Differentiation Mathlab: Week 2 Task Available now – due next Tuesday

Differentiating both sides of the equation wrt to t Area of base=5m A large tank is being filled with water at a set rate. Find an equation in terms of l for the rate at which the volume of water will change with time. Volume of water = m 3 Now if we know a value for i.e. the rate the volume of water is increasing (that’s just the flow rate of the tap e.g. 1000cm 3 /second) we can substitute in to find

Problem: An airplane is flying in a straight direction and at a constant height of 5000 meters. The angle of elevation of the airplane from a fixed point of observation is a. The speed of the airplane is 500 km/hr. What is the rate of change of angle a when it is at 25 degrees? Horizontal velocity, We need a relationship between angle a and distance x. => We want when a=25 degrees Differentiate both sides wrt t