DIFFERENTIATION AND INTEGRATION PART 1. Mr C s IB Standard Notes
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1 DIFFERENTIATION AND INTEGRATION PART 1 Mr C s IB Standard Notes
2 In this PDF you can find the following: 1. Notation 2. Keywords Make sure you read through everything and the try examples for yourself before looking at the solutions 3. Basic Methods 4. Standard Derivatives and Integrals
3 NOTATION Writing derivatives and integrals Functions are typically written two different ways
4 NOTATION Writing derivatives and integrals y = e x + sin 2x + y =..
5 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + Or f(x)=.
6 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + We have different ways to write derivatives
7 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + 1 st Derivative
8 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = f (x) = 1 st Derivative
9 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = f (x) = 2 nd Derivative (differentiate again)
10 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = d 2 y dx 2 = f (x) = f (x) = 2 nd Derivative (differentiate again)
11 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = d 2 y dx 2 = f (x) = f (x) = 3 nd Derivative (and again)
12 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = f (x) = 3 nd Derivative (and again) d 2 y dx 2 = d 3 y dx 3 = f (x) = f (3) (x) =
13 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = f (x) = 4 th Derivative (yet again) d 2 y dx 2 = d 3 y dx 3 = f (x) = f (3) (x) =
14 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = d 2 y dx 2 = d 3 y dx 3 = d 4 y dx 4 = f (x) = f (x) = f (3) (x) = f (4) (x) = 4 th Derivative (yet again)
15 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = d 2 y dx 2 = d 3 y dx 3 = d 4 y dx 4 = f (x) = f (x) = f (3) (x) = f (4) (x) = And so on
16 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + dy dx = d 2 y dx 2 = d 3 y dx 3 = d 4 y dx 4 = d n y dx n = f (x) = f (x) = f (3) (x) = f (4) (x) = f (n) (x) = And so on
17 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + When we integrate we always write it the same way
18 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + y dx = f(x) dx = When we integrate we always write it the same way
19 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + y dx = f(x) dx = Make sure to include the variable you are integrating with respect to
20 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + y dx = f(x) dx = Make sure to include the variable you are integrating with respect to
21 NOTATION Writing derivatives and integrals y = e x + sin 2x + f(x) = e x + sin 2x + y dx = f(x) dx = This is the same as the variable in the equation (in our case x)
22 NOTATION Writing derivatives and integrals Sometimes the question may use different lettered variables
23 NOTATION Writing derivatives and integrals A(t) = e t + 2t 2 Sometimes the question may use different lettered variables
24 NOTATION Writing derivatives and integrals A(t) = e t + 2t 2 Just make sure to write the correct derivative or integral
25 NOTATION Writing derivatives and integrals A(t) = e t + 2t 2 da dt = Just make sure to write the correct derivative or integral
26 NOTATION Writing derivatives and integrals A(t) = e t + 2t 2 da dt = A dt = Just make sure to write the correct derivative or integral
27 KEYWORDS When do I differentiate? Make sure to look out for certain keywords which will let you know to differentiate
28 KEYWORDS When do I differentiate? Find derivative, dy/dx, f (x) You might be explicitly told to differentiate...that s pretty straight forward
29 KEYWORDS When do I differentiate? Find derivative, dy/dx, f (x) Gradient You might need to find the gradient of a function for a particular value of x
30 KEYWORDS When do I differentiate? Find derivative, dy/dx, f (x) Gradient Equation of Tangent or Normal If you want to work out normals, you must first find the gradient and then take the negative reciprocal (3 1 3 )
31 KEYWORDS When do I differentiate? Find derivative, dy/dx, f (x) Gradient Equation of Tangent or Normal Differentiation tells you how one variable changes with respect to another, e.g. differentiate speed w/r to time and get acceleration Rate of Change
32 KEYWORDS When do I differentiate? Find derivative, dy/dx, f (x) Gradient Equation of Tangent or Normal You might need to find stationary or turning points of a curve (where gradient = 0) Rate of Change Stationary points (max/min)
33 KEYWORDS When do I differentiate? Find derivative, dy/dx, f (x) Gradient Equation of Tangent or Normal Finding the maximum or minimum value for a function is it s optimal solution. You often see questions like this with population models. Rate of Change Stationary points (max/min) Finding Optimum Solutions (best, largest, etc.)
34 KEYWORDS When do I differentiate? Find derivative, dy/dx, f (x) Gradient Equation of Tangent or Normal Rate of Change Stationary points (max/min) Finding Optimum Solutions (best, largest, etc.)
35 KEYWORDS When do I integrate? There are fewer examples of keywords to look for when integrating
36 KEYWORDS When do I integrate? Find integral, Quite often you will simply be told to integrate, or given the integration symbol
37 KEYWORDS When do I integrate? Find integral, Find the Area You may be asked to find a particular area (e.g. beneath a curve, bounded between two curves, etc.)
38 KEYWORDS When do I integrate? Find integral, Find the Area Volumes of Revolution And don t forget there is a formula for integrating to find the volume produced when a shape is rotated around the x-axis
39 KEYWORDS When do I integrate? Find integral, Find the Area Volumes of Revolution Don t forget to square the function first. You can multiply by π either at the beginning or end, it doesn t make a difference.
40 KEYWORDS When do I integrate? Find integral, Find the Area Volumes of Revolution
41 DIFFERENTIATION TECHNIQUES Powers of x
42 DIFFERENTIATION TECHNIQUES Powers of x Any expression consisting of terms using x n can be differentiated the same way
43 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x Any expression consisting of terms using x n can be differentiated the same way
44 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x You can differentiate terms one at a time, and they all follow the same rule
45 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x Multiply by the power Subtract one from the power f (x)= 4x
46 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x Sometimes you might have to rewrite using powers and rules of indices f (x)= 4x
47 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x = x1 2 x 2x+1 x Sometimes you might have to rewrite using powers and rules of indices f (x)= 4x
48 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x And then finally differentiate = x1 2 x = x 1 2 f (x)= 4x
49 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x = x1 2 x = x 1 2 f (x)= 4x 1 2 x 3 2
50 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x x 1 2 x 2x+1 x You can also split up fractions to make life easier (all terms on top divide by everything on the bottom) x 1 2 f (x)= 4x 1 2 x 3 2
51 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x x 1 2 x 2x x x 3 x 1 2 f (x)= 4x 1 2 x 3 2
52 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x x 1 2 x 2x x x 3 x 1 2 2x 2 +x 3 f (x)= 4x 1 2 x 3 2
53 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x x 1 2 x 2x x x 3 x 1 2 2x 2 +x 3 f (x)= 4x 1 2 x x 3 3x 4
54 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x x 1 2 x 2x x x 3 x 1 2 2x 2 +x 3 f (x)= 4x 1 2 x x 3 3x 4
55 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x x 1 2 x 2x+1 x x x x 3 When you differentiate a constant it disappears (7 is 7x 0 ), multiplying by 0 makes it all 0 x 1 2 2x 2 +x 3 f (x)= 4x 1 2 x x 3 3x 4 +0
56 DIFFERENTIATION TECHNIQUES Powers of x e.g. 2x 2 + x x 2x+1 x x 1 2 x 2x x x 3 x 1 2 2x 2 +x 3 f (x)= 4x 1 2 x x 3 3x 4
57 INTEGRATION TECHNIQUES Powers of x
58 INTEGRATION TECHNIQUES Powers of x Integration is the reverse of differentiation
59 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx
60 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx To differentiate you: x power, -1 from power Therefore to integrate you: +1 to power, power
61 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx To differentiate you: x power, -1 from power Therefore to integrate you: +1 to power, power
62 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx To differentiate you: x power, -1 from power Therefore to integrate you: +1 to power, power
63 INTEGRATION TECHNIQUES Powers of x e.g. 5x x4 dx To differentiate you: x power, -1 from power Therefore to integrate you: +1 to power, power
64 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx Integrating a constant just puts an x on the end of it 4 is 4x x1 5 4 x4
65 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx Integrating a constant just puts an x on the end of it 4 is 4x x1 5 4 x4 +4x
66 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx At this point it looks as though we are correct. The integral we have can be differentiated and give the original expression 5 4 x4 +4x
67 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx However we also must include a constant term that might have differentiated to x4 +4x
68 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx However we also must include a constant term that might have differentiated to x4 +4x
69 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx 5 4 x4 +4x +C
70 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx This is our constant of integration, C 5 4 x4 +4x +C Always remember to write it at the end of an indefinite integral
71 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx 5 4 x4 +4x +C Including limits
72 INTEGRATION TECHNIQUES Powers of x e.g. 5x x4 +4x +C Including limits dx The one time you do not need to include C, is when you are evaluating the integral (known as a definite integral)
73 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx The limits tell you numbers to substitute into your final expression as x. 5 4 x4 +4x +C Including limits 2 e.g. 5x dx
74 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1
75 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx You first evaluate by substituting the top number for x 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1
76 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx You first evaluate by substituting the top number for x 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2)
77 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx You first evaluate by substituting the top number for x 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2) 28
78 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx And then the bottom number 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2) 28
79 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx And then the bottom number 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2) (1)4 +4(1)
80 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx And then the bottom number 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2) (1)4 +4(1) 2 3 4
81 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx And subtract the result 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2) (1)4 +4(1) 2 3 4
82 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx And subtract the result 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2) (1)4 +4(1) 2 3 4
83 INTEGRATION TECHNIQUES Powers of x e.g. 5x dx And subtract the result 5 4 x4 +4x +C Including limits 2 e.g. 5x dx = 5 4 x4 + 4x 2 1 = 5 4 (2)4 +4(2) (1)4 +4(1) =
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