35 - Performing Integration, differentiation using sage. |
35 - Performing Integration, differentiation using sage. |
36 - Defining matrices. |
36 - Defining matrices. |
37 - Defining Symbolic functions. |
37 - Defining Symbolic functions. |
38 - Simplifying and solving symbolic expressions and functions. |
38 - Simplifying and solving symbolic expressions and functions. |
39 |
39 |
40 * Questions 1 |
40 * Question 1 |
41 - Define the following expression as symbolic |
41 - Define the following expression as symbolic |
42 expression in sage. |
42 expression in sage. |
43 |
43 |
44 - x^2+y^2 |
44 - x^2+y^2 |
45 - y^2-4ax |
45 - y^2-4ax |
46 |
46 |
47 * Solutions 1 |
47 * Solution 1 |
48 #+begin_src python |
48 #+begin_src python |
49 var('x,y') |
49 var('x,y') |
50 x^2+y^2 |
50 x^2+y^2 |
51 |
51 |
52 var('a,x,y') |
52 var('a,x,y') |
53 y^2-4*a*x |
53 y^2-4*a*x |
54 #+end_src python |
54 #+end_src python |
55 * Questions 2 |
55 * Question 2 |
56 - Find the values of the following constants upto 6 digits precision |
56 - Find the values of the following constants upto 6 digits precision |
57 |
57 |
58 - pi^2 |
58 - pi^2 |
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59 - euler_gamma^2 |
59 |
60 |
60 |
61 |
61 - Find the value of the following. |
62 - Find the value of the following. |
62 |
63 |
63 - sin(pi/4) |
64 - sin(pi/4) |
64 - ln(23) |
65 - ln(23) |
65 |
66 |
66 * Solutions 2 |
67 * Solution 2 |
67 #+begin_src python |
68 #+begin_src python |
68 n(pi^2,digits=6) |
69 n(pi^2,digits=6) |
69 n(sin(pi/4)) |
70 n(sin(pi/4)) |
70 n(log(23,e)) |
71 n(log(23,e)) |
71 #+end_src python |
72 #+end_src python |
72 * Question 2 |
73 * Question 3 |
73 - Define the piecewise function. |
74 - Define the piecewise function. |
74 f(x)=3x+2 |
75 f(x)=3x+2 |
75 when x is in the closed interval 0 to 4. |
76 when x is in the closed interval 0 to 4. |
76 f(x)=4x^2 |
77 f(x)=4x^2 |
77 between 4 to 6. |
78 between 4 to 6. |
78 |
79 |
79 - Sum of 1/(n^2-1) where n ranges from 1 to infinity. |
80 - Sum of 1/(n^2-1) where n ranges from 1 to infinity. |
80 |
81 |
81 * Solution Q1 |
82 * Solution 3 |
82 #+begin_src python |
83 #+begin_src python |
83 var('x') |
84 var('x') |
84 h(x)=3*x+2 |
85 h(x)=3*x+2 |
85 g(x)= 4*x^2 |
86 g(x)= 4*x^2 |
86 f=Piecewise([[(0,4),h(x)],[(4,6),g(x)]],x) |
87 f=Piecewise([[(0,4),h(x)],[(4,6),g(x)]],x) |
87 f |
88 f |
88 #+end_src python |
89 #+end_src python |
89 * Solution Q2 |
90 |
90 #+begin_src python |
91 #+begin_src python |
91 var('n') |
92 var('n') |
92 f=1/(n^2-1) |
93 f=1/(n^2-1) |
93 sum(f(n), n, 1, oo) |
94 sum(f(n), n, 1, oo) |
94 #+end_src python |
95 #+end_src python |
95 |
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96 |
96 |
97 * Questions 3 |
97 * Question 4 |
98 - Differentiate the following. |
98 - Differentiate the following. |
99 |
99 |
100 - x^5*log(x^7) , degree=4 |
100 - sin(x^3)+log(3x), to the second order |
|
101 - x^5*log(x^7), to the fourth order |
101 |
102 |
102 - Integrate the given expression |
103 - Integrate the given expression |
103 |
104 |
104 - x*sin(x^2) |
105 - x*sin(x^2) |
105 |
106 |
106 - Find x |
107 - Find x |
107 - cos(x^2)-log(x)=0 |
108 - cos(x^2)-log(x)=0 |
108 - Does the equation have a root between 1,2. |
109 - Does the equation have a root between 1,2. |
109 |
110 |
110 * Solutions 3 |
111 * Solution 4 |
111 #+begin_src python |
112 #+begin_src python |
112 var('x') |
113 var('x') |
113 f(x)= x^5*log(x^7) |
114 f(x)= x^5*log(x^7) |
114 diff(f(x),x,5) |
115 diff(f(x),x,5) |
115 |
116 |
119 var('x') |
120 var('x') |
120 f=cos(x^2)-log(x) |
121 f=cos(x^2)-log(x) |
121 find_root(f(x)==0,1,2) |
122 find_root(f(x)==0,1,2) |
122 #+end_src |
123 #+end_src |
123 |
124 |
124 * Question 4 |
125 * Question 5 |
125 - Find the determinant and inverse of : |
126 - Find the determinant and inverse of : |
126 |
127 |
127 A=[[x,0,1][y,1,0][z,0,y]] |
128 A=[[x,0,1][y,1,0][z,0,y]] |
128 |
129 |
129 * Solution 4 |
130 * Solution 5 |
130 #+begin_src python |
131 #+begin_src python |
131 var('x,y,z') |
132 var('x,y,z') |
132 A=matrix([[x,0,1],[y,1,0],[z,0,y]]) |
133 A=matrix([[x,0,1],[y,1,0],[z,0,y]]) |
133 A.det() |
134 A.det() |
134 A.inverse() |
135 A.inverse() |
135 #+end_src |
136 #+end_src |
136 * Summary |
137 * Summary |
137 - We learnt about defining symbolic |
138 - We learnt about defining symbolic expression and functions. |
138 expression and functions. |
139 - Using built-in constants and functions. |
139 - Using built-in constants and functions. |
140 - Using <Tab> to see the documentation of a function. |
140 - Using <Tab> to see the documentation of a |
141 - Simple calculus operations . |
141 function. |
142 - Substituting values in expression using substitute function. |
142 |
143 - Creating symbolic matrices and performing operation on them . |
143 * Summary |
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144 - Simple calculus operations . |
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145 - Substituting values in expression |
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146 using substitute function. |
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147 - Creating symbolic matrices and |
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148 performing operation on them . |
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149 |
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150 * Thank you! |
144 * Thank you! |
151 #+begin_latex |
145 #+begin_latex |
152 \begin{block}{} |
146 \begin{block}{} |
153 \begin{center} |
147 \begin{center} |
154 This spoken tutorial has been produced by the |
148 This spoken tutorial has been produced by the |