Lab5temp 151 22C Jupyter Notebook 10 4 22 1 30 PM MATH 151 Lab 5 Put team members names and section number here In 1 from sympy import from sympy plotting import plot plot parametric Question 1 In 2 from sympy import x symbols x f exp x 1 x 2 print The first derivative of y is y diff f x the second is y The first derivative of y is y 2 x exp x x 2 1 exp x the second is y x 2 4 x 3 exp x the third is y x 2 6 x 7 exp x the fourth is y x 2 8 x 13 exp x t he fifth is y x 2 10 x 21 exp x the sixth is y x 2 12 x 31 exp x the seventh is y x 2 14 x 43 exp x and the eigth is y x 2 16 x 57 exp x In 3 print The formula for the nth derivative of f is exp x x 2 n 2 x n n 1 1 The formula for the nth derivative of f is exp x x 2 n 2 x n n 1 1 In 1 from sympy import x symbols x f exp x 1 x 2 print Derivative fifty is exp x x 2 50 2 x 50 50 1 Derivative fifty is x 2 100 x 2451 exp x Question 2 http localhost 8888 notebooks Downloads Lab5temp 151 22C ipynb 3g Page 1 of 5 1a 1b 1c In 14 A symbols A B symbols B f A sin k t B sin k t deriv diff f t 2 print f f print 4f 25f 0 s 4 deriv 25 f value solve s k print values of k are value 0 and value 2 therefore all members of y A sin kt B cos kt is also a solution Lab5temp 151 22C Jupyter Notebook 10 4 22 1 30 PM In 15 from sympy import k symbols k t symbols t y cos k t derivative diff y t 2 f 4 derivative 25 y value solve f k print values of k are value 0 and value 1 values of k are 5 2 and 5 2 f A sin k t B sin k t 4f 25f 0 values of k are 5 2 and 5 2 therefore all members of y A sin kt B cos kt is also a solution Question 3 In 17 t symbols t f t 2 2 t 1 9 derivative diff f t print Derivative is derivative Derivative is 18 t 2 9 2 t 1 10 9 t 2 8 2 t 1 9 http localhost 8888 notebooks Downloads Lab5temp 151 22C ipynb 3g Page 2 of 5 2a 2b 3a 3b Lab5temp 151 22C Jupyter Notebook 10 4 22 1 30 PM In 18 print Derivative simplified is derivative simplify Derivative simplified is 45 t 2 8 2 t 1 10 In 21 function derivative simplify values solve function t print Horizontal tangent line is at the point t values 0 Horizontal tangent line is at the point t 2 In 22 t symbols t f 2 t 1 5 t 2 t 2 4 derivative diff f t print Derivative is derivative Derivative is 2 t 1 5 8 t 4 t 2 t 2 3 10 2 t 1 4 t 2 t 2 4 In 33 print Derivative simplified is derivative simplify print The derivative is expanded and factored Derivative simplified is 2 t 1 4 t 2 t 2 3 10 t 2 10 t 4 2 t 1 2 t 1 20 The derivative is expanded and factored 3c 3d 3e 3f 3g http localhost 8888 notebooks Downloads Lab5temp 151 22C ipynb 3g Page 3 of 5 In 34 print Derivative factored is derivative factor print This is more simplified than the simplified deirvative powers and multiplications are simplified grouped Derivative factored is 2 2 t 1 4 t 2 t 2 3 13 t 2 5 t 8 This is more simplified than the simplified deirvative powers and multiplications are simplified grouped Lab5temp 151 22C Jupyter Notebook 10 4 22 1 30 PM In 29 function derivative factor values solve function t print Factored is more useful because you can identify using the brackets where the derivative function would equal 0 The powers are cancelled and one bracket remians 4 which is equated to 0 and solved Therefore the factored form would be quicker to solve print Horizontal tangent line is at the point t values 0 Factored is more useful because you can identify using the bracke ts where the derivative function would equal 0 The powers are ca ncelled and one bracket remians 4 which is equated to 0 and sol ved Therefore the factored form would be quicker to solve Horizontal tangent line is at the point t 1 2 Question 4 In 36 W symbols W symbols symbols F W sin cos derivative diff F print Rate of change with respet to is derivative Rate of change with respet to is W cos sin sin cos 2 In 42 value solve derivative print value 0 2 atan sqrt 2 1 1 4a 4b 4c http localhost 8888 notebooks Downloads Lab5temp 151 22C ipynb 3g Page 4 of 5 Lab5temp 151 22C Jupyter Notebook 10 4 22 1 30 PM In 61 F 60 0 6 sin cos plot F pi pi ylim 0 70 print Graph of F with vertical aysmptotes print Estimate of when dF d 0 is 0 5 60 50 Graph of F with vertical aysmptotes Estimate of when dF d 0 is 0 5 4d In 60 derivative diff F value solve derivative print Value of when dF d 0 is value 0 Value of when dF d 0 is 0 540419500270584 http localhost 8888 notebooks Downloads Lab5temp 151 22C ipynb 3g Page 5 of 5
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