Lab3temp_151_24A

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 1/14 MATH 151 Lab 3 Put team members' names and section number here. Question 1 1a In [128… import sympy as sp from sympy.plotting import ( plot , plot_parametric ) In [ ]: Section 404 - Group 3 Lisa Pham , Coby Holley , Bradley White , Owen Koss , Redlin Krueger , Garret Worden In [2]: matplotlib notebook

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 2/14 In [11]: # Start your code here #x = symbols('x') import numpy as np import matplotlib.pyplot as plt from sympy import limit , Symbol , sqrt , sin , pi # function definition def f ( x ): return np . sqrt ( x ** 3 + x ** 2 ) * np . sin ( np . pi / x ) + 2 def g ( x ): return np . full_like ( x , - 2 ) def h ( x ): return np . full_like ( x , 2 ) # domain x = np . linspace ( - 0.8 , 0.8 , 100 ) # x values f_values = f ( x ) g_values = g ( x ) h_values = h ( x ) # plot plt . plot ( x , f_values , label = 'f(x)' ) plt . plot ( x , g_values , label = 'g(x)' ) plt . plot ( x , h_values , label = 'h(x)' )

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 4/14 1b

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 5/14 lim_{x->0} g(x) = -2 lim_{x->0} h(x) = 2 lim_{x->0} f(x) = -2 1c In [15]: # limits lim_g = g ( 0 ) lim_h = h ( 0 ) lim_f = lim_g print ( "lim_{x->0} g(x) =" , lim_g ) print ( "lim_{x->0} h(x) =" , lim_h ) print ( "lim_{x->0} f(x) =" , lim_f ) x_sym = Symbol ( 'x' ) f_expr = sqrt ( x_sym ** 3 + x_sym ** 2 ) * sin ( pi / x_sym ) + 2 In [16]: # limits lim_g = g ( 0 ) lim_h = h ( 0 ) lim_f = lim_g print ( "lim_{x->0} g(x) =" , lim_g ) print ( "lim_{x->0} h(x) =" , lim_h ) print ( "lim_{x->0} f(x) =" , lim_f ) x_sym = Symbol ( 'x' ) f_expr = sqrt ( x_sym ** 3 + x_sym ** 2 ) * sin ( pi / x_sym ) + 2

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 7/14 print ( "The left sided limit when x is approaching 1 is: " , left_limit_of right_limit_of_1 = sp . limit ( f ( 1 + 0.0001 ), x , 1 , '+' ) print ( "The right sided limit when x is approaching 1 is: " , right_limit_ print ( "As the limit approaches 1 for f(x) from the right and the left doe print () #Limit of 3 left_limit_of_3 = sp . limit ( f ( 3 ), x , 3 , '-' ) print ( "The left sided limit when x is approaching 3 is: " , left_limit_of_ right_limit_of_3 = sp . limit ( f ( 3 ), x , 3 , '+' ) print ( "The right sided limit when x is approaching 3 is: " , right_limit_o print ( "As the limit approaches 3 for f(x) from the right and the left doe print () #Limit of 5 left_limit_of_5 = sp . limit ( f ( 5 ), x , 5 , '-' ) print ( "The left sided limit when x is approaching 5 is: " , left_limit_of_ right_limit_of_5 = sp . limit ( f ( 5 ), x , 5 , '+' ) print ( "The right sided limit when x is approaching 5 is: " , right_limit_o print ( "As the limit approaches 5 for f(x) from the right and the left doe

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 8/14 The left sided limit when x is approaching 1 is: -2 The right sided limit when x is approaching 1 is: -2.99980000000000 As the limit approaches 1 for f(x) from the right and the left does not e qual each other f(x) is not continous. The left sided limit when x is approaching 3 is: 1 The right sided limit when x is approaching 3 is: 1 As the limit approaches 3 for f(x) from the right and the left does equal each other f(x) is continous. The left sided limit when x is approaching 5 is: 7.00000000000000 The right sided limit when x is approaching 5 is: 7.00000000000000 As the limit approaches 5 for f(x) from the right and the left does equal each other f(x) is continous. 2d In [4]: matplotlib notebook In [127… import numpy as np import matplotlib.pyplot as plt import sympy as sp def f ( x ): if x <= 1 : return - 2 elif 1 < x <= 3 : return 2 * x - 5

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 10/14 Question 3 3a

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 11/14 1.0000063497741163 3b In [8]: import math n = 99999 sum ([ math . log ( i + 1 ) for i in range ( n )]) / ( n * math . log ( n ) - n ) #proves that any (non integer) number with stirling's formula will approxi Out[8]: In [126… import numpy as np import matplotlib.pyplot as plt from sympy import factorial , sqrt , pi # Define the factorial function f(x) def f ( x ): return factorial ( x ) # Define Stirling's approximation function g(x) def g ( x ): return sqrt ( 2 * pi * x ) * ( x / np . e ) ** x # Generate x values x_values = np . linspace ( 0.1 , 5 , 100 ) # Compute y values for both functions f_values = [ f ( xi ) for xi in x_values ] g_values = [ g ( xi ) for xi in x_values ]

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 13/14 In [ ]:

2/21/24, 3:18 PM Lab3temp_151_24A localhost:8888/nbconvert/html/Lab3temp_151_24A.ipynb?download=false 14/14 In [ ]: