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Problem 4 (7 points) Consider the integral (a) (2 points) Find an upper bound $|f''(x)| \leq K_2$ for $x \in [1, \infty)$ where $f(x) = e^x/x$. Save the value as K2. In []: # Use the variable name K2 # YOUR SOLUTION GOES HERE In []: assert sha1(str(K2).encode('utf8')).hexdigest() == '042bb56f0af253c4571e3218a4cbbcca22176a75', "That is not the expected value." print("Problem 4 Test 1: Success!") (b) (5 points) Write a function called Ei which takes input parameters b and max_error and returns the approximation of the integral given by the trapezoid method with N subintervals which guarantees the error is less than max_error. In []: #YOUR SOLUTION GOES HERE def Ei(b,max_error): In []: "Check that Ei returns the correct type." assert type(Ei(2,0.001)) == np.float64 print("Problem 4 Test 2: Success!") In []: "Check that Ei returns the correct values." assert Ei(1,0.01) == 0 print("Problem 4 Test 3: Success!") In []: "Check that Ei returns the correct values." epsilon = 10e-14 assert abs(Ei(5,0.001) - 0.21833350347126337) < epsilon

          Problem 4 (7 points)
Consider the integral
(a) (2 points) Find an upper bound $|f''(x)| \leq K_2$ for $x \in [1, \infty)$ where $f(x) = e^x/x$. Save the value as K2.
In []: # Use the variable name K2
# YOUR SOLUTION GOES HERE
In []: assert sha1(str(K2).encode('utf8')).hexdigest() == '042bb56f0af253c4571e3218a4cbbcca22176a75', "That is not the expected value."
print("Problem 4 Test 1: Success!")
(b) (5 points) Write a function called Ei which takes input parameters b and max_error and returns the approximation of the integral
given by the trapezoid method with N subintervals which guarantees the error is less than max_error.
In []: #YOUR SOLUTION GOES HERE
def Ei(b,max_error):
In []: "Check that Ei returns the correct type."
assert type(Ei(2,0.001)) == np.float64
print("Problem 4 Test 2: Success!")
In []: "Check that Ei returns the correct values."
assert Ei(1,0.01) == 0
print("Problem 4 Test 3: Success!")
In []: "Check that Ei returns the correct values."
epsilon = 10e-14
assert abs(Ei(5,0.001) - 0.21833350347126337) < epsilon

Problem 4 (7 points)
Consider the integral
(a) (2 points) Find an upper bound |f”(x)| ≤ K2 for x ∈ [1, ∞) where f(x) = e^x/x. Save the value as K2.
In []: # Use the variable name K2
# YOUR SOLUTION GOES HERE
In []: assert sha1(str(K2).encode('utf8')).hexdigest() == '042bb56f0af253c4571e3218a4cbbcca22176a75', "That is not the expected value."
print("Problem 4 Test 1: Success!")
(b) (5 points) Write a function called Ei which takes input parameters b and maxerror and returns the approximation of the integral
given by the trapezoid method with N subintervals which guarantees the error is less than maxerror.
In []: #YOUR SOLUTION GOES HERE
def Ei(b,maxerror):
In []: "Check that Ei returns the correct type."
assert type(Ei(2,0.001)) == np.float64
print("Problem 4 Test 2: Success!")
In []: "Check that Ei returns the correct values."
assert Ei(1,0.01) == 0
print("Problem 4 Test 3: Success!")
In []: "Check that Ei returns the correct values."
epsilon = 10e-14
assert abs(Ei(5,0.001) - 0.21833350347126337) < epsilon

Added by Autumn F.

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