write test cases for a function that computes the left Riemann's sum

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```python # define a function to compute the left Riemann sum def left_riemann_sum(f, a, b, n): """ Computes the left Riemann sum for the function f over the interval [a, b] with n subintervals. Parameters: f (function): The function to integrate. a (float): The start of the interval. b (float): The end of the interval. n (int): The number of subintervals. Returns: float: The left Riemann sum. """ # calculate the width of each subinterval dx = (b - a) / n # initialize the sum to zero sum = 0.0 # loop over all subintervals for i in range(n): # compute the x-value at the left end of the subinterval x = a + i * dx # add the area of the rectangle to the sum sum += f(x) * dx # return the computed sum return sum ``` ```python # main function that tests left_riemann_sum def test_left_riemann_sum(): # define a simple function for testing: f(x) = x f_test = lambda x: x # test the left_riemann_sum function with a linear function f(x) = x over the interval [0, 1] with 1 subinterval # the exact answer should be 0.0 because the area of the rectangle is 0.0 * 1 = 0.0 assert left_riemann_sum(f_test, 0, 1, 1) == 0.0 # test the left_riemann_sum function with the same linear function over the interval [0, 1] with 2 subintervals # the exact answer should be 0.25 because the areas of the rectangles are (0.0 * 0.5) + (0.5 * 0.5) = 0.25 assert left_riemann_sum(f_test, 0, 1, 2) == 0.25 ``` ∻Explanation∻ ⚹The left_riemann_sum function calculates the left Riemann sum by summing the areas of rectangles under the curve of the function f, using the left endpoints of subintervals. The test_left_riemann_sum function defines test cases to verify the correctness of the left_riemann_sum function.⚹ ∻Step-by-step Instruction∻ ‖ ▪ ⋮ Define the left_riemann_sum function that takes a function f, interval [a, b], and number of subintervals n as arguments and computes the left Riemann sum. ‖ ‖ ▪ ⋮ Implement the computation by dividing the interval into n subintervals, calculating the width of each subinterval, and summing the areas of the rectangles. ‖ ‖ ▪ ⋮ Define the test_left_riemann_sum function to test the left_riemann_sum function with specific use cases. ‖ ‖ ▪ ⋮ Use a simple linear function f(x) = x for testing because its integral can be easily calculated manually, providing clear expected results for comparison. ‖ ‖ ▪ ⋮ Write assertions to compare the computed left Riemann sum with the expected results for the given test cases. ‖ ∻Time Complexity∻ ⚹The time complexity of the left_riemann_sum function is O(n) because it involves a loop that iterates n times to sum the areas of n rectangles.⚹ ∻Space Complexity∻ ⚹The space complexity of the left_riemann_sum function is O(1) because it uses a fixed amount of space; it does not allocate any additional space that grows with the input size n.⚹

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