MATH 1MP3 * TEST 2

1. Multiple choice questions: circle ONE answer. No justification is needed.

(a) [2] For which of the following does Python return True when the command is executed.

(I) print({1, 2, 3} == {3, 3, 2, 3, 1})

(II) print(1 <= {1, 2, 3})

(III) print({3, 2} <= {1, 2, 3})

(A) none

(B) I only

(C) II only

(D) III only

(E) I and II

(F) I and III

(G) II and III

(H) all three

(b) [2] What is the output of the following Python code?

p = [1, 2, 3]
a = 4

def eval(p, a):
    result = 0
    for i in range(0, len(p)):
        result += p[i] * a ** i
    return result

print(eval(p, a))

(A) 49

(B) 57

(C) 24

(D) 33

(E) 47

(F) 53

(G) 29

(H) none of these

2. The geometric series $\sum_{i=0}^{\infty} ar^i = a + ar + ar^2 + ...$ converges to $s = \frac{a}{1 - r}$ when $|r| < 1$.

(a) [2] Write a code that computes the partial sum of the geometric series when $a = 1$ and $r = \frac{1}{4}$, stopping when the number of terms reaches max_terms (i.e., if max_terms=15, then you would compute the sum of the first 15 terms in the series).

(b) [1] Determine the exact sum of the series in part (a).

(c) [2] Write a code to compute the partial sum of the series in part (a), stopping when either max_terms is reached or the difference between the partial sum and the theoretical sum from part (b) is less than 0.001.

3. Look at the following code:

import numpy as np

def f(x):
    return np.log(x) + x

a = 0.1
b = 0.9

while abs(b - a) >= 0.1:
    mid = (a + b) / 2
    if f(mid) == 0:
        print('solution is', mid)
    if f(a) * f(mid) = 0:
        b = mid
    if f(mid) * f(b) = 0:
        a = mid

print(a, b)

(a) [2] How many times will the while loop be executed? Support your answer with appro- priate calculations.

(b) [2] What does the output in line 17 represent in the context of the task this code is trying to accomplish?

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