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MATH1013 Quizzes

Quiz 1: The Bisection method
Question 1 Questions
Solve the equation 3x2 = 5x + 2 to 2 decimal places.
a)
x = 1.00, x = 0.67
 b)
x = 0.33, x = 2.00
c)
x = 1.00, x = 0.67
 d)
x = 0.06, x = 1.73

Choice (a) is incorrect
Choice (b) is correct!
We need to solve 3x2 5x 2 = (3x + 1)(x 2) = 0.
Choice (c) is incorrect
Choice (d) is incorrect
Solve the equation 2x2 = 4x + 1 to 2 decimal places.
a)
x = 3.22, x = 0.78
 b)
x = 1.71, x = 0.29
c)
x = 2.22, x = 0.22
 d)
x = 2.22, x = 0.22

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
We need to solve 2x2 + 4x 1 = 0 and using the quadratic formula we find x = 4 ±16 + 8 4 = 4 ±24 4 therefore, x = 1 + 6 2 orx = 1 6 2 therefore, x = 0.2247,x = 2.2247.
Choice (d) is incorrect
Consider the two graphs sketched below.
 
PIC
Which of the equations could they be used to solve ?
a)
e2x = 2x + 3
 b)
ex = 2x 3
c)
e2x + 3 = 2x + 3
 d)
e2x = 2x + 3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The two graphs are y = e2x and y = 2x + 3. Their points of intersection (dotted in the sketch) are the solutions of the equation e2x = 2x + 3.
Consider the two graphs sketched below.
PIC
Which of the following equations could they be used to solve ?
a)
sin2x = x3 3x + 2
 b)
sin2x + x3 3x + 2 = 0
c)
sinx = x3 3x + 2 2
 d)
sinx = x2 3x + 2

Choice (a) is correct!
The two graphs are y = sin2x and y = x3 3x + 2. Their intersection solves the equation sin2x = x3 3x + 2.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Which of the following functions has roots which are solutions of the equation e2x+5 = x3 5x + 6 ?
a)
f(x) = 2x + 5 lnx3 + ln5x ln6
 b)
f(x) = e2x+5 x3 + 5x 6
c)
f(x) = e2x+5 x3 + 5x + 6
 d)
f(x) = 2x + 5 + ln(x3 + 5x 6)

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
How many solutions are there of the equation sin(2x 5) 2 = (3x 4)2 in ?
a)
None
 b)
One
c)
Two
 d)
Infinitely many

Choice (a) is correct!
sin(2x 5) 2 1 but (3x 4)2 0, for all x hence there are no solutions.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
How many solutions are there to the equation ln(x + 1) = x2 2x 3 in ?
a)
None
 b)
One
c)
Two
 d)
Infinitely many

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The easiest way to see this is by graphing y = ln(x + 1) and y = x2 2x 8 on the same set of axes.
PIC
We see that there are only two possible solutions, one near x = 1 and one near x = 4.3.
Choice (d) is incorrect
How many solutions are there to the equation x3 + x + 1 = cosx in ?
a)
None
 b)
One
c)
Two
 d)
Infinitely many

Choice (a) is incorrect
Choice (b) is correct!
Consider f(x) = x3 + x + 1 cosx. f(0) = 0 so x = 0 is solution. Now f(x) = 3x2 + sinx + 1 > 0 for all x , since 1 sinx 1 and 3x2 + 1 1. Therefore the function is always increasing and so there can be only the one root, at x = 0.
Choice (c) is incorrect
Choice (d) is incorrect
Use the bisection method three times on the function f(x) = x2 sinx 1 to determine where f(x) changes sign on the interval 2 < x < 0.
a)
f(x) changes sign on the interval 0.75 x 0.5
 b)
f(x) changes sign on the interval 0.25 x 0
c)
f(x) changes sign on the interval 1 x 0.75
 d)
We cannot use this method as f(x) does not change sign on this interval.

Choice (a) is correct!
The first iteration tells us that the change of sign is between -1 and 0. The second iteration tells us that the change of sign is between -1 and -0.5. The third iteration tells us that the change of sign occurs between -0.75 and -0.5.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Use the dissection method on f(x) = x3 cosx + 1 to determine where the function changes sign on the interval 1.1 < x < 0.1.
a)
f(x) changes sign on the interval 0.2 x 0.1
 b)
f(x) changes sign on the interval 0.9 x 0.8
c)
f(x) changes sign on the interval 0.4 x 0.3
 d)
f(x) changes sign on the interval 0.5 x 0.4

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
This is the correct response since f(0.5) < 0 and f(0.4) > 0.