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MATH1003 Quizzes

Quiz 5: Properties of Logs and Exponentials
Question 1 Questions
Which of the following are defined for all real values of x ?
(1)log10ex(2)(x2 + 1)x(3)(x2 1)x
Exactly one option must be correct)
a)
(1) and (2) and (3)
b)
(2) and (3) only.
c)
(1) and (2) only.
d)
(1) and (3) only.
e)
None of them

Choice (a) is incorrect
Remember that ab = eb ln a and so ab makes sense only when a > 0.
Choice (b) is incorrect
Remember that ab = eb ln a and so ab makes sense only when a > 0.
Choice (c) is correct!
Choice (d) is incorrect
Remember that ab = eb ln a and so ab makes sense only when a > 0.
Choice (e) is incorrect
Which option is a simplified version of the expression ln(5e2x) + eln(5x) ? Exactly one option must be correct)
a)
ln10 + lnx + 5x, for all x > 0
b)
ln5 + 10x, for all x
c)
ln2 + lnx + ln5, for all x > 0
d)
7x + ln5, for all x
e)
None of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
The expression only makes sense when x > 0 and so the answer is 7x + ln5, for all x > 0
If y = ln(e2x + 4), what is x in terms of y ? Exactly one option must be correct)
a)
x = 1 2ln(e2y 4)
b)
x = (ln(e2y 4))2
c)
x = 1 2(2y ln4)
d)
x = 1 4ln(ey 4)
e)
x = 1 2ln(e2y 2)

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
If y = 3xex , select the option which equals dy dx. Exactly one option must be correct)
a)
3xex (x + 1)ex ln3
b)
(xex + ex)ln3
c)
xex3xex1
d)
xex3x+ex2
e)
None of the above

Choice (a) is correct!
Choice (b) is incorrect
Try writing y as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Choice (c) is incorrect
Try writing y as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Choice (d) is incorrect
Try writing y as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Choice (e) is incorrect
Use logarithmic differentiation to find dy dx when y = x3 + x4(ln(sinx))2 1 + sinx Exactly one option must be correct)
a)
y 3x2 + 4x3 2(x3 + x4) + 2cotx cosx 1 + sinx
b)
y 3x2 + 4x3 (x3 + x4) + cotx lnsinx cosx 1 + sinx
c)
y 3x2 + 4x3 2(x3 + x4) + 2cotx lnsinx 1 1 + sinx
d)
y 3x2 + 4x3 2(x3 + x4) + 2cotx lnsinx cosx 1 + sinx
e)
None of the above

Choice (a) is incorrect
Taking natural logs of both sides of the equation for y gives
lny = 1 2ln(x3 + x4) + 2ln(lnsinx)) ln(1 + sinx).
Now differentiate both sides with respect to x.
Choice (b) is incorrect
Taking natural logs of both sides of the equation for y gives
lny = 1 2ln(x3 + x4) + 2ln(lnsinx)) ln(1 + sinx).
Now differentiate both sides with respect to x.
Choice (c) is incorrect
Taking natural logs of both sides of the equation for y gives
lny = 1 2ln(x3 + x4) + 2ln(lnsinx)) ln(1 + sinx).
Now differentiate both sides with respect to x.
Choice (d) is correct!
Choice (e) is incorrect
Which values of t satisfy simultaneously the pair of inequalities ln|t| < 1 and t2 > 1 ? Exactly one option must be correct)
a)
All t such that 1 < t < e.
b)
All t in the intervals (e,1) or (1,e).
c)
All t such that 1 e < t < e.
d)
All t 1.
e)
All t such that e < t < 1 e or 1 e < t < e

Choice (a) is incorrect
Remember that the solution set of t2 > 1 will contain negative as well as positive numbers!
Choice (b) is correct!
Choice (c) is incorrect
Remember that if ln|t| < 1, then 1 < ln|t| < 1.
Choice (d) is incorrect
Remember that if ln|t| < 1, then 1 < ln|t| < 1 and that the solution set of t2 > 1 will contain negative as well as positive numbers!
Choice (e) is incorrect
Remember that if ln|t| < 1, then 1 < ln|t| < 1 and that the solution set of t2 > 1 will contain negative as well as positive numbers!
If y = 3x 22x and z = 32x + 2x, what is yz ? Exactly one option must be correct)
a)
3x + 2x 62x + 6x
b)
3x 2x + 6x
c)
62x 64x
d)
3x 2x + 6x
e)
None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
In fact, yz = 3x 2x + 6x(1 63x).
Solve the equation ln(ex 1) = 0 for x and enter your answer correct to two decimal places.

Correct!
Taking exponentials gives ex 1 = 1, after which we obtain ex = 2, and then x = ln4.
Incorrect. Please try again.
Have you taken exponentials of each side of the equation? This gives ex 1 = 1, from which you can obtain the unique solution for x after further manipulations.
Two of the following have identical derivatives. Tick the pair that do. (Zero or more options can be correct)
a)
6x x2 + cos2x + 2sec2x
b)
1 2sinxcosx 2(x 3) + 12
c)
1 2sin2x + 2tan2x (x 3)2
d)
cosxsinx + sec2x x2
e)
2sec2x + 3 + sinxcosx + 6x x2

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be False.
There is at least one mistake.
For example, choice (e) should be True.
Correct!
  1. False
  2. False
  3. True
  4. False
  5. True
The equation 2xy + y3x = 1 defines y implicitly as a function of x near the point with coordinates (0,1). What is the value of dy dx at this point? Exactly one option must be correct)
a)
2ln2
b)
1 ln2
c)
1 + 2ln2 2
d)
1 ln2
e)
ln2 2

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect