Thursday, August 13, 2015

Analysis Quiz 13: Brain Power Enrichment Quiz (II)

Quiz 13: Brain Power Enrichment Quiz (II)

The diagram below represents the graph y=x and the coordinate points (2, 2), (4, 4) and (6, 6).

Use the diagram below to answer question 1.

Question 1: What is the relation between the points (2, 2), (4, 4) and (6, 6)?

A. (4, 4) is the standard deviation of (2, 2) and (6, 6).

B. (4, 4) is the average of (2, 2) and (6, 6).

C. (4, 4) is the distance between (2, 2) and (6, 6).


Apparently 4, 4) is the midpoint of (2, 2) and (6, 6). The midpoint between two points is also the average of those two point.

Therefore B is the answer.

The diagram below represents the graph y=ln(x) and the coordinate points (2, ln(2)), (4, ln(4)) and (6, ln(6)).

Use the diagram below to answer question 2.

Question 2: Which of the following is correct?

A.The gradient of the curve at (2, ln(2)) is less than the gradient of the curve at (6, ln(6)).

B. Distance between (2, ln(2)) and (4, ln(4))=Distance between (4, ln(4)) and (6, ln(6))

C.The arch length of the curve between (2, ln(2)) and (4, ln(4)) is the same as the arc length between (4, ln(4)) and (6, ln(6)).

D. None of the above.


If you put a ruler that act as the tangent line to the curve of $y=\ln x$, you would notice the tangent line at  (2, ln(2)) is steeper than the tangent line at  (6, ln(6)), therefore A is a wrong observation.

The distance formulas tell us:

Distance between $(2,\,\ln(2))$ and $(4,\,\ln(4))=\sqrt{4+\ln^2(2)}\approx 2.11671$

Distance between $(4,\,\ln(4))$ and $(6,\,\ln(6))=\sqrt{4+\ln^2(1.5)}\approx 2.04069$

Therefore, B is also a wrong information.

If you're familiar with how the graph of $y=\ln x$ behaves (which you really should), you can tell off the top of your head that its graph slowly grows to positive infinity as $x$ increases. This suggests the arc length of the curve between $(2,\,\ln(2))$ and $(4,\,\ln(4))$ is larger than the arc length between $(4,\,\ln(4))$ and $(6,\,\ln(6))$. Thus, C is out of the question to be the correct answer.

We can conclude by now that none of the above statement is correct. Therefore D is our choice.

Question 3: Do you think there is any difference in value between the following pair of numbers?

$\dfrac{\ln 2+\ln 6}{2}$ versus $\ln\left(\dfrac{2+6}{2}\right)$


A. Yes.

B. No.


From the graph of $y=\ln x$, we can see that the spot $\left(\dfrac{2+6}{2},\,\ln\left(\dfrac{2+6}{2}\right)\right)$ on the curve $y=\ln x$ lies above the point $\left(\dfrac{2+6}{2},\,\dfrac{\ln 2+\ln 6}{2}\right)$, this literally means $\ln\left(\dfrac{2+6}{2}\right)\gt \dfrac{\ln 2+\ln 6}{2}$.

Therefore the answer for question $3$ is A.

Question 4: Why is there the difference between $\dfrac{\ln 2+\ln 6}{2}$ versus $\ln\left(\dfrac{2+6}{2}\right)$?

A. I'm certain they are the same.

B. Because the graph of y=ln(x) is a strictly increasing function.

C. Because the graph of y=ln(x) is an increasing function.

D. Because x grows faster than ln(x) as x → +∞.


A is certainly not the correct answer because those two values are not the same.

Now, to pick the real answer for this question, we need to be very clear with the definition of strictly increasing and increasing function.

A function $f(x)$ is said to be strictly increasing on an interval $I$ if $f(b)>f(a)$ for all $b\gt a$, where a,b in I. On the other hand, if $f(b)\ge f(a)$ for all $b\gt a$, the function is said to be (non-strictly) increasing.

But upon contemplating for more, we realized that the graph of y=x (as shown in picture 1 above) that y=x is also a strictly increasing function, and yet, $\dfrac{2+6}{2}=\dfrac{f(2)+f(6)}{2}=4$.

Therefore, it is the last option that says "$x$ grows faster than $\ln(x)$ as $x \rightarrow \infty$" is the correct answer.

Question 5: What theorem explains the phenomenon described in question 4?

A. AM-GM inequality.

B. Markov's inequality.

C. Cauchy–Schwarz inequality.

D. Jensen's inequality.


Jensen's inequality is the answer as it states if the graph of $f$ is concave, like $y=\ln x$, then we have:

[MATH]\ln\left(\dfrac{1}{n}\sum_{i=1}^{n} x_i\right)\ge \dfrac{1}{n}\left(\sum_{i=1}^{n} \ln (x_i)\right)[/MATH]

Question 6: Do you think the questions above have prepared you well to prove $\sqrt{1000}+\sqrt{999}-\sqrt{998}>\sqrt{998}-\sqrt{997}+\sqrt{996}$.

A. Of course.

B. No.


We don't know the answer unless we tried, by following the hints that we have gotten after answering the previous questions.

Note that the graph of $y=\sqrt{x}$ behaves just like $y=\ln x$ where $y$ grows slowly as $x$ increases, thus we have:

$\sqrt{\dfrac{996+998}{2}}\gt \dfrac{\sqrt{996}+\sqrt{998}}{2}$

$\sqrt{997}\gt \dfrac{\sqrt{996}+\sqrt{998}}{2}$

$2\sqrt{997}\gt \sqrt{996}+\sqrt{998}$

$\sqrt{997}\gt \sqrt{996}-\sqrt{997}+\sqrt{998}$(*)

On the other hand, it's so straightforward to notice that:

$\sqrt{1000}+\sqrt{999}\gt \sqrt{998}+\sqrt{997}$

Rearrange it so we get:

$\sqrt{1000}+\sqrt{999}-\sqrt{998}\gt \sqrt{997}$(**)

Now, combining both inequalities we got from (*) and (**), we have easily proved that:


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