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#1 Re: Help Me ! » [ASK] Proof of Some Quadratic Functions » 2019-04-08 08:30:41

Thank you Alg Num Theory, The corrected last statement can be proved as follows:

which completes the proofs. The link which Alg Num Theory has provided contains another correct proof of the theorem, if you would like.

#2 Re: Help Me ! » [ASK] Proof of Some Quadratic Functions » 2019-04-07 21:20:37

Hi Monox D. I-Fly, I hope you recover soon. The proofs are very simple just use the quadratic formula as follows:


which proves (1)

which proves (2)

which proves (3)

which proves (4)

which proves (5). Finally we have:

but we have:

Therefore the last statement is correct only when a=b=c=1, otherwise there are something wrong with its proclaims. I hope that answers your question.

#3 Re: Help Me ! » Interpreting Graphs » 2019-04-04 20:52:20

To err on the caution side, I think the bob bundy suggestion is the more rational one, since the rate of change measure the amount of variance in a quantity therefore there are no plausible reason to distinguish between the negative and positive rate of changes if they have the same absolute value and hence it is more suitable that the answer of 4 to be fixed to “B and at the starting of D”. I will amend the answer above.

#4 Re: Help Me ! » Interpreting Graphs » 2019-04-04 19:59:33

Hi bob bundy, I think the greatest is B since it is possitive and any possitive value is geater than the negtive value. In fact, the geatest rate of change is in the B part of the curve and the least rate of change is at the start of section D. On the other hand, regardless of the previous argument there is a logical reason to interpret the answer in this manner since otherwise the stituation will be inconclusive which enforce that interpretation.

#5 Re: Help Me ! » Interpreting Graphs » 2019-04-04 19:39:53

I have to have an issue with you ganesh. All the answers are incorrect except the first one and partly the answer of question three. This is so because the rate of change of the temperature at a particular point on the temporal curve of the temperature is the slop of the curve at that point. With this fact on our mind, since the slop of the curve at C is zero the first answer is correct. However, since all the curve parts are line segments (which have constant slops) except the D part which is a continuous curve and hence with changing slop, then the correct answer for question 2 is D. Now since the slop of the curve in the D and C parts are negative and zero, respectively, while it is positive in the A and B parts then the answer of the third question is A and B. Finally, since the slop of the curve is negative in D and Zero in C while the slop of B greater than that of A and since the absolute value of the rate of change at the starting point of D is equal to the slop in B then the correct answer of Four is B and the starting point of D. Therefore in summary the answer will be as follows:

1- C
2- D
3- A and B
4- B and at the starting of segment D

I hope that will help. By the way ganesh your answer for question five is correct.

#6 Re: Help Me ! » Just asking a simple question » 2019-04-03 07:07:21

Hi monie27, you should make more practice with the nomal distribution problem since this question is a standatd question about this distribution. However, there are to ways to solve this question the first one by calculating the probabilities directly from the probability density formula, but the second and more common way to calculate the required probabilities is by using the normal distribution tables. I will solve the problem by the second method as an illustration as below:
You first have to deduce the standard deviation which is the squre root of the variance, so from the given infronation its value is 10 ml. Second you have to measure the filling amount of coffee in terms of the standard deviation after putting the normal distribution in the standard form (i.e. putting the average of the filling amount of coffee equals to zero). Therefore, applying the previous procedures, the required answers are:

a- (235-230)/10=0.5 ⇒ 1-0.6915=0.3085
b- (245-230)/10=1.5 ⇒ 0.9332-0.6915=0.2417
c- 350×0.6915=242.025 cups

Good luck

#7 Re: Help Me ! » helpppp » 2019-03-13 07:15:37

Hi monie27, the answer is as follows: Let's the probabilities of arriving and departing on time are B and A, respectively.  Then for the conditional probability, it can be proven that we have:


but we have

however since we have:

finally we have:

Therefore the answers are:
1-0.08
2-0.78
3-0.85
4-0.91
I hope that will help.

#8 Re: Help Me ! » Geometry Problem » 2019-03-03 10:22:01

If both circles are in the opposite sides of the line l then the distance OP will be the addition of both radii, but if both circles are on the same side of that line then the distance will be the difference between them hence the possible leangths are 8 and 2 respectively.

#9 Re: Help Me ! » [ASK] Exponents and Roots » 2019-02-27 20:46:14

Hi phrontister, you are right I forgot to put the parentheses about the numerator when I performed the calculation (your answer is the correct one). Thank you to mention that (I have changed the number)

#10 Re: Help Me ! » [ASK] Exponents and Roots » 2019-02-27 04:15:09

You did not get the correct answer because non of them is, in fact, a correct answer!! This can be demonstrated by substitution as follows:


however we have

which is the required proof. Sorry but this is the true situation!!

#11 Re: Help Me ! » I need help with a motion in two dimensions problem. » 2019-02-20 08:41:28

If you would like, the following is also another solution. In the velocity triangle described in the second solution if you drop a vertical line on the resultant (the velocity of the boat with respect to Earth) and labeled the obtained two sections of the resultant and the perpendicular line segment

and
respectively. you would get:

but you would get also

so from equation (2) it follows that

and hence substituting into equation (1) you would have:

which are also the required results.

#12 Re: Help Me ! » I need help with a motion in two dimensions problem. » 2019-02-20 07:03:18

If you are interested, as an alternative solution you can draw the velocity triangle such that the river flows toward the positive x axis (the east direction) and the south direction to the negative y axis. In this case if you drop a vertical line from the head of the resultant vector (the velocity of the boat relative to Earth) on the x axis and take the reflection of that triangle about the x axis, you will have an identical triangle in which the Cartesian coordinates of the resultant is:


from which the magnitude of this velocity (the speed) is given by:

however, from the law of cosines we have:

Solving equations (1) and (2) simultaneously would give the required solutions which are identical to those found by the previous mothed.

#13 Re: Help Me ! » I need help with a motion in two dimensions problem. » 2019-02-20 04:30:56

The problem is simple, but you have made a mistake in assuming that the velocity triangle is a right triangle while in the question the instructor emphasizes that the boat is heading 60 degree south of east, so the triangle is in fact a 45-120-15 triangle and to solve the problem you can use the law of sines (I think you know what that means since you have reached this point in your course). Therefore the solution goes as follows (I will utilize the notation which you have used):


and for the velocity of the boat with respect to the river we would have:

which are exactly the answers given in the book. Good luck!

#14 Re: Help Me ! » Inequality and arbitrary largeness of numbers » 2019-02-13 02:00:33

If they are complex then they can be negative or positive numbers. However, in your problem they assume that the numbers are integers and any integer is a real number.

#15 Re: Help Me ! » Really easy inequality question » 2019-02-13 01:34:24

Hi Dr. Francis Hung, this can be proven easily by observing that a,b and c are greater than or equal to zero since the square root of a negative number is not a real number and the inequality emphasizes that a,b and c are in fact real numbers. However, since the three terms in the left side of the given inequality are identical it is sufficient to prove that each term is less than or equal to 1 hence the prove goes as follows:


remember that

#16 Re: Help Me ! » Inequality and arbitrary largeness of numbers » 2019-02-08 07:51:27

Dear Amartyanil, yes it is greater than or equal to zero. Any sum of square numbers is greater than or equal to zero.

#17 Re: Help Me ! » helpppp » 2019-02-07 15:12:02

Hi Derek, it is straightforward for the final person to deduce the correct color of her hat. This can be proven as follows: For clarity let me call "the first person" from back (Dom) the first person and the next one "the second person" and so no. Since there are just three black hats so if the second, third and forth persons wearing them then the first person would be certain that his hat is white (remember that there are just three black hats and they know this fact). Therefore some of the hats on the other persons are black and some are while or all of them are white. The second person know the previous conditions from the answer of the first person, but if the two hats of the third and forth persons are black then he would be sure (from the information he had got from the first person and from his observation) that his hat is white, However, his uncertainty insured the third person and the forth person that not all of their hats are back. At this stage the third person would be sure that his hat most be white if the front person has black hat (since he know all the answers of the previous students) and the froth and final student "Amy" knew all of this information therefore she is certain that her hat is white.

#18 Re: Help Me ! » Inequality and arbitrary largeness of numbers » 2019-02-04 17:34:23

The problem and the deduction is very simple. How did you come across this problem? smile Anyway the explanation of this problem is stated below:

It can be shown that the distance of any point (a, b, c, d) in the Euclidean four-dimensional space from the origin (using the Cartesian coordinate system) is given by:


N.B. This formula of the distance in the Euclidean space can be either obtained by definition or deduced using Pythagoras theorem. Now it goes without saying that if one of the integers a, b, c and d becomes arbitrarily large the distance of the corresponding point (according to the last formula) will be arbitrarily large from the origin. However, the point P=(a, b, c, d) (where all of the components are integers and at least two of them unequal) is defined recursively as follows:

Therefore to prove the proclaimed statement (that the distance of these points from the origin gets arbitrarily large), it is sufficient to show that at least one of the numbers a, b, c and d must get arbitrarily large. So the last task can be proven as follows. Suppose that after n iterations we get:

however from the definition of these points (stated above) we have:

Now from the first assumption we have:

and hence

The last inequality holds since all of the terms are squared. Therefore, we get:

however, we have

and hence substituting in the first inequality we get:

so continuing in this process recursively we get at last:

Therefore since by definition at least one of the integers in the right of the last inequality in not zero and all of the terms are squared, the proclaimed statement is established and the related proposition is true. I hope that answers your question.

#19 Re: Help Me ! » Last step » 2019-01-27 07:04:52

Your answer is correct, but you have made a mistake and misunderstood the book. The point of inflection is not x in fact any point is an ordered pain (x,y) so the value of x which you have found is the first component of the point of inflection and the second component (which is y) can be found by substituting x=6 in the original equation, so we have:


hence the point of inflection is (6,2/9) which is the answer found in your book. I hope that will help.

#20 Re: Help Me ! » Integration » 2019-01-26 21:12:54

This problem can be solved along several paths. However, I will take the integration path since this is the natural answer of the question. The identity can be deduced as follows:


but we know that:

hence

However, when x=0 we have:

and hence

which is the required result. I hope that answer your question.

#21 Re: Help Me ! » Can someone help me write a proof for my idea? » 2019-01-13 11:30:08

The second proposition can be proven or disproven as follows:

If we assume that p is a prime number then we would have:


but we know that p is a prime hence the last equation is impossible except for the prime number 3 and hence:

Therefore the only integer which is a prime and satisfies both the conditions, stated in the proposition, is the prime 3. However, if we assume that p is not prime we should exclude all of the integers which are not a multiple of 3 for the same argument stated above. So this will produce the following expression which satisfies the second condition of the proposition (that the number is not factorable by 3):

Testing the n values from 1 to 40 did not produce any pseudoprime that satisfies the second condition (the Fermat primality test). However, if a value of the latter expression at any given integer n turns out to be pseudoprime then it will provide a counterexample that disproves the proposition. On the other hand, if it is proven to have no pseudoprime up to a particular value of n then the previous argument constitutes a proof of the proposition for all integers below the value of the expression corresponding to the related value of n.

#22 Re: Help Me ! » Limit of $n \int_0^1\frac{x^{n}}{x^2+3x+2} \,dx$ » 2018-12-15 11:28:03

In fact the "exact" solution can be obtained by the following argument:


since 0<x<1, but we have:

which is the exact value of the limit. (sorry for the confusion)

#23 Re: Help Me ! » Limit of $n \int_0^1\frac{x^{n}}{x^2+3x+2} \,dx$ » 2018-12-15 06:05:51

In the previous solution, we have taken the average of the denominator as the weight in the summation which provides a good approximation of the limit, but it does not produce the exact solution. To get a more exact solution we should take as the weight in the summation the average of the whole fraction in the integrand. This follows from the following argument:


but we have:

Therefore, we get

and hence

#24 Re: Help Me ! » Limit of $n \int_0^1\frac{x^{n}}{x^2+3x+2} \,dx$ » 2018-12-15 01:44:15

Since 0<x<1 we notice that:


we would have

Therefore we have:

so the limit is between 1/6 and 1/2. However, if we calculate the average value of the function in the denominator, we would get:

but the average, by definition, is the amount that produces the same sum if each term in the summation is replaced by it, therefore we have:

and hence

which is the required limit. Notice that it is between 1/6 and 1/2 as predicted. I hope that answers your question.

Best wishes

#25 Re: Help Me ! » Quasilinear 2nd order PDEs with inital data » 2018-10-30 09:44:53

Hi Emman22, the solution is as follow:


and hence

which is the required solution. Your solution is also correct, but it required some manipulation to reach the desired form.

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