Math Is Fun Forum

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.

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.

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:

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)

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

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):

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:

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.

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:

The second proposition can be proven or disproven as follows:

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

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:

Hi Emman22, the solution is as follow: