Apink!'s methods exam question thread!

[Apink!]

Hey guys,

I'm needing a lot of help! Completed many practice exams (well not that many) and came up with many questions that I couldn't answer. Here are the questions that I would really appreciate if someone clarified them for me!

First:

Consider the family of curves given by p(x) = |ln (x-1)| + |ln(x-2)| +c, where c is a real constant. Give the exact coordinates of the sharp point of p in terms of c.

I was having a very difficult time with this one, perhaps because my foundation in modulus function is shaky. I thought that I would find the interception point between y= ln(x-1) +ln(x-2) +c and y = -ln(x-1) - ln(x-2)+ c to find the x-coordinate based on my previous knowledge of linear modulus functions which seemed to intercept each other at the sharp point. But that didn't work. So I decided to find the x-int of y = ln(x-1) + ln(x-2) +c which also did not work. Could someone please tell me how I can solve this, logically (i.e// without guessing on the CAS) and explain why my previous attempts failed

Secondly:

A chocolate factory makes teddy bear statues. The weight of the statues are normally distributed with a mean of 1000 grams and a standard deviation of 4 grams. To reduce the number of overweight statues to 10% of rejects, the machine can be modified to change the standard deviation while retaining the mean at 1000 grams. Find, to the nearest whole number, what the new standard deviation should be

For this question, the solutions use a guess- and- check method which I highly dislike. Because firstly: it's annoying, secondly: it's not logical in anyway (and I'm the kind of a person who needs to see logic in problem solving) and thirdly, if you guess wrongly, god knows how much time you're gonna wrestle with this question. So does anyone have a logical way of solving this?

Thirdly,

The average rate of change of f(x) = 3x²+2x+k over the interval [0,2] is equal to twice the average value of f(x) over the same interval. The value of k equals?

I highly believe that some sort of error occurred here, because I think the answer is -2 but the answer booklet tells me it's +2. Any ideas here?

Fourthly,

A restaurant is well regarded for its lobster dishes. It purchases its lobster directly from the local seafood supplier who is not always reliable with the weight.
It is found that the weight of lobsters purchased follow a normal distribution with mean u kg and standard deviation σ kg. It is known that 25% of the lobsters have weights which differ from u by at most m kg.
Find the probability that a randomly chosen lobster has a weight which exceeds ukg by at most 3m kg. Give your answer correct to four decimal places

This question caused a headache for me. Whoever can clarify this, I am forever indebted to you. This is what did:
Pr (u-m <X<u+m) = 0.25, and we are looking for:
Pr (u<X< u+3m) =?
But the answer tells me that
I'm supposed to look for this--> Pr (X< u+3m) =? But this doesn't make much sense, because the question asks to find prob that "lobster has a weight which exceeds ukg by at most 3m kg" but  Pr (X< u+3m) is referring to the probability of weights which are smaller that u+3m which may also be below u kg. Can anyone clarify?

Fifthly:

If g'(x) and h'(x) exist, and g'(x) > h'(x) for all x, then the graph of y= g(x) and the graph of y= h(x)

A. Do not intersect
B. Intersect exactly once
C. Could intersect more than once
D. Intersect no more than once
E. Have a common tangent at each point of intersection

For this question, I think that the answer is B but the answer booklet tells me that the correct answer is D which is it doesn't have to intersect, but it would intersect no more than once. Could someone explain to me what it may not intersect at all? Because I think for it to not intersect they have to be parallel, i.e// have same gradient and it tells us that g'(x) > h'(x) which tells us that they can't be parallel


Thank you so much for reading, and hopefully you can help me!

[Orb]

Hey guys,

I'm needing a lot of help! Completed many practice exams (well not that many) and came up with many questions that I couldn't answer. Here are the questions that I would really appreciate if someone clarified them for me!

First:

Consider the family of curves given by p(x) = |ln (x-1)| + |ln(x-2)| +c, where c is a real constant. Give the exact coordinates of the sharp point of p in terms of c.

I was having a very difficult time with this one, perhaps because my foundation in modulus function is shaky. I thought that I would find the interception point between y= ln(x-1) +ln(x-2) +c and y = -ln(x-1) - ln(x-2)+ c to find the x-coordinate based on my previous knowledge of linear modulus functions which seemed to intercept each other at the sharp point. But that didn't work. So I decided to find the x-int of y = ln(x-1) + ln(x-2) +c which also did not work. Could someone please tell me how I can solve this, logically (i.e// without guessing on the CAS) and explain why my previous attempts failed

Secondly:

A chocolate factory makes teddy bear statues. The weight of the statues are normally distributed with a mean of 1000 grams and a standard deviation of 4 grams. To reduce the number of overweight statues to 10% of rejects, the machine can be modified to change the standard deviation while retaining the mean at 1000 grams. Find, to the nearest whole number, what the new standard deviation should be

For this question, the solutions use a guess- and- check method which I highly dislike. Because firstly: it's annoying, secondly: it's not logical in anyway (and I'm the kind of a person who needs to see logic in problem solving) and thirdly, if you guess wrongly, god knows how much time you're gonna wrestle with this question. So does anyone have a logical way of solving this?

Thirdly,

The average rate of change of f(x) = 3x²+2x+k over the interval [0,2] is equal to twice the average value of f(x) over the same interval. The value of k equals?

I highly believe that some sort of error occurred here, because I think the answer is -2 but the answer booklet tells me it's +2. Any ideas here?

Fourthly,

A restaurant is well regarded for its lobster dishes. It purchases its lobster directly from the local seafood supplier who is not always reliable with the weight.
It is found that the weight of lobsters purchased follow a normal distribution with mean u kg and standard deviation σ kg. It is known that 25% of the lobsters have weights which differ from u by at most m kg.
Find the probability that a randomly chosen lobster has a weight which exceeds ukg by at most 3m kg. Give your answer correct to four decimal places

This question caused a headache for me. Whoever can clarify this, I am forever indebted to you. This is what did:
Pr (u-m <X<u+m) = 0.25, and we are looking for:
Pr (u<X< u+3m) =?
But the answer tells me that
I'm supposed to look for this--> Pr (X< u+3m) =? But this doesn't make much sense, because the question asks to find prob that "lobster has a weight which exceeds ukg by at most 3m kg" but  Pr (X< u+3m) is referring to the probability of weights which are smaller that u+3m which may also be below u kg. Can anyone clarify?

Fifthly:

If g'(x) and h'(x) exist, and g'(x) > h'(x) for all x, then the graph of y= g(x) and the graph of y= h(x)

A. Do not intersect
B. Intersect exactly once
C. Could intersect more than once
D. Intersect no more than once
E. Have a common tangent at each point of intersection

For this question, I think that the answer is B but the answer booklet tells me that the correct answer is D which is it doesn't have to intersect, but it would intersect no more than once. Could someone explain to me what it may not intersect at all? Because I think for it to not intersect they have to be parallel, i.e// have same gradient and it tells us that g'(x) > h'(x) which tells us that they can't be parallel


Thank you so much for reading, and hopefully you can help me!

1) I'm assuming it's just the cusp that you're talking about, you'll see that c doesn't actually change the cusp, it just moves the graph up/down
Using the cas you get x=3

[studentofthenow]

Hey guys,

I'm needing a lot of help! Completed many practice exams (well not that many) and came up with many questions that I couldn't answer. Here are the questions that I would really appreciate if someone clarified them for me!

First:

Consider the family of curves given by p(x) = |ln (x-1)| + |ln(x-2)| +c, where c is a real constant. Give the exact coordinates of the sharp point of p in terms of c.

I was having a very difficult time with this one, perhaps because my foundation in modulus function is shaky. I thought that I would find the interception point between y= ln(x-1) +ln(x-2) +c and y = -ln(x-1) - ln(x-2)+ c to find the x-coordinate based on my previous knowledge of linear modulus functions which seemed to intercept each other at the sharp point. But that didn't work. So I decided to find the x-int of y = ln(x-1) + ln(x-2) +c which also did not work. Could someone please tell me how I can solve this, logically (i.e// without guessing on the CAS) and explain why my previous attempts failed

Secondly:

A chocolate factory makes teddy bear statues. The weight of the statues are normally distributed with a mean of 1000 grams and a standard deviation of 4 grams. To reduce the number of overweight statues to 10% of rejects, the machine can be modified to change the standard deviation while retaining the mean at 1000 grams. Find, to the nearest whole number, what the new standard deviation should be

For this question, the solutions use a guess- and- check method which I highly dislike. Because firstly: it's annoying, secondly: it's not logical in anyway (and I'm the kind of a person who needs to see logic in problem solving) and thirdly, if you guess wrongly, god knows how much time you're gonna wrestle with this question. So does anyone have a logical way of solving this?

Thirdly,

The average rate of change of f(x) = 3x²+2x+k over the interval [0,2] is equal to twice the average value of f(x) over the same interval. The value of k equals?

I highly believe that some sort of error occurred here, because I think the answer is -2 but the answer booklet tells me it's +2. Any ideas here?

Fourthly,

A restaurant is well regarded for its lobster dishes. It purchases its lobster directly from the local seafood supplier who is not always reliable with the weight.
It is found that the weight of lobsters purchased follow a normal distribution with mean u kg and standard deviation σ kg. It is known that 25% of the lobsters have weights which differ from u by at most m kg.
Find the probability that a randomly chosen lobster has a weight which exceeds ukg by at most 3m kg. Give your answer correct to four decimal places

This question caused a headache for me. Whoever can clarify this, I am forever indebted to you. This is what did:
Pr (u-m <X<u+m) = 0.25, and we are looking for:
Pr (u<X< u+3m) =?
But the answer tells me that
I'm supposed to look for this--> Pr (X< u+3m) =? But this doesn't make much sense, because the question asks to find prob that "lobster has a weight which exceeds ukg by at most 3m kg" but  Pr (X< u+3m) is referring to the probability of weights which are smaller that u+3m which may also be below u kg. Can anyone clarify?

Fifthly:

If g'(x) and h'(x) exist, and g'(x) > h'(x) for all x, then the graph of y= g(x) and the graph of y= h(x)

A. Do not intersect
B. Intersect exactly once
C. Could intersect more than once
D. Intersect no more than once
E. Have a common tangent at each point of intersection

For this question, I think that the answer is B but the answer booklet tells me that the correct answer is D which is it doesn't have to intersect, but it would intersect no more than once. Could someone explain to me what it may not intersect at all? Because I think for it to not intersect they have to be parallel, i.e// have same gradient and it tells us that g'(x) > h'(x) which tells us that they can't be parallel


Thank you so much for reading, and hopefully you can help me!

By any chance is the answer to secondly 9?

[Swagadaktal]

for fifthly, you need to picture ( i sketched out 2 diff graphs, just random graphs), and as g'(x) is always greater than h'(x) u can always  assume that they are parallel ( never intersect), however, there is at least 1 point of intersection as when you anti-diff, you have the +c . This + C creates 1 point of intersection.

[Swagadaktal]

for fourthly you know that pr(u-m<x<u+m) = 0.25, therefore pr(x>u+m) = 0.375
=> (u+m-u)/o = invNorm(0.625) , u find m in terms of o, to find out 3m you multiply that ans by 3. Then you find out how many standard deviations it is away (for 3m), so you use pr(Z>3*(whatever ans u just found before)) and you can find the probability... soz dont have cas with me rn

[Swagadaktal]

ooh didnt read the question properly, from the last step there you gotta find pr(u<x<3m), you find pr(z>3m) and then you minus it from 1/2, then that's pr(u<x<3m)