Calculus homework help!!!!

[shinningstar14]

/My first week at college and i don't understand my homework. Please help me. Solution? Hints? Suggestion? Anything will be extremely helpful!!!!

1) Using the fact that 75=2.60-45, compute cos(5?/12).

2) The negation of a statement A is a statement B which is true whenever A is false, and false whenever A is true. For example: the negation of "all students in Suzan's class are old" is "some student in Suzan's class is young" (and not "all students in Suzan's class are young").

Write the negation of the following statements:

1. Some offices in this building are dirty.
2. No-one is allowed past this point.

3) If f(x)=ex+1 and g(x)=x2+1, what is the range of the composite function f(g(x))?

Please respond asap!

 

[mmm4444bot]

the fact that 75 = 2.60 - 45 Do you really believe this "fact" ?

compute cos(5?/12) Since 5?/12 = 2?/3 - ?/4, you can use this identity:

cos(A - B) = cos(A) cos(B) + sin(A) sin(B)



Some offices in this building are dirty. What scenario would make this statement false ?

No-one is allowed past this point. What scenario would make this statement false ?



f(x) = ex + 1 Typing ex means e times x. Do you mean one of the following, instead ?

e^x + 1

e^(x + 1)

g(x) = x2 + 1 I'm guessing that you intend x^2 + 1. Is that correct ?

Click HERE to read about standard conventions for clearly typing math expressions using a keyboard.

Your posted exercises are all precalculus review.

What do you already know about function domain and range ? What do you already know about composite functions ?

Please respond to my seven questions above, and we'll go from there.

Cheers ~ Mark


PS: When you have some time, please read the post titled, "Read Before Posting".

 

[shinningstar14]

Since this was my first post, i didn't know how this worked. I typed it in a hurry; therefore, i had alot of mistakes. I am soorrryyy and i will read that post too. I will rewrite the questions:

1) Using the fact that 75=2*60(2 times 60)-45, compute cos(5?/12).

2) Some offices in this building are dirty. What scenario would make this statement false ?
No-one is allowed past this point. What scenario would make this statement false ?

I DON'T get it! So help me. i took IB Mathematics I & II and i never studied negations.

3) ) If f(x)=e^x + 1 and g(x)=x^2 + 1 what is the range of the composite function f(g(x))?
I think f(g(x))= e^x^2 + 1.

These are chapter 1 of my calculus book so it is technically review of pre-calculus and intro to calculus. Thanks for help

 

[mmm4444bot]

I am soorrryyy

I DON'T get it!

So help me.

I don't know what you need.

Please tell me how I can help you.

Do you need examples to jog your memory ?

EG:

sin(7?/12)

7?/12 radians = 105°

Since 105° = 60° + 45° we can treat sin(7?/12) as sin(60° + 45°) and use an identity to evaluate it because 60° and 45° are special angles for which we know the sine ratios.

(If you forgot sine and cosine for the special angles, look them up.)

sin(60° + 45°) = sin(60°) cos(45°) + cos(60°) sin(45°)

= (?3/2)(?2/2) + (1/2)(?2/2)

= ?6/4 + ?2/4

Therefore:

sin(7?/12) = (?6 + ?2)/4

Now, if any part of this example does not make sense, please ask a specific question, and we'll go from there.

Otherwise, you have the identity to use for cos(5?/12).

Also, let me know where you're stuck on the other exercises.

Cheers ~ Mark

 

[mmm4444bot]

i never studied negations.

The exercise tells what they are.

"Some offices in this building are dirty".

Okay, I'll try a different approach.

Let's say that you need to rent an office in a building, and the landlord is walking you through the building. Your monthly rent will include a cleaning service.

As you're walking through the building, checking it out, a gossipy secretary that works in the building whispers in your ear, "Some offices in this building are dirty. The landlord is supposed to have them cleaned."

Now, what if the gossipy secretary is lying? Then what's true?

If the secretary's statement is false, then none of the offices are dirty. Does that make sense?

So, the negation of "Some of the offfices in this building are dirty" is "None of the offices in this building are dirty".

Please ask specific questions, if you need more help.

Cheers!

 

[mmm4444bot]

3) ) If f(x)=e^x + 1 and g(x)=x^2 + 1 what is the range of the composite function f(g(x))?

I do not understand the significance of the part in red.

I think f(g(x))= e^x^2 + 1. Is there supposed to be another plus one ?

f(g(x)) = e^(x^2 + 1) + 1

We don't really need to know the algebraic definition of f(g(x)).

We look at the range of g(x) because that set is the domain we use for f(x).

What is the range of x^2 + 1 ?

Hint: The graph is a parabola that opens upward with vertex at (0, 1).

 

[shinningstar14]

Alright, so this is what i did for number 1

1) Using the fact 75= 2*65-45, computer cos(5?/12).

sin(5?/12), 5?/12 radians = 75°

sin(30° + 45°) = sin(30°) cos(45°) + cos(30°) sin(45°)

= (?3/2)(?2/2) + (1/2)(?2/2)

= ?6/4 + ?2/4

So, sin(5?/12) = (?6 + ?2)/4

Your example really helped.

For number 2,

I have a question, there is only suppose to be one negation for two statements? or two different ones?

Two different ones would be:

1) None of the offices in this building are dirty.

2) Only few people are allowed past this point.

I am right? Correct me if i am wrong. :?:

For number 3,

f(g(x)) = e^(x^2 + 1) + 1, so the range of the composite function f(g(x)) is the interval [e^2,?) or [e,?)???

Thanks for all your help. Mark

 

[Deleted member 4993]

Alright, so this is what i did for number 1

1) Using the fact 75= 2*65-45, <<< How is this a fact?

and where did you use this dubious fiction in your derivation?


computer cos(5?/12).

sin(5?/12), 5?/12 radians = 75°

sin(30° + 45°) = sin(30°) cos(45°) + cos(30°) sin(45°)

= (?3/2)(?2/2) + (1/2)(?2/2)

= ?6/4 + ?2/4

So, sin(5?/12) = (?6 + ?2)/4

Your example really helped.

For number 2,

I have a question, there is only suppose to be one negation for two statements? or two different ones?

Two different ones would be:

1) None of the offices in this building are dirty.

2) Only few people are allowed past this point.

I am right? Correct me if i am wrong. :?:

For number 3,

f(g(x)) = e^(x^2 + 1) + 1, so the range of the composite function f(g(x)) is the interval [e^2,?) or [e,?)???

Thanks for all your help. Mark

 

[shinningstar14]

My teacher got it from somewhere. Idk. It was part of the question. Btw, your response didn't really help me

 

[mmm4444bot]

compute cos(5?/12) Since 5?/12 = 2?/3 - ?/4, you can use this identity:

cos(A - B) = cos(A) cos(B) + sin(A) sin(B)

My example used an identity for sin(A + B).

You need to use the identity that I gave you above for cos(A - B).

75 = 120 - 45, so them's the A and B.

cos(5?/12) = cos(120° - 45°) = cos(A - B) above.

 

[mmm4444bot]

f(g(x)) = e^(x^2 + 1) + 1, so the range of the composite function f(g(x)) is the interval [e^2,?) or [e,?)???

I was hoping that you would have answered my question.

So, is the lowest number in the range e or e^2, you ask ?

Well, the lowest number in the range comes from inputing the lowest number in the domain, in this exercise.

Did you do that ?

I can't see any work.

 

[shinningstar14]

This is what i have for number 1:

cos(5?/12), 5?/12 radians = 75°

cos(120 - 45) = cos(120°) cos(45°) + sin(120°) sin(45°)

= (-1/2)(?2/2) + (1/2)(1/2)

= - ?2/4 + ?2/4

So, cos(5?/12) = 0

It is really this easy, i feel stupid. And please tell me if this is the right answer?

Is my answer to number 2 correct? Please answer me.

There is only suppose to be one negation for two statements? or two different ones?

Two different ones would be:

1) None of the offices in this building are dirty.

2) Only few people are allowed past this point.

I am right? Correct me if i am wrong

For number 3, the lowest value is -1

f(g(x)) = e^(x^2 + 1) + 1

= e^((-1)^2 + 1) + 1

=e^(1 + 1) + 1

=e^(2) + 1

so the range is [e^2,?). Yeah, i got it!!!! All because of u though

 

[Deleted member 4993]

This is what i have for number 1:

cos(5?/12), 5?/12 radians = 75°

cos(120 - 45) = cos(120°) cos(45°) + sin(120°) sin(45°)

= (-1/2)(?2/2) + (1/2)(1/2)

= - ?2/4 + ?2/4 <<< That should be 1/4

So, cos(5?/12) = 0 <<< Incorrect

It is really this easy, i feel stupid. And please tell me if this is the right answer?

Is my answer to number 2 correct? Please answer me.

There is only suppose to be one negation for two statements? or two different ones?

Two different ones would be:

1) None of the offices in this building are dirty.

2) Only few people are allowed past this point.

I am right? Correct me if i am wrong

For number 3, the lowest value is -1

f(g(x)) = e^(x^2 + 1) + 1

= e^((-1)^2 + 1) + 1

=e^(1 + 1) + 1

=e^(2) + 1

so the range is [e^2,?). Yeah, i got it!!!! All because of u though

 

[shinningstar14]

SORRYYYY, IT WAS A TYPO/ MISTAKE

1) cos(5?/12), 5?/12 radians = 75°

cos(120 - 45) = cos(120°) cos(45°) + sin(120°) sin(45°)

= (-1/2)(?2/2) + (?3/2)(?2/2)

= - ?2/4 + ?6/4

So, cos(5?/12) = (-?2 + ?6)/4

 

[mmm4444bot]

cos(120°) cos(45°) + sin(120°) sin(45°)

(-1/2)(?2/2) + (1/2)(1/2)

Except for the ratios in red, the line above is correct.

Is my answer to number 2 correct? Please answer me.

There is only suppose to be one negation for two statements?

I read it as two separate exercises. I looked at the original post again, and I'm not sure that I can help you with the negations. They seem to say that the negative of some is all, whereas I take the negative of some to be none.

For number 3, the lowest value is -1

-1 is not in the range of x^2 + 1

 

[BigGlenntheHeavy]

\(\displaystyle Given: \ f(x) \ = \ e^x+1 \ and \ g(x) \ = \ x^2+1, \ find \ range \ of \ f[g(x)].\)

\(\displaystyle e^x \ > \ 0, \ \implies \ e^x+1 \ > \ 0+1 \ = \ 1 \ and \ x^2 \ \ge \ 0 \ \implies \ x^2+1 \ \ge \ 0+1 \ =1.\)

\(\displaystyle Hence \ range \ of \ f(x) \ = \ (1,\infty) \ and \ range \ of \ g(x) \ = \ [1,\infty).\)

\(\displaystyle Now, \ let \ h(x) \ = \ f[g(x)] \ = \ f(x^2+1) \ = \ e^{x^2+1}+1.\)

\(\displaystyle Ergo, \ range \ of \ h(x) \ = \ [e+1,\infty), \ see \ graph.\)

\(\displaystyle What \ are \ the \ domains \ of \ f(x), \ g(x), \ and \ h(x)?\)

[attachment=0:34p7g1yj]eee.jpg[/attachment:34p7g1yj]

 

[mmm4444bot]

SORRYYYY, IT WAS A TYPO/ MISTAKE

Sloppy typing wastes time; you can save time by proofreading your posts before submitting. Use the [Preview] button, to proofread your posts.

cos(5?/12) = (-?2 + ?6)/4

This is correct.

 

[cristine27]

Calculus is my weakness in all math subjects. So whenever I have homework in calculus, I find it difficult to solve it. But the feeling it gives whenever I finish one problem is exhilarating, at least I know I did my best.

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