1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
tester [92]
2 years ago
14

Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = 7 cos2(x) − 14 sin(x), 0 ≤ x ≤ 2π (a) Find the int

erval on which f is increasing. (Enter your answer using interval notation.)
Mathematics
1 answer:
Troyanec [42]2 years ago
3 0

Answer:

[\frac{\pi}{2},\frac{3\pi}{2}]

Step-by-step explanation:

Let me first state that I am assuming your function is

f(x)=7cos^2(x)-14sin(x)

If this is incorrect, then disregard this whole answer/explanation.

In order to find where the function is increasing or decreasing, we need to first find the first derivative, set it equal to 0, and then factor to find the values that cause the derivative to equal 0.  This is where you expect to find a max or a min value in the function itself.  But this function is not going to be easily solved for 0 once we find the derivative unless we make it in terms of either sin or cos right now, before taking the first derivative.  

Let cos^2(x)=1-sin^2(x)

This is a Pythagorean trig identity, and I'm assuming that if you're in calculus solving for the intervals of increasing and decreasing values that you have, at one time, used trig identities.

Rewriting:

f(x)=7(1-sin^2(x))-14sin(x) which simplifies to

f(x)=7-7sin^2(x)-14sin(x) and in order of descending values of x:

f(x)=-7sin^2(x)-14sin(x)+7

Now we can find the derivative.  For the first term, let u = sin(x), therefore,

f(u)=u^2, u' = cos(x), and f'(u) = 2u.  The derivative is found by multiplying f'(u) by u', which comes out to 2sin(x)cos(x)

The derivative for the next 2 terms are simple, so the derivative of the function is

f'(x)=-7[2sin(x)cos(x)]-14cos(x) which simplifies down to

f'(x)=-14sin(x)cos(x)-14cos(x)

We will set that equal to zero and solve for the values that cause that derivative to equal 0.  But first we can simplify it a bit.  You can factor out a -14cos(x):

f'(x)=-14cos(x)(sin(x)+1)

By the Zero Product Property, either

-14cos(x) = 0 or sin(x) + 1 = 0

Solving the first one for cos(x):

cos(x) = 0

Solving the second one for sin(x):

sin(x) = -1

We now look to the unit circle to see where, exactly the cos(x) = 0.  Those values are

\frac{\pi}{2},\frac{3\pi}{2}

The value where the sin is -1 is found at

\frac{3\pi}{2}

We set up a table (at least that's what I advise my students to do!), separating the intervals in ascending order, starting at 0 and ending at 2pi.

Those intervals are

0 < x < \frac{\pi}{2}, \frac{\pi}{2}, and \frac{3\pi}{2}

Now pick a value that falls within each interval and evaluate the derivative at that value and determine the sign (+ or -) that results.  You don't care what the value is, only the sign that it carries.  For the first interval I chose

f'(\frac{\pi}{4})=- so the function is decreasing here (not what you wanted, so let's move on to the next interval).

For the next interval I chose:

f'(\pi)=+ so the function is increasing here.

For the last interval I chose:

f'(\frac{7\pi}{4})=-

It appears that the only place this function is increasing is on the interval

[\frac{\pi}{2},\frac{3\pi}{2}]

You might be interested in
Holly begins with 1 penny in her piggy bank. She adds 2 pennies on day 1, 4 pennies on day 2, and 8 pennies on day 3. She contin
muminat

Answer:

All the options are correct.

Step-by-step explanation:

Holly begins with 1 penny in her piggy bank. She adds 2 pennies on day 1, 4 pennies on day 2, and 8 pennies on day 3. She continues this pattern for 30 days.

The pattern which she choses is 2^n where n denotes the number of days.

A)

The graph of this sequence would pass the vertical line test.

" Vertical line test: A function can only have one output, y, for each unique input, x " .

Here also y=2^n and x=n.

B)

Each day corresponds to a unique amount of pennies in this sequence.

Since the sequence is increasing, hence option B is correct.

C)

The graph of this sequence would pass the horizontal line test.

A test use to determine if a function is one-to-one.

As for each n we get a unique value.

Hence, option C is correct.

D)

Each amount of pennies corresponds to a unique day in this sequence.

This option is also true as each  amount is a function of number of days.

5 0
3 years ago
Read 2 more answers
What's the answer was for <br> 1/2r+2(3/4r-1)=1/4r+6
Simora [160]
1/2r+2(3/4r-1)=1/4r+6
2r-2=1/4r+6
2r=1/4r+8
7/4r=8
7r=32
r=32/7
5 0
2 years ago
F(x)=2x^2-x+1 find the vertex
Galina-37 [17]

The Answer fam is (1/4,7/8)

5 0
3 years ago
Evaluate the expression for x = 3, y=13 , and z = 5.
hoa [83]

Answer:

-437

Step-by-step explanation:

x=3,y=13,z=5

12x-3y4x-z

12(3)-3(13)(4)(3)-5

36-468-5

-437

5 0
3 years ago
Can someone explain to me how to find mean/median of a data set real quick?
PIT_PIT [208]
Mean is were you add up all of the numbers and how many numbers are there thats how much you divide by and median is were you line up the numbers from least to greatest then try to find the middle number <span />
3 0
3 years ago
Read 2 more answers
Other questions:
  • Some rational numbers are not integers? true or false
    10·1 answer
  • Will give brainliest!
    12·2 answers
  • Plz HELP<br> .........../....
    8·1 answer
  • What is the value of x in the diagram?
    9·2 answers
  • Which of the following rational functions is graphed below?
    13·1 answer
  • 9+8=17 what number sentence shows the commutative property of addition
    11·2 answers
  • Factor this polynomial expression.
    6·1 answer
  • What is the distance between (-5,-5) and (-9, -2)?
    10·1 answer
  • What’s the answer to this question??
    12·1 answer
  • A farmer sells 7.7 kilograms of apples and pears at the farmer's market.
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!