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Leviafan [203]
3 years ago
15

3 sin^{2} x +cos 2x= (5/4)answer in radians

Mathematics
1 answer:
Vera_Pavlovna [14]3 years ago
6 0

Answer:

I believe it's 0.540717

Step-by-step explanation:

3(sin(2))x+(cos(2))(x)=5/4

Simplify: 2.311745x=5/4

Divide: 2.311745x/2.311735=5/4/2.311745

x=0.540717

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How do you solve this limit of a function math problem? ​
hram777 [196]

If you know that

e=\displaystyle\lim_{x\to\pm\infty}\left(1+\frac1x\right)^x

then it's possible to rewrite the given limit so that it resembles the one above. Then the limit itself would be some expression involving e.

For starters, we have

\dfrac{3x-1}{3x+3}=\dfrac{3x+3-4}{3x+3}=1-\dfrac4{3x+3}=1-\dfrac1{\frac34(x+1)}

Let y=\dfrac34(x+1). Then as x\to\infty, we also have y\to\infty, and

2x-1=2\left(\dfrac43y-1\right)=\dfrac83y-2

So in terms of y, the limit is equivalent to

\displaystyle\lim_{y\to\infty}\left(1-\frac1y\right)^{\frac83y-2}

Now use some of the properties of limits: the above is the same as

\displaystyle\left(\lim_{y\to\infty}\left(1-\frac1y\right)^{-2}\right)\left(\lim_{y\to\infty}\left(1-\frac1y\right)^y\right)^{8/3}

The first limit is trivial; \dfrac1y\to0, so its value is 1. The second limit comes out to

\displaystyle\lim_{y\to\infty}\left(1-\frac1y\right)^y=e^{-1}

To see why this is the case, replace y=-z, so that z\to-\infty as y\to\infty, and

\displaystyle\lim_{z\to-\infty}\left(1+\frac1z\right)^{-z}=\frac1{\lim\limits_{z\to-\infty}\left(1+\frac1z\right)^z}=\frac1e

Then the limit we're talking about has a value of

\left(e^{-1}\right)^{8/3}=\boxed{e^{-8/3}}

# # #

Another way to do this without knowing the definition of e as given above is to take apply exponentials and logarithms, but you need to know about L'Hopital's rule. In particular, write

\left(\dfrac{3x-1}{3x+3}\right)^{2x-1}=\exp\left(\ln\left(\frac{3x-1}{3x+3}\right)^{2x-1}\right)=\exp\left((2x-1)\ln\frac{3x-1}{3x+3}\right)

(where the notation means \exp(x)=e^x, just to get everything on one line).

Recall that

\displaystyle\lim_{x\to c}f(g(x))=f\left(\lim_{x\to c}g(x)\right)

if f is continuous at x=c. \exp(x) is continuous everywhere, so we have

\displaystyle\lim_{x\to\infty}\left(\frac{3x-1}{3x+3}\right)^{2x-1}=\exp\left(\lim_{x\to\infty}(2x-1)\ln\frac{3x-1}{3x+3}\right)

For the remaining limit, write

\displaystyle\lim_{x\to\infty}(2x-1)\ln\frac{3x-1}{3x+3}=\lim_{x\to\infty}\frac{\ln\frac{3x-1}{3x+3}}{\frac1{2x-1}}

Now as x\to\infty, both the numerator and denominator approach 0, so we can try L'Hopital's rule. If the limit exists, it's equal to

\displaystyle\lim_{x\to\infty}\frac{\frac{\mathrm d}{\mathrm dx}\left[\ln\frac{3x-1}{3x+3}\right]}{\frac{\mathrm d}{\mathrm dx}\left[\frac1{2x-1}\right]}=\lim_{x\to\infty}\frac{\frac4{(x+1)(3x-1)}}{-\frac2{(2x-1)^2}}=-2\lim_{x\to\infty}\frac{(2x-1)^2}{(x+1)(3x-1)}=-\frac83

and our original limit comes out to the same value as before, \exp\left(-\frac83\right)=\boxed{e^{-8/3}}.

3 0
3 years ago
Use the distributive property to express 32 + 64
dolphi86 [110]
2(16+32) should be correct. goodluck!
4 0
2 years ago
Read 2 more answers
A plane is a flying at an altitude of 23,760 feet. Which expression can be used to find the altitude on miles.
Zina [86]

Answer:

D. 23,760÷5,280 assuming that 5,5,280 is a typo

Step-by-step explanation:

If you divide it will give you the height in miles because there are 5,280 feet in a mile.

6 0
2 years ago
Beth purchased a computer, and its value depreciated exponentially each year. Beth will sketch a graph of the situation, where x
Katyanochek1 [597]

What is x?

It is the number of years since purchase. Can this be negative?

NO! Years cannot be negative. So we disregard negative x-axis.


What is y?

It is the value of the computer at that year. A computer starts off with a positive value (y axis) and depreciates to 0 after some years. Can this be negative?

NO! At 0 value, the computer is worth nothing! There is nothing known as <em>negative </em>value. So we disregard negative y-axis.


The graph that Beth will sketch will have to be in the 1st quadrant (positive x-axis & positive y-axis).

ANSWER: Quadrant I only

3 0
3 years ago
Please help me.
oksano4ka [1.4K]

Answer:

5 Soup can

Step-by-step explanation:

If 3 soup can cost $3.75

Then 1 soup can will cost $3.75/3

So, how many can soup will cost $6.25

= $6.26/(cost of 1 soup can)

= $6.25/($3.75/3)

=$6.25 × 3÷ 3.75

= 5 cans of soup

3 0
3 years ago
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