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Sphinxa [80]
3 years ago
14

The graph below shows two polynomial functions, f(x) and g(x):

Mathematics
2 answers:
pentagon [3]3 years ago
5 0
One glance at the graphs should be enuf to tell you that one (the red one) is the graph of a parabola with positive leading coeff. and that the other is the gaph of an odd function which here happens to be y = x^3, also with a pos. lead. coeff.
neonofarm [45]3 years ago
3 0
The answer is '<span>f(x) is an odd degree polynomial with a positive leading coefficient'.

An odd degree polynomial with a positive leading coefficient will have the graph go towards negative infinity as x goes towards negative infinity, and go towards infinity as x goes towards infinity.

An even degree polynomial with a negative leading coefficient will have the graph go towards infinity as x goes toward negative infinity, and go towards negative infinity as x goes toward infinity.

g(x) would have a a positive leading coefficient with an even degree, as the graph goes towards infinity as x goes towards either negative or positive infinity.
</span>
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Help me with trigonometry
poizon [28]

Answer:

See below

Step-by-step explanation:

It has something to do with the<em> </em><u><em>Weierstrass substitution</em></u>, where we have

$\int\, f(\sin(x), \cos(x))dx = \int\, \dfrac{2}{1+t^2}f\left(\dfrac{2t}{1+t^2}, \dfrac{1-t^2}{1+t^2} \right)dt$

First, consider the double angle formula for tangent:

\tan(2x)= \dfrac{2\tan(x)}{1-\tan^2(x)}

Therefore,

\tan\left(2 \cdot\dfrac{x}{2}\right)= \dfrac{2\tan(x/2)}{1-\tan^2(x/2)} = \tan(x)=\dfrac{2t}{1-t^2}

Once the double angle identity for sine is

\sin(2x)= \dfrac{2\tan(x)}{1+\tan^2(x)}

we know \sin(x)=\dfrac{2t}{1+t^2}, but sure,  we can derive this formula considering the double angle identity

\sin(x)= 2\sin\left(\dfrac{x}{2}\right)\cos\left(\dfrac{x}{2}\right)

Recall

\sin \arctan t = \dfrac{t}{\sqrt{1 + t^2}} \text{ and } \cos \arctan t = \dfrac{1}{\sqrt{1 + t^2}}

Thus,

\sin(x)= 2 \left(\dfrac{t}{\sqrt{1 + t^2}}\right) \left(\dfrac{1}{\sqrt{1 + t^2}}\right) = \dfrac{2t}{1 + t^2}

Similarly for cosine, consider the double angle identity

Thus,

\cos(x)=  \left(\dfrac{1}{\sqrt{1 + t^2}}\right)^2- \left(\dfrac{t}{\sqrt{1 + t^2}}\right)^2 = \dfrac{1}{t^2+1}-\dfrac{t^2}{t^2+1} =\dfrac{1-t^2}{1+t^2}

Hence, we showed \sin(x) \text { and } \cos(x)

======================================================

5\cos(x) =12\sin(x) +3, x \in [0, 2\pi ]

Solving

5\,\overbrace{\frac{1-t^2}{1+t^2}}^{\cos(x)} = 12\,\overbrace{\frac{2t}{1+t^2}}^{\sin(x)}+3

\implies \dfrac{5-5t^2}{1+t^2}= \dfrac{24t}{1+t^2}+3 \implies  \dfrac{5-5t^2 -24t}{1+t^2}= 3

\implies 5-5t^2-24t=3\left(1+t^2\right) \implies -8t^2-24t+2=0

t = \dfrac{-(-24)\pm \sqrt{(-24)^2-4(-8)\cdot 2}}{2(-8)} = \dfrac{24\pm 8\sqrt{10}}{-16} =  \dfrac{3\pm \sqrt{10}}{-2}

t=-\dfrac{3+\sqrt{10}}{2}\\t=\dfrac{\sqrt{10}-3}{2}

Just note that

\tan\left(\dfrac{x}{2}\right) =  \dfrac{3\pm 8\sqrt{10}}{-2}

and  \tan\left(\dfrac{x}{2}\right) is not defined for x=k\pi , k\in\mathbb{Z}

6 0
2 years ago
josh flips a colored chip that is red on one side and black on the other a total of 20 times. what is the probability of the 21t
kiruha [24]
As long as the mass of the chip is evenly distributed on each side, whatever times you flip the chip,

The probability will always be 50%
3 0
3 years ago
Juan and his father went on a driving trip.
Jet001 [13]

Answer:

162

Step-by-step explanation:

260/2=130

130+32=162

8 0
3 years ago
Read 2 more answers
Find the area of a triangle with sides of length 6 and 7 and included angle 74°. (round your answer to one decimal place.)
Yanka [14]
We use this formula:
<span>Area = ½ • side 1 • sine (A) • side 2
</span>Area = <span>½ • 6 • sine (74) • 7
</span><span>Area = <span>21 • sine (74)
Area = 21*0.96126
Area = </span></span><span><span><span>20.18646 </span> </span> </span>
Area = 20.2 (rounded)
Source:
http://www.1728.org/triang.htm


5 0
3 years ago
Three subtracted from twice a number equals 31 more than three times the number. find the number
Ksivusya [100]

Answer:

34

Step-by-step explanation:

Let the no. be x,

2x-3=3x+31

3x-2x= -3-31

x= -34

5 0
3 years ago
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