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Vitek1552 [10]
3 years ago
7

I can't seem to figure this one out. Could someone show how to do it and the answer? Thanks in advance!

Mathematics
1 answer:
vagabundo [1.1K]3 years ago
5 0

Answer:

\displaystyle (-\infty, -\frac{1}{6})

Step-by-Step explanation:

We have the function:

\displaystyle y=\int_{1}^{x}\frac{1}{3+t+3t^2}\, dt

And we want to find the interval for which y is concave upwards.

Therefore, we will need to find the second derivative of y, find its inflection points (where y''=0), and test for values.

So, let's take the derivative of both sides with respect to x. So:

\displaystyle y^\prime=\frac{d}{dx}\Bigg[\int_{1}^{x}\frac{1}{3+t+3t^2}\, dt\Bigg]

By the Fundamental Theorem of Calculus:

\displaystyle y^\prime=\frac{1}{3+x+3x^2}

So, we will take the derivative again. Hence:

\displaystyle y^\prime^\prime=\frac{d}{dx}\Big[\frac{1}{3+x+3x^2}\Big]=\frac{d}{dx}\Big[(3+x+3x^2)^{-1}\Big]

We will use the chain rule. Let:

\displaystyle u=x^{-1}\text{ and } v=3+x+3x^2

Differentiate:

\displaystyle y^\prime^\prime=-(3+x+3x^2)^{-2}(1+6x)

Rewrite:

\displaystyle y^\prime^\prime=-\frac{6x+1}{(3+x+3x^2)^2}

So, points of inflection, where the concavity changes, is whenever the second derivative is 0 or undefined.

We can see that the second derivative will never be undefined since the denominator can never equal 0.

So, our only possible inflection points are when it's equal to 0. Hence:

\displaystyle 0=-\frac{6x+1}{(3+x+3x^2)^2}

Multiplying both sides by the denominator gives:

0=-(6x+1)

Then it follows that:

\displaystyle x=-\frac{1}{6}

So, our only possible point of inflection is at <em>x=-1/6. </em>

We will test for values less than and greater than this inflection point.

Testing for <em>x=-1</em>, we see that:

\displaystyle y^\prime^\prime=-\frac{6(-1)+1}{3+(-1)+3(-1)^2}=1>0

Since the result is positive, y is concave up for all values less than -1/6.

And testing for <em>x=0</em>, we see that:

\displaystyle y^\prime^\prime=-\frac{6(0)+1}{3+(0)+3(0)^2}=-\frac{1}{3}

Since the result is negative, y is concave down for all values greater than -1/6.

Therefore, the interval for which <em>y</em> is concave up is:

\displaystyle (-\infty, -\frac{1}{6})

Note that we use parentheses instead of brackets since at exactly <em>x=-1/6</em>, our graph is neither concave up nor concave down.

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Two isosceles triangles are shown above. If b = 180-4a and a=35, What is the value of c?
Ymorist [56]

Answer:

a=35 given

b=40  

c=110

I couldn't complete see c. Please look at the picture to see what I assumed it to be.

Step-by-step explanation:

Hmmm... I guess a and b are not vertical.

We are given b=180-4a and a=35 so b=180-4(35)=180-140=40.

So b=40.

Isosceles triangles always have congruent base angles.  So let's call both of the base angles in the bottom triangle x.

That means x+x+40=180.

We need to solve this for x.

Combine like terms:

2x+40=180

Subtract 40 on both sides:

2x=140

Divide both sides by 2:

x=140/2

Simplify:

x=70

So I'm assuming that c and it's adjacent angle are sitting on a straightedge together which means 70+c=180.

70+c=180

Subtract 70 on both sides:

c=180-70

Simplify:

c=110

4 0
3 years ago
Help me out a little
konstantin123 [22]
I believe your answer is (1,5)

I graphed the equations and the ordered pairs and (1,5) is the only point that fell on both of the shaded areas.

hope this helps !!
8 0
2 years ago
Exponential 2^2/2^-3​
kenny6666 [7]

Answer: 32

Step-by-step explanation:

2^2 is 4

2^-3 = 1/8

and 4/1/8 is 32

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3 years ago
HELP ME PLEASE I HAVE BEEN DOING THIS FOR TWO HOURS, I BEG OF YOU‍♀️‍♀️
BabaBlast [244]

Answer:

  • The first factor is itself the sum of two terms
  • The solution is the product of two factors

Step-by-step explanation:

The math symbols of the expression include parentheses, a plus sign, and a multiplication symbol. The order of operations tells you to do the "plus" operation inside parentheses first. The numbers involved in a "plus" operation, or sum, are called "terms." The result is their sum.

Then the second operation in the order of operations is to do the multiplication of the sum on the left of the multiplication symbol by the number to the right of the multiplication symbol. The numbers involved in multiplication are called "factors". The result of multiplication is called a "product."

Matching up this vocabulary with what you see, you realize ...

  • the first factor is the sum of two terms
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6 0
3 years ago
Raheem made 14 ounces of curry sauce which contained (among other things), 3.25 ounces of spices. He wants to make a larger batc
wolverine [178]

Answer:

19.5 ounces of spices

Step-by-step explanation:

Raheem made 14 ounces of curry sauce which contained (among other things), 3.25 ounces of spices. He wants to make a larger batch for a family dinner and decides he will need a total of 84 ounces of sauce altogether. How many ounces of spices will he need to complete his recipe?

The question above is calculated as:

14 ounces of sauce = 3.25 ounces of spices

84 ounces of sauce = x ounces of spices

Cross Multiply

14x = 84 × 3.25

x = 84 × 3.25/14

x = 19.5 ounces of spices

Hence, for 84 ounces of curry sauce he would need 19.5 ounces of spices

6 0
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