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2 years ago

Use finite differences to identify the degree of the polynomial that best describes the data. 9 10 11 12 13 14

1 answer:

4 0

The forward differences for this data is 1, 1, 1, 1, 1, 1 (since 10 - 9 = 1, 11 - 10 = 1, etc). Since we only need one iteration of differences, a linear polynomial will fit the data exactly.

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Evaluate the integral, show all steps please!

Aloiza [94]

Answer:

$\displaystyle \int \dfrac{1}{(9-x^2)^{\frac{3}{2}}}\:\:\text{d}x=\dfrac{x}{9\sqrt{9-x^2}} +\text{C}$

Step-by-step explanation:

Fundamental Theorem of Calculus

$\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))$

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

$\displaystyle \int \dfrac{1}{(9-x^2)^{\frac{3}{2}}}\:\:\text{d}x$

Rewrite 9 as 3² and rewrite the 3/2 exponent as square root to the power of 3:

$\implies \displaystyle \int \dfrac{1}{\left(\sqrt{3^2-x^2}\right)^3}\:\:\text{d}x$

Integration by substitution

$\boxed{\textsf{For }\sqrt{a^2-x^2} \textsf{ use the substitution }x=a \sin \theta}$

$\textsf{Let }x=3 \sin \theta$

$\begin{aligned}\implies \sqrt{3^2-x^2} & =\sqrt{3^2-(3 \sin \theta)^2}\\ & = \sqrt{9-9 \sin^2 \theta}\\ & = \sqrt{9(1-\sin^2 \theta)}\\ & = \sqrt{9 \cos^2 \theta}\\ & = 3 \cos \theta\end{aligned}$

Find the derivative of x and rewrite it so that dx is on its own:

$\implies \dfrac{\text{d}x}{\text{d}\theta}=3 \cos \theta$

$\implies \text{d}x=3 \cos \theta\:\:\text{d}\theta$

Substitute everything into the original integral:

$\begin{aligned}\displaystyle \int \dfrac{1}{(9-x^2)^{\frac{3}{2}}}\:\:\text{d}x & = \int \dfrac{1}{\left(\sqrt{3^2-x^2}\right)^3}\:\:\text{d}x\\\\& = \int \dfrac{1}{\left(3 \cos \theta\right)^3}\:\:3 \cos \theta\:\:\text{d}\theta \\\\ & = \int \dfrac{1}{\left(3 \cos \theta\right)^2}\:\:\text{d}\theta \\\\ & = \int \dfrac{1}{9 \cos^2 \theta} \:\: \text{d}\theta\end{aligned}$

Take out the constant:

$\implies \displaystyle \dfrac{1}{9} \int \dfrac{1}{\cos^2 \theta}\:\:\text{d}\theta$

$\textsf{Use the trigonometric identity}: \quad\sec^2 \theta=\dfrac{1}{\cos^2 \theta}$

$\implies \displaystyle \dfrac{1}{9} \int \sec^2 \theta\:\:\text{d}\theta$

$\boxed{\begin{minipage}{5 cm}\underline{Integrating $\sec^2 kx$}\\\\$\displaystyle \int \sec^2 kx\:\text{d}x=\dfrac{1}{k} \tan kx\:\:(+\text{C})$\end{minipage}}$

$\implies \displaystyle \dfrac{1}{9} \int \sec^2 \theta\:\:\text{d}\theta = \dfrac{1}{9} \tan \theta+\text{C}$

$\textsf{Use the trigonometric identity}: \quad \tan \theta=\dfrac{\sin \theta}{\cos \theta}$

$\implies \dfrac{\sin \theta}{9 \cos \theta} +\text{C}$

$\textsf{Substitute back in } \sin \theta=\dfrac{x}{3}:$

$\implies \dfrac{x}{9(3 \cos \theta)} +\text{C}$

$\textsf{Substitute back in }3 \cos \theta=\sqrt{9-x^2}:$

$\implies \dfrac{x}{9\sqrt{9-x^2}} +\text{C}$

Learn more about integration by substitution here:

brainly.com/question/28156101

brainly.com/question/28155016

4 0

2 years ago

Pls Help meeeee UWU (im gonna die)

Ronch [10]

Answer:

Step-by-step explanation:

When you sub in the numbers you get 20 divided by 2

6 0

4 years ago

What is the difference between a centroid, orthocenter, and a circumcenter?

Lisa [10]

Answer:

Take an example in a triangle:

The orthocenter is the intersection point of 3 altitudes.

The centroid is the intersection point of 3 medians.

The circumcenter is the intersection point of 3 bisectors.

Centroid, orthocenter, and circumcenter are collinear.

Hope this helps!

4 0

3 years ago

9.3488 rounded to the nearest tenth?

Tanya [424]

Note the place values:

9 ones place value

. (decimal point)

3 tenths place value

4 hundredths place value

8 thousandths place value

8 ten thousandths place value

You are solving for the tenth place value. Look at the digit directly right of the tenths place value. It is the hundredths place value, which is 4.

Note that when rounding, if the number next to what you are solving for, has a value of 5 or greater, you round up. If it has a value of 4 and less, you round down. In this case, it is a 4, so round down.

9.3488 rounded to the nearest tenth is 9.3

9.3 is your answer.

4 0

3 years ago

Read 2 more answers

Need help solving this

Alex_Xolod [135]

Answer:

15,625 feet and either 56.469 seconds or 87.719 seconds

Step-by-step explanation:

strap in because we need calculus buddy. we're given the displacement equation but we want velocity because at the bullets max height it will pause briefly before coming down and this implies zero velocity. derivative of that equation is -32t+1000. set this equal to zero and find t to see how long this takes. 31.25 seconds to reach zero velocity (the tip of the parabolic motion). now plug that t back into the displacement equation to find height after 31.25 seconds and it's like 15,625 which is ridiculously high. now it has to come down so gravity will be taking over here so it has it's own special equations and we know the equation for that is like x=1/2at^2 or something. we know x is 15,625. let's find time it takes to get down by solving that for t. 15,625÷4.9=t^2 so t=56.469 or something. now its unclear if they want the entire time elapsed from firing the gun or just the falling to ground time so if it's the entire time let's go ahead and add that initial 31.25 seconds to the 56.498 seconds so like 87.719 total seconds.

3 0

3 years ago