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2 years ago

The recursive formula for an arithmetic sequence is:

Mathematics

1 answer:

gogolik [260]2 years ago

3 0

the answer should be B-4

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Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curve.

const2013 [10]

Answer:

$T(t) = ( -sin(t^2), cos(t^2) )\\\\N(t) = T'(t) / |T'(t) | = (-cos(t^2) , -sin(t^2))$

$T(t) =r'(t)/|r'(t)| = (2t/ \sqrt{4t^2 + 36} , -6/\sqrt{4t^2 + 36},0)$

$N(t) =T(t)/|T'(t)| = (3/(9 + t^2)^{1/2} , t/(9 + t^2)^{1/2},0)$

Step-by-step explanation:

Remember that for any curve $r(t)$

The tangent vector is given by

$T(t) = \frac{r'(t) }{| r'(t)| }$

And the normal vector is given by

$N(t) = \frac{T'(t)}{|T'(t)|}$

For this case, using the chain rule

$r'(t) = ( -10*2tsin(t^2) , 102t cos(t^2) )\\$

And also remember that

$|r'(t)| = \sqrt{(-10*2tsin(t^2))^2 + ( 10*2t cos(t^2) )^2} \\\\ = \sqrt{400 t^2*( sin(t^2)^2 + cos(t^2) ^2 })\\=\sqrt{400t^2} = 20t$

Therefore

$T(t) = r'(t) / |r'(t) | = ( -10*2tsin(t^2) , 10*2t cos(t^2) )/ 20t\\\\ = ( -10*2tsin(t^2)/ 20t , 10*2t cos(t^2) / 20t )\\= ( -sin(t^2), cos(t^2) )$

Similarly, using the quotient rule and the chain rule

$T'(t) = ( -2t cos(t^2) , -2t sin(t^2))$

And also

$|T'(t)| = \sqrt{ ( -2t cos(t^2))^2 + (-2t sin(t^2))^2} = \sqrt{ 4t^2 ( ( cos(t^2))^2 + ( sin(t^2))^2)} = \sqrt{4t^2} \\ = 2t$

Therefore

$N(t) = T'(t) / |T'(t) | = (-cos(t^2) , -sin(t^2))$

Notice that

1. $|N(t)| = |T(t) | = \sqrt{ cos(t^2)^2 + sin(t^2)^2 } = \sqrt{1} = 1$

2. $N(t)*T(T) = cos(t^2) sin(t^2 ) - cos(t^2) sin(t^2 ) = 0$

Simlarly

$r'(t) = (2t,-6,0) \\$

and

$|r'(t)| = \sqrt{(2t)^2 + 6^2} = \sqrt{4t^2 + 36}$

Therefore

$T(t) =r'(t)/|r'(t)| = (2t/ \sqrt{4t^2 + 36} , -6/\sqrt{4t^2 + 36},0)$

Then

$T'(t) = (9/(9 + t^2)^{3/2} , (3 t)/(9 + t^2)^{3/2},0)$

and also

$|T'(t)| = \sqrt{ ( (9/(9 + t^2)^{3/2} )^2 + ( (3 t)/(9 + t^2)^{3/2})^2 + 0^2 }\\= 3/(t^2 + 9 )$

And since

$N(t) =T(t)/|T'(t)| = (3/(9 + t^2)^{1/2} , t/(9 + t^2)^{1/2},0)$

6 0

3 years ago

Simplify the folllowing: (xy^3 z^4)

allsm [11]

Answer:

Remove parentheses. xy³ z^4

4Step-by-step explanation:

3 0

3 years ago

Read 2 more answers

12. A flat, square roof needs a square patch in the corner to seal a leak. The side length of

NISA [10]

Answer:

Step-by-step explanation:

Area of the good part= area of whole roof - area of bad part

Since the roof is a square roof, the area will be calculated using the formulae for area of a square

Area of a square =length²

Area of good part =(x+12)²-x²

A= {(x+12)(x+12)} - x²

A = (x²+12x+12x+144) -x²

Open the bracket and rearrange the equation

A= x²-x²+24x+144

A=(24x+144) ft

5 0

3 years ago

Could anyone help me find the value of x for this question?

kirza4 [7]

Can we get a picture

4 0

3 years ago

X and y are both bigger than 30. x and y have HCF 21 and LCM 1617. Find x and y.

Sunny_sXe [5.5K]

The values of x and y are 99 and 343

The HCF of numbers is the highest common factors of the numbers

The LCM of numbers is the lowest common multiple of the numbers

The HCF and the LCM are given as:

$HCF = 21$

$LCM = 1617$

Multiply the HCF and the LCM

$HCF \times LCM = 21 \times 1617$

$HCF \times LCM = 33957$

The product of the numbers x and y equals the product of the HCF and the LCM.

So, we have:

$x \times y = 33957$

<h3>Prime Factors</h3>

Express 33957 as a prime factor

$x \times y = 3^2 \times 7^3 \times 11$

Rewrite the equation as

$x \times y = 99 \times 7^3$

$x \times y = 99 \times 343$

By comparison:

$x = 99$

$y= 343$

Hence, the values of x and y are 99 and 343