find the second derivative of the parametric equation x(t)=sin t and y(t)=cos t​

Answers

Answer 1

Presumably you mean the second derivative of y with respect to x, d²y/dx².

Compute the first derivative. By the chain rule,

[tex]\dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex]

Differentiate the two parametric equations with respect to t :

[tex]x = \sin(t) \implies \dfrac{dx}{dt} = \cos(t)[/tex]

[tex]y = \cos(t) \implies \dfrac{dy}{dt} = -\sin(t)[/tex]

Then the first derivative is

[tex]\dfrac{dy}{dx} = \dfrac{-\sin(t)}{\cos(t)} = -\tan(t)[/tex]

Now, dy/dx is a function of t, so we can denote it by, say, dy/dx = f(t). Then by the chain rule, the second derivative will be

[tex]\dfrac{d^2y}{dx^2} = \dfrac{df}{dx} = \dfrac{df}{dt} \times \dfrac{dt}{dx} = \dfrac{\frac{df}{dt}}{\frac{dx}{dt}}[/tex]

Differentiating f(t) :

[tex]f(t) = -\tan(t) \implies \dfrac{df}{dt} = -\sec^2(t)[/tex]

Then the second derivative is

[tex]\dfrac{d^2y}{dx^2} = \dfrac{-\sec^2(t)}{\cos(t)} = \boxed{-\sec^3(t)}[/tex]

and since y = cos(t), we can go on to say

[tex]\dfrac{d^2y}{dx^2} = -\dfrac1{y^3}[/tex]


Related Questions

PLEASE HELP ILL GIVE FIVE STARS AND GIVE BRAINLIEST AND GIVE THANKS PLEASE HELP
Ray is x years old. His brother Ron is y years older than Ray.
What will the difference of their ages be in 5 years?

Answers

ray is x years old

Ron is x+y years old

in 5 years ron will be 5 years older than now, so x+y+5 years old

 orAnswer:

x - x + y + 5 - 5 = y  or   x + y - x = y

Step-by-step explanation:

The difference now,

x + y - x = y

The difference in 5 years,

(x + y + 5) - (x + 5) = x - x + y + 5 - 5

x - x + y + 5 - 5 = y

The difference in their ages never changes

This figure represents a garden box that is to be filled with soil.

How much soil will it take to fill the garden box?

Enter your answer in the box.

Answers

Answer:

1728 ft³

Step-by-step explanation:

Lower part:

volume = 19 ft × 12 ft × 6 ft = 1368 ft³

Upper part:

volume = 10 ft × 12 ft × 3 ft = 360 ft³

Total Volume = 1728 ft³

Find the area of a square with a slide length of 7/12

Answers

[tex]\text{Area of square} = \text{Side}^2 = \left(\dfrac{7}{12}\right)^2 = \dfrac{49}{144}=0.3402\overline{7}~ \text{Units}^2[/tex]

In a discount basis loan, interest is "pald" at the time the loan is made by deducting the Interest amount from the loan
amount.
“True
False”

Answers

Answer:

true

hope that helps

Step-by-step explanation:

Create a table, chart, or diagram that you can use to help you decide which hypothesis test you need to use (selecting from: population proportion, a population mean, two proportions from independent samples, two means from dependent samples, and two means from independent samples). Explain your table, chart, or diagram so that someone new to statistics could use it to select the appropriate hypothesis test.

Answers

A hypothesis testing is a statistical inference that uses data from a sample to draw conclusions.

What is hypothesis testing?

It should be noted that hypothesis testing simply means a statistical act where an assumption is made regarding a parameter.

In this case, a hypothesis testing is a statistical inference that uses data from a sample to draw conclusions. The null hypothesis predicts no effect while the alternative hypothesis states the research predictions.

Learn more about hypothesis on:

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Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=−7

y=x^2, x=y^2

Answers

Answer:

X=23>45_78
X=2^3< /7= 8
Y= 8x^
comclucion :
coronita pliss

(c) 111 IT IIISH (Find the value of):
6
E (4n -2)
1 =2

Answers

Answer: 70

Step-by-step explanation:

Suppose f(x) = x². Find the graph of f(x-4)
4
Click on the correct answer.
1
2 3
Click on each graph to enlarge it.
graph 1
graph 2
graph 3
graph 4
What’s the answer ?

Answers

Answer:

  1

Step-by-step explanation:

Translation of a function h units to the right is accomplished by replacing x in the function definition with (x-h).

Here, x has been replaced by x-4, so we expect to see the graph look like the original function graph shifted 4 units to the right.

That matches Graph 1.

Please help me on Mathematics

✨Nonsense answer Report✨
✨Rude answer Report ✨

✨ Correct answer Brainlest ✨
✨ Complete answer Brainlest ✨

Answers

Answer:

1. 20 km/hr (60km / 3h)

2. 70 m/hr (170m / 2h)

3. 6.5 hr (260cm / 40km)

4. 4.5 s (72cm / 16cm)

5. 184 km (46km * 4h)

Step-by-step explanation:

We'll use d = distance, t = time, s = speed.

Speed over time can be found by dividing distance by time

s = d / t

We can take this formula and manipulate it to find either distance or time if we know two of the values. Keeping in mind division and multiplication cancel each other.

If we wanted to find d for instance we can multiply both sides by t to get

s * t = d

For time we can divide both sides of the above formula by s to get

t = d / s

Answer:

1. 20 km/h

2. 70 m/h

3. 6.5 h

4. 4.5 s

5. 184 km

Step-by-step explanation:

For first two questions, you can use the following formula:

[tex]\displaystyle \large{s = \frac{d}{t}}[/tex]

I denote ‘s’ be speed, ‘d’ be distance and ‘t’ be time. Some books may use ‘v’ for speed instead of ‘s’ but I use ‘s’ variable instead so it makes the equation easier to memorize.

Therefore, speed = distance/time

( 1 ) The first question is given both distance be 60 km and time be 3 hours. Therefore:

[tex]\displaystyle \large{s=\frac{60}{3}}\\\displaystyle \large{s=20 \ \ \sf{km / h}}[/tex]

So for the first question, the speed is 20 km/h

( 2 ) The second is given both distance be 140 m and time be 2 hours. Therefore:

[tex]\displaystyle \large{s=\frac{140}{2}}\\\displaystyle \large{s=70 \ \ \sf{m / h}}[/tex]

Therefore, speed is 70 m/h

For the third and four questions, we will be finding time. From the speed formula:

[tex]\displaystyle \large{s=\frac{d}{t}}[/tex]

Multiply both sides by t.

[tex]\displaystyle \large{st=d}[/tex]

Divide both sides by s.

[tex]\displaystyle \large{t=\frac{d}{s}}[/tex]

Therefore, time = distance/speed

( 3 ) The question is given speed be 40 km/h and distance be 260 km. Therefore:

[tex]\displaystyle \large{t=\frac{260}{40}}\\\displaystyle \large{t=\frac{13}{2} \ \ \sf{h}}\\\displaystyle \large{t = 6\frac{1}{2} \ \ \sf{h}}\\\displaystyle \large{t= 6.5 \ \ \sf{h}}[/tex]

Therefore, time is 6.5 h

( 4 ) The question is given speed be 16 cm/s and distance be 72 cm. Therefore:

[tex]\displaystyle \large{t=\frac{72}{16}}\\\displaystyle \large{t=\frac{9}{2} \ \ \sf{s}}\\\displaystyle \large{t=4\frac{1}{2} \ \ \sf{s}}\\\displaystyle \large{t=4.5 \ \ \sf{s}}[/tex]

Therefore, time is 4.5 s

For the last question, we have to find distance. From the speed formula:

[tex]\displaystyle \large{s=\frac{d}{t}}[/tex]

Multiply both sides by t.

[tex]\displaystyle \large{st=d}\\\displaystyle \large{d=st}[/tex]

Therefore, distance = speed * time

( 5 ) The question is given time be 4 hours and speed be 46 km/h. Therefore:

[tex]\displaystyle \large{d=46 \cdot 4}\\\displaystyle \large{ d=184 \ \ \sf{km}}[/tex]

Therefore, distance is 184 km.

Question: If [tex]$f(x)=\cos \left[2 \tan ^{-1}\left(\sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)\right]$[/tex]
Options:
[tex](a) $f^{\prime}(x)(x-1)^{2}-2(f(x))^{2}=0$\\\(b) $f^{\prime}(x)(x-1)^{2}+2(f(x))^{2}=0$\\(c) $f^{\prime}(x)(x+1)^{2}+2(f(x))^{2}=0$\\(d) $f^{\prime}(x)(x+1)^{2}-2(f(x))^{2}=0$[/tex]

Answers

Note that √((1 - x)/x) is defined only as long as 0 < x ≤ 1.

Consider a right triangle with reference angle θ such that

[tex]\cot(\theta) = \sqrt{\dfrac{1-x}x}[/tex]

In other words, on an appropriate domain,

[tex]\theta = \cot^{-1}\left(\sqrt{\dfrac{1-x}x}\right)[/tex]

In such a triangle, you would find that

[tex]\sin(\theta) = \sqrt x[/tex]

so f(x) reduces a bit to

[tex]f(x) = \cos\left(2 \tan^{-1}(\sqrt x)\right)[/tex]

Now consider another triangle with reference angle ɸ such that

[tex]\tan(\phi) = \sqrt x \implies \phi = \tan^{-1}(\sqrt x)[/tex]

In this triangle, you would find

[tex]\cos(\phi) = \dfrac1{\sqrt{1+x}}[/tex]

Recalling the double angle identity for cosine, it follows that

[tex]f(x) = \cos(2\phi) = 2 \cos^2(\phi) - 1 = \dfrac{1-x}{1+x}[/tex]

Differentiating with respect to x yields

[tex]f'(x) = -\dfrac2{(1+x)^2}[/tex]

while

[tex]f(x)^2 = \dfrac{(1-x)^2}{(1+x)^2}[/tex]

It follows that

[tex]f'(x)(x-1)^2 - 2f(x)^2 = 0[/tex]

so B is the correct choice.

Due in a MINUTE!!! Need help!

Answers

Answer:

GH = 15

Step-by-step explanation:

In a trapezoid, the length of the median is one-half the sum of the lengths of the bases. Therefore:

[tex]9x - 3 = \frac{1}{2} (19 + 5x + 1)[/tex]

[tex]18x - 6 = 5x + 20[/tex]

[tex]13x = 26[/tex]

[tex]x = 2[/tex]

[tex]9(2) - 3 = 18 - 3 = 15[/tex]

what types of atoms are there in a cell

Answers

Answer:

carbon (C), hydrogen (H), nitrogen (N), and oxygen (O)—make up 96.5% of an organism's weight.

Step-by-step explanation:

hope this helps.

H because it’s a metal source all the giving power products and produce


6. The angles of triangle ABC measure 30 degrees, 40 degrees, and 110 degrees. Will
its circumcenter fall inside the triangle, on the triangle, or outside the
triangle? Explain your reasoning.

Answers

Answer: Outside the triangle

Step-by-step explanation:

Because one of the angles has a measure greater than 90 degrees, we know that the triangle must be obtuse. Therefore, the circumcenter must fall outside the triangle because in an obtuse triangle, the circumcenter is always located outside of the triangle.

Round your answer to the nearest hundredth.

Answers

Answer:

33.69°

Step-by-step explanation:

Hope this helps! Please let me know if you need more help or think my answer is incorrect. Brainliest would be MUCH appreciated. Also remember to rate answers to help other students, While leaving thanks to answers that help you. Have a great day!

Answer:

33.69°

Step-by-step explanation:

Let's take ∡CBA as y

Tan y =  [tex]\frac{opposite}{adjacent}[/tex]

         =  [tex]\frac{6}{9}[/tex]

         = 0.6666

y = 33.69°

Find the measure of each angle in the isosceles trapezoid.

Answers

Answer:

J=118

L=62

M=62

Step-by-step explanation:

In an isoseles trapezoid, the top angles are congruent to each other and the bottom angles are congruent to each other. Opposite angles are supplementary, and all angles together add up to 360.

3. Evaluate the following:
a.) What value of Syields the coordinate point (1, 3)?
b.) What value of Syields the coordinate point (3, 1)?
c.) x(710)=
d.) y(V10+ V5)= = B С
e.) x(2)=
f.) y(2.5)= ​

Answers

Answer: See below

Step-by-step explanation:

Question 3:

[tex]a) \ $S=1$ \ yields \ $(1,3)$ \\ \\b) \ $S=3$ \ yields \ $(3,1)$ \\ \\ c) \ $x(\sqrt{10})=\sqrt{10}$ \\ \\d) $y(\sqrt{10}+\sqrt{5})=1$ \\ \\e) $x(2)=2$ \\ \\f) $y(2,5)=1$ \\[/tex]

Challenge:

[tex]\begin{aligned}&x(s)=s \quad ,s \in[0,4] \\&y(s)=\left\{\begin{array}{cc}3 s & , s \in[0,1] \\5-2 s &, s \in[1,2] \\1 & s \in[2,4]\end{array}\right.\end{aligned}[/tex]

Write an algebraic expression for the following word phrase. The quotient of 69 and x

Answers

Hello!

_____

The quotient of two numbers means we divide one number by another number.

Example :-

The quotient of 6 and a number:-

6÷x

According to the example above, the algebraic expression looks like so:-

69÷x

note:-

Hope everything is clear; if you need any explanation/clarification, kindly let me know, and I will comment and/or edit my answer :)

How to solve problems in different ways and how to do them correct and fast so that way I kneo

Answers

Answer:

[tex]Hi!~I~am~beinggreat78~and~I~will~help~you.\\[/tex]

To solve problems, you first must find the source of the conflict, and attempt to let the situation play out, or use a strategy yourself. To solve math problems, go by PEMDAS.

Parentheses 1stExponents 2ndMultiply 3rdDivide 4thAdd 5thSubtract 6th

With this strategy, PEMDAS, you can solve math problems quickly, and like a pro!

Can someone please help me with this

Answers

Answer:

1, 4, 6.

Step-by-step explanation:

[tex]\left(2n+8\right)\left(n+4\right)=2n^2+16n+32[/tex]

1. [tex]2(x+4)^2=2n^2+16n+32[/tex]

2. [tex]2\left(n+4\right)\left(n+4\right)=2(n+4)^2[/tex]

3. [tex]16n+32=16n+32[/tex]

4. [tex]2n^2+16n+32=2n^2+16n+32[/tex]

5. [tex]2n^2+8n+32=2n^2+8n+32[/tex]

6. [tex]2(n^2+8n+16)=2n^2+16n+32[/tex]

As you can see from the answers it is 1, 4, and 6.

Alan bought 120 pens at the cost of €2 per pen. He sold 72 of them for €2.50 per pen and the remaining at the price of €2 per pen. Find his percentage profit or loss.

Answers

Answer:

His profit was 15%

Step-by-step explanation:

120*2€ = 240€

72*2.5€ = 180€

120-72 = 48

48*2€ = 96€

And now a simple proportion:

240€ = 100%

276€ = x

x = (276*100)/240 %

x = 115%

115%-100% = 15%

His profit was 15%

PLEASE HELP ILL GIVE FIVE STARS AND GIVE BRAINLIEST AND GIVE THANKS PLEASE HELP
Ray is x years old. His brother Ron is y years older than Ray.
What will the sum of their ages be in 5 years?

Answers

Answer: X+Y+5

Step-by-step explanation:

There is no single answer you have to write an equation.

When K is negative the parabola _____

Answers

Answer:

Is shifted downwards

Step-by-step explanation:

K determines where the parabola is on the y axis basically. If the k value was positive it would be shifted upwards and if it was negative it would be shifted downwards.

use the equation f(x)=0.003x²-0.083x+2.514. Predict the value for x+20 years.
When x=20 years,f(x)=?

Answers

Answer:

2.054

Step-by-step explanation:

put the value of x in f(x)

-28;a)__; -20; -16;b)__



help me please​

Answers

Answer:

It ether -36 or a-28

Convert the unit of length. Round the answer to one decimal place if necessary. 170 cm ≈ _____ ft ?

Answers

Answer:

5.56 ft

Step-by-step explanation:

30.48 cm per foot

170 cm ÷ 30.48 cm/ft = 5.56 ft

Hope this helps!

If you were to roll two dice, what is the probability that the sum of the numbers showed on the dice is greater than 5?

Answers

The probability that an outcome is greater than 5 is the likelihood of the outcome

The probability that the sum of the numbers showed on the dice is greater than 5 is 13/18

How to determine the probability?

The sample size of a roll of two dice is:

n(S) = 36

The number of outcomes that have a sum greater than 5 is:

Greater than 5 = 26

So, the required probability is:

P = 26/36

Simplify

P = 13/18

Hence, the probability that the sum of the numbers showed on the dice is greater than 5 is 13/18

Read more about probability at:

https://brainly.com/question/251701

precio antes de impuestos 1563.60 impuesto 5.4%

Answers

Answer:

precio antes De impuestos 1563.60 impuestos 5.4%

The lengths of two sides of a triangle are 10 and 24, and the third side is x. How many whole number values are possible for x.

Answers

Answer:

[tex]20[/tex].

Step-by-step explanation:

In any triangle, the sum of the lengths of any two sides should be strictly greater than the length of the third side. For example, if the length of the three sides are [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex]:

[tex]a + b > c[/tex],

[tex]a + c > b[/tex], and

[tex]b + c > a[/tex].

In this question, the length of the sides are [tex]10[/tex], [tex]24[/tex], and [tex]x[/tex]. The length of these sides should satisfty the following inequalities:

[tex]10 + 24 > x[/tex],

[tex]10 + x > 24[/tex], and

[tex]24 + x > 10[/tex].

Since [tex]x > 0[/tex], the inequality [tex]24 + x > 10[/tex] is guarenteed to be satisfied.

Simplify [tex]10 + 24 > x[/tex] to obtain the inequality [tex]x < 35[/tex].

Similarly, simplify [tex]10 + x > 24[/tex] to obtain the inequality [tex]x > 14[/tex].

Since [tex]x[/tex] needs to be a whole number, the greatest [tex]x[/tex] that satisfies [tex]x < 35[/tex] would be [tex]34[/tex]. Similarly, the least [tex]x\![/tex] that satisfies [tex]x > 14[/tex] would be [tex]15[/tex]. Thus, [tex]x\!\![/tex] could be any whole number between [tex]15\![/tex] and [tex]34\![/tex] (inclusive.)

There are a total of [tex]34 - 15 + 1 = 20[/tex] distinct whole numbers between [tex]15[/tex] and [tex]34[/tex] (inclusive.) Thus, the number of possible whole number values for [tex]x[/tex] would be [tex]20[/tex].

7, 10, 13, 23, 36, 55, 67
Pt A
Which of these puts the correct classification of the numbers as prime or composite?
A) Prime: 10, 13, 36, 67
Composite:7, 23, 55
B) Prime:7, 15, 55, 67
Composite:10, 36, 23
C)Prime:10, 36, 55
Composite:7, 13, 23, 67
D) Prime:7, 13, 23, 67
Composite:10, 36, 55
Pt B
Which of these are composite numbers?
A)11 and 12
B)28 and 29
C)31 and 32
D)44 and 45

Answers

Answer:

Pt A: D. Pt B: D

Step-by-step explanation:

You can multiply 5x2 which easily equals 10. 6x6 equals 36 and you cannot multiply anything other than one and itself for 7, 13, 23, or 67. for part B, you can multiply 4x 11 to equal 44 and you can multiply 9x5 to equal 45 so it would be composite

44 and 45 are composite numbers.

What are prime and composite numbers?

A prime number has only two factors, itself and one.

A composite number has at least three factors, and sometimes a lot more.

1.  Prime:7, 13, 23, 67

Composite:10, 36, 55

As, 7, 13, 23 and 67 have only two factors : one and number itself

and 10, 36, 55 have more than 2 factors like 2.

2. 44 and 45 both have more than 2 factors.

Learn more about this concept here:

https://brainly.com/question/303182

#SPJ5

Stacy buys a bagel assortment that has 6 oat bagels for every 2 blueberry bagels. Naomi buys a bagel assortment that has 9 oat bagels for every 5 blueberry bagels. Each girl bought 10 blueberry bagels.

Stacy’s Bagels
Oat 6 12 18 24 30 36
Blueberry 2 4 6 8 10 12

Naomi’s Bagels
Oat 9 18 27 36 45 54
Blueberry 5 10 15 20 25 30

Find the number of oat bagels that each girl bought.

Stacy bought
oat bagels, and Naomi bought
oat bagels.

Answers

Answer: Stacy: 30 Naomi: 18

Step-by-step explanation:

Stacy- ratio of oat to blueberry is 6:2 we want to find x:10

10 is 5 times 2, and whatever you do to the bottom you have to do to the top, and 6 times 5 is 30


Naomi- ratio of oat to blueberry is 9:5 we want to find x:10

The 10 is 2 times 5, and whatever you do to the bottom you have to do to the top, and 9 times 2 is 18

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