Let f and g be the functions defined by f(t) = 2t2 and g(t) = t3 + 4t.
1) Determine f'(t) and g′(t).
2) Let p(t) = 2t2 (t3 + 4t) and observe that p(t) = f(t) ⋅ g(t). Re-write the formula for p by distributing the 2t2 term. Then, compute p′(t) using the sum and constant multiple rules.
3) p′(t) = f′(t) ⋅ g′(t).
A. True
B. False
4) Let q(t) = t3 + 4t2/2t2 and observe that q(t) = g(t)/f(t). Rewrite the formula for q by dividing each term in the numerator by the denominator and simplify to write q as a sum of constant multiples of powers of t. Then, compute q′(t) using the sum and constant multiple rules.
5) q′(t) = g′(t)/f'(t).
A. True
B. False

Answers

Answer 1

Answer:

1) [tex]f'(t)=4t,\ g'(t)=3t^2+4[/tex]

2) [tex]p(t) =2t^5+8t^3[/tex]

[tex]p'(t)=10t^4+24t^2[/tex]

3) False

4)[tex]q(t) =\dfrac{1}{2}t+2t^{-1}[/tex]

[tex]q'(t)=\dfrac{1}{2}-\dfrac{2}{t^2}[/tex]

5) False

Step-by-step explanation:

Given that:

[tex]f(t) = 2t^2[/tex] and [tex]g(t) = t^3 + 4t[/tex]

Formula:

[tex]1. \dfrac{d}{dx}x^n=nx^{n-1}[/tex]

[tex]2. \dfrac{d}{dx}C.f(x)=C.f'(x)\ \{\text{C is a constant}\}[/tex]

1) Using above formula:

[tex]f'(t)=2\times 2 t^{2-1}=4t[/tex]

[tex]g'(t)=3t^{3-1}+4\times 1 t^{1-1}=3t^2+4[/tex]

2) [tex]p(t) =2t^2(t^3+4t)[/tex]

Rewriting the formula by distributing the [tex]2t^2[/tex] term:

[tex]p(t) =2t^2.t^3+2t^2.4t=2t^5+8t^3[/tex]

[tex]p'(t) = 10t^4+24t^2[/tex]

3) By using answers of part (1):

[tex]f'(t).g'(t)=12t^3+16t[/tex]

[tex]p'(t) = 10t^4+24t^2[/tex]

Therefore it is False that [tex]p'(t) = f'(t).g'(t)[/tex]

4) [tex]q(t)=\dfrac{t^3+4t}{2t^2}[/tex]

Writing by distributing:

[tex]q(t)=\dfrac{t^3}{2t^2}+\dfrac{4t}{2t^2}\\\Rightarrow q(t) =\dfrac{t}{2}+\dfrac{2}{t}\\\Rightarrow q(t) =\dfrac{1}{2}t+2t^{-1}[/tex]

Using the formula:

[tex]q'(t)=\dfrac{1}{2}t^{1-1}+2\dfrac{-1}{t^2}\\\Rightarrow q'(t)=\dfrac{1}{2}-\dfrac{2}{t^2}[/tex]

(5)By using answers in part (1):

[tex]\dfrac{g'(t)}{f'(t)}=\dfrac{3t^2+4}{4t}=\dfrac{3}{4}t+\dfrac{1}t[/tex]

[tex]q'(t)=\dfrac{1}{2}-\dfrac{2}{t^2}[/tex]

Therefore, it is False that:

[tex]q'(t)=\dfrac{g'(t)}{f'(t)}[/tex]


Related Questions

determine the slope of the line given this two points (-2,4) and (1,6) with solution

Answers

Answer:

Slope = y2 - y1/x2 - x1

Slope = 6-4 / 1--2

Slope = 2/3

Step-by-step explanation:

hope that will help you

The slope of the line will be 2/3.

The formula for the slope will be:

= (y2 - y1) / (x2 - x1)

where,

y2 = 6

y1 = 4

x2 = 1

x1 = -2

We will slot the values into the equation which will be:

= (y2 - y1) / (x2 - x1)

= (6-4) / (1- -2)

= 2/(1+2)

= 2/3

Therefore, the slope is 2/3.

Read related link on:

https://brainly.com/question/24645618

WILL BE MARKED BRAINLYEST
Ray OJ bisects Angle LOK.

Measurement of Angle LOK = 82 degrees.

Solve for x.

Answers

Answer:

x = 14

Step-by-step explanation:

<LOJ = 3x

<KOJ = (2x + 12)°

<LOK = 82°

m<LOJ + m<KOJ = m<LOK (angle addition postulate)

3x + 2x + 12 = 82 (substitution)

5x + 12 = 82

5x = 82 - 12 (Subtraction property of equality)

5x = 70

x = 70/5 (division property of equality)

x = 14

Angle b= 61, angle c=57, and =2x what does x =

Answers

Answer:

can you provide more information

Step-by-step explanation:

Find 6/11/-3. Write in simplest form

Answers

Answer:

Bit confusing but I'll try my best.

Step-by-step explanation:

6/11 diving by -3.

-2/11.

Alex would like to buy one pack of colored pencils

Answers

Answer:

good for alex

Step-by-step explanation:

17. Create an equation that requires distribution on both sides to solve and has a solution of -2.

Answers

Answer:

googl really helps sorry

A radio station requires DJs to play 2 commercials for every 14 songs they play. What is the unit rate of songs to commercials? songs The unit rate is 1 commercial​

Answers

The correct answer for this question is that the unit rate is 5:1. We can work this out by simplifying down the 2 commercials to every 10 songs ration, so that 10:2 becomes 5:1. We can work this out by halving the original ratio to simplify it.

how many different choices of car does a person have if a particular model comes in 11 colors and 4 styles

Answers

Answer:

44 options

Step-by-step explanation:

In order to solve this problem you would simply need to multiply the options specifics together. In this scenario, these are the color and style. Therefore since the individual can choose from 11 colors and 4 styles you simply multiply 11 by 4. This is because every single color chosen has 4 styles to choose from,  meaning that all 4 styles also have 11 colors to choose from and vice versa.

11 * 4 = 44

Therefore, we see that an individual looking to buy a car has a total of 44 options to choose from.

Help me please I’m timed

Answers

Answer:

a=3 b=4

Step-by-step explanation:

a on the power of 3 is 8

3 on the power of 4 is 81

multiply them and you get 648

16) 56 is what percent of 40?
%

Answers

140

Step-by-step explanation:

56:40*100 =

(56*100):40 =

5600:40 = 140

Write a number that has a 5 that is 1/10 the value of the 5 in 5.348

Answers

Answer:

0.5

Step-by-step explanation:

5.348

5 is in ones place

5/10=0.5

faud's model racing care drives an average speed of 3 feet per second. Fuad records the distance and times in a table. At this rate, how long will it take the car to travel 21 feet?

Answers

Answer:

well 3 × 7 = 21

Step-by-step explanation:

1. 0+3=3

2. 3+3=6

3. 6+3=9

4. 9+3=12

5. 12+3=15

6. 15+3=18

7. 18+3=21

So the answer is 7

Factor completely.
16-49x^2=16−49x
2
=
(Can someone help me plz)

Answers

X=0,1
Pls mark brainliest

help me solve please.​

Answers

X=1/2 because if you set the equations equal to each other you would get that x-value

Shakira is baking cookies to put in packages for a fundraiser. Shakira has made 86 chocolate chip cookies and 42 sugar cookies.
Shakira wants to create identical packages of cookies to sell, and she must use all of the cookies.
What is the greatest number of identical packages that Shaquira can make?

Answers

Answer:

she can make 42 identical packages

Step-by-step explanation:

Answer:

Assuming there can only be 2 cookies per identical package, She can make 42 identical packages which contains 2 chocolate chip cookies and 1 sugar cookie.

86 / 2 = 43 chocolate chips cookies

42 / 1 = 42 sugar cookies

With the sugar cookies not having the capability to match the requirements we will assume for this answer that an "Identical Package" is composed of 2 chocolate chip cookies, and a single sugar cookie.

-I hope this is the answer you are looking for, please feel free to post your questions of brainly.com in the future.

Can anyone solve this please ?

Answers

The answer could be -2 or -2f

$20.00, 10% tax, 15% tip
Urgently helpp!

Answers

Answer:

Tax=$2

15% before tax= $3

15% after= $3.30

Answer:

I know the tip but can't help with tax

Can someone help please

Answers

L = 31
M = 90
N = 59

L + M + N = 180
(3x-5) + 90 + (5x-1) = 180

Combine

8x + 84 = 180
8x = 96
X = 12

Replace X with 12 for N and L

L = (3x -5) = 36 -5 = 31
N = (5x - 1) = 60 - 1 = 59

The list price of a toolbox is $127.35. The chain discount is 10/5. What is the final net price?

Answers

Answer:

[tex]125.35[/tex]

Step-by-step explanation:

[tex]The \: final \: net \: price \:is \: 127.35 - \frac{10}{5} =\\ 127.35 - 2 = 125.35[/tex]

if you gave someone an apple if you had 500 apples and then another 2x apples and then 3x apples next 456x how much left man?????????????????????????

Answers

Answer:

31

Step-by-step explanation:

Hope this helps!!!

Solve the logarithmic equation by writing in exponential form or by graphing.

log (7 − 2x) = −1

Answers

Answer:

10^-1 = 7 - 2x

Step-by-step explanation:

Segment addition pleas help in struggling

Answers

Answer:

x = -8

Step-by-step explanation:

using what's given, you can make the following equation:

3 + 2x + 22 = x + 17

3 + 22 + 2x - x = 17

25 + x = 17

x = 17 - 25

x = -8

Answer:

Step-by-step explanation:

3 + 2x + 22= x + 17

2x + 25 = x + 17

x + 25 = 17

x = -8

3 + 2(-8) + 22 = -8 + 17

25 - 16 = 9

9 = 9

Select the term that is suggested by the fugure

Answers

Answer:

im pretty sure its line segment if

Step-by-step explanation:

if its not im so sorry!!

Please help I’m in a hurry

Answers

Answer:

I believe it is A 20,

however id wait for another answer to confirm lol

A cause I believe and I am right

PLZ HELP THIS IS DUE SOON!!!!!!! Divide. Round to the nearest hundredth if necessary.
To the nearest hundredth, 80.88÷8.425 is_______

Answers

Answer:

10

Step-by-step explanation:

when you divide 80.88 by 8.425 you get 9.6

and since youre rounding, you can round up 6 because its higher than 5

and your final answer should be 10.

hoped this helped! :)

Express this number in scientific notation. 53,010,00053,010,000

Answers

Answer:

[tex]5.30100005301 * 10^{15}[/tex]

in abook 5/8 of the pages have pictures on them given that 70 pages have at least one picture on them work out the total numb in the pages​

Answers

Answer:

112 pages

Step-by-step explanation:

Setup a ratio of 5/8 to 70/x and use the butterfly method to solve the equation and get 112. (I know I'm late)

347 is what in expanded form

Answers

Answer:

300+40+7

Step-by-step explanation:

Pls help i dont understand :( will give Brainliest + 25 Points

Answers

Answer:

A

Step-by-step explanation:

plug the points into the slope formula

slope = (y2 - y1)/(x2 - x1)  Then you get the slope of -0.5

(5 - -1))/(-6-6)

6/-12

-0.5

Plug a point in to the equation of y = -0.5x + b

-1 = -0.5(6) +b

2 = b

Solve and show your work.

12-[2 + 6 ÷ (10 - 2³)]

Answers

Answer:

7

Step-by-step explanation:

[tex]2^{3} = 8[/tex]  Plug 8 in: 12 - [2 + 6 ÷ (10 - 8)]10 - 8 = 2Plug 2 in: 12 - [2 + 6 ÷ (2)]6 ÷ 2 = 3Plug 3 in: 12 - [2 + 3]2 + 3 = 5Plug 5 in: 12 - 512 - 5 = 7

I hope this helps!

Other Questions
Can someone do my ela assignment What is the value of -8(17-12)/-2(8-(-2)) 48x3 +128x2 75x 200 factor As a seed plant grows, the roots grow downward due to the force of Not sure if you can help but please try!ScenarioYou look outside your bedroom window and see clear, sunny skies and calm weather. You overhear a weather forecast on the news that says a high pressure system will be in the area for the next few days. You are wondering what your friend who lives in Kentucky was going to do this week. You decide to text her and check it out. You were surprised to learn that it was raining where she lived and could turn into an ice/snow storm later. No school tomorrow for her!!!! LUCKY!!!External DataYou are located at the yellow X on the map below. Your friend is located at the black X on the map belowPrompt:Write a scientific explanation that predicts what the weather would be like in your area and your friends area over the next few days. Which statement describes thylakoids?Glucose is formed here.Water molecules are formed here.Only photosystem II functions here.Photosystems I and II function here. Generally accepted accounting principles require companies to use only one factor overhead rate for product costing True or False? What should you consider when choosing evidence for your literary analysis essay? (Choose all that apply.)A- Have I described the evidence accuratelyB- Is the evidence historically importantC- Is the evidence relevant to my claim?D- Will the evidence make my essay interesting? What is the major threats to Philippines' aquatic biomes? As a student how can you help in preventing or combating this/these threat/s? One atom k40 has how many neutrons A place where two or more bones come together On the graph Y= -3x+4? How can a candidate win the popular vote and lose the election? When citing a source you should use*A.an exclamation point B.quotations C.a semi colon or colonD. a period at the end of the sentence Which of the following is a natural source of electricity?A deep minesB ligtheningC coalD car batteries Any natural number and 0 means? A basic characteristic of American democracy is the right to vote, because it allows citizens to - EASY POINTS READ HE PASSAGE AND ANSWER THE QUESTION ^^ Help please fast!!!!1.The factored form of a quadratic function is f(x)=(x-p)(x-q).The standard form of a quadratic function is fX)=ax^2+bx+c. The factoredform for a quadratic function tells us the x-intercepts of the quadratic function, while the standard form for a quadratic function tells us theyintercept of the quadratic function.If p=2 and q=-3, use what you have learned about multiplying polynomials to1) write the factored form of the quadratic function f(x)2) write the standard form for the quadratic function f(x) answer for immediate brainliest