Mathematics College

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

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

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

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.

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

Answers

Answer:

googl really helps sorry

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!!

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

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

Answers

Answer:

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

16) 56 is what percent of 40?
%

Answers

140

Step-by-step explanation:

56:40*100 =

(56*100):40 =

5600:40 = 140

$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

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

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.

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

Answers

Answer:

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

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:

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.

Answers

Answer:

Bit confusing but I'll try my best.

Step-by-step explanation:

6/11 diving by -3.

-2/11.

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

Alex would like to buy one pack of colored pencils

Answers

Answer:

good for alex

Step-by-step explanation:

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)

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:

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:

Step-by-step explanation:

Hope this helps!!!

Can anyone solve this please ?

Answers

The answer could be -2 or -2f

Solve and show your work.

12-[2 + 6 ÷ (10 - 2³)]

Answers

Answer:

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!

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

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