Why should we learn the limit definition of the derivative when we could just learn shortcut rules for differentiation?

Answers

Answer 1

Answer:

Step-by-step explanation:

I think this is a very essential question to ask. While a lot of people want shortcuts to get things done(i used to too, when I was a student), shortcuts doesn't always cut it short. The reason is just as simple as that of any other subject that deals with theory.

Learning the limit definition gives ample knowledge about how the whole derivation works in general. Yes, quite alright, you can get the basic differentiation done when you don't learn that, but as someone who has gone far in calculus, I'd tell you for free, a time will come during computation where you'd get stuck because you never had that prior knowledge of definition.

Further along the line in calculus, you would be doing derivatives where you'd have to apply your knowledge of the definition of limit, and how would you be able get that done if you hadn't already learnt it?

Exactly!


Related Questions

consider the figure below. which of the following lines are parallel? justify your answer.

Answers

Answer:

1. r1 || r2 because of the same side or consecutive interior theorem

2. l1 || l2 because they are corresponding angles

The angle formed between two parallel lines having a common transversal have several common properties

The correct responses are;

1.  Lines r₁ and r₂ are parallel

2. Lines, l₁ and l₂ are parallel

lines r₁ and r₂, are parallel

The reason why the given lines are parallel are as follows:

1. The given figures are presented as shown in the attached diagram

From the diagram, we have that the same side interior angles formed by the common transversal l₂ the lines r₁ and r₂ are 63° and 117°, which are supplementary angles (63° + 117° = 180°)

Therefore, lines r₁ and r₂ are parallel given that same side interior angles formed by two parallel lines having the same transversal are supplementary

2. From the attached diagram, we have;

The alternate exterior angles formed between the lines, l₁ and l₂ that have the common transversal r₁ are equal, therefore;

Lines, l₁ and l₂ are parallel, given that the alternate exterior angles formed by two parallel lines having a common transversal are equal

From the diagram, we also have that the other angle of the linear pair angle with the 63° angle is 117°, and the second angle of the linear pair angle is both either corresponding angle and alternate interior angle with the given 117°

Therefore, lines r₁ and r₂, are parallel given that corresponding angles and the alternate interior angles formed between parallel lines having a common transversal are equal

Learn more about the angles formed between parallel lines having a common transversal here:

https://brainly.com/question/15937977

A car travels at an average speed of 48 miles per hour. How long does it take to travel 180 miles?

Answers

Answer:

Step-by-step explanation:

It will take 3hr and 36 min to travel 180 miles

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[tex]s = \frac{distance \: traveled}{time \: of \: the \: distance \: traveled} \\ [/tex]

[tex]48 = \frac{180}{t} \\ [/tex]

Multiply sides by t

[tex]48 \times t = \frac{180}{t} \times t \\ [/tex]

[tex]48t = 180[/tex]

Divided sides by 48

[tex] \frac{48}{48} t = \frac{180}{48} \\ [/tex]

[tex]t = \frac{4 \times 9 \times 5}{4 \times 3 \times 4} \\ [/tex]

[tex]t = \frac{3 \times 5}{4} \\ [/tex]

[tex]t = \frac{15}{4} \\ [/tex]

[tex]t = \frac{16 - 1}{4} \\ [/tex]

[tex]t = \frac{16}{4} - \frac{1}{4} \\ [/tex]

[tex]t = 4 - 0.25[/tex]

[tex]t = 3.75 \: \: \: \: hours[/tex]

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A remote control car travels at a rate of 10 feet per second. How many inches per second is this? (1 feet = 12 inches)

Answers

10 feet would be 120 inches

Today 4 friends went out for lunch. Their total bill was $24.80, including tax and gratuity. They decided to split the bill equally and each paid with a $10 bill. How much money will each person get back

Answers

Answer:

each person would getb3 dollars back

Step-by-step explanation:

since each friend pays $10 and there are 4 friends the total is 40 but now you have to do $40.00 minus $25.00 because of tax and gratuity so when you do that you get $15.00 now you have to do 15 ÷ 4 and that equals $3.75 and since you cant divide 75 coins to 4 people we dont add that but now you have your answer each friend gets $3.00

The price of a train ticket is 155.00 What would it be if you had a 75% discount

Answers

Answer:116.25

Step-by-step explanation:


It would be 38.75


Hope this helps ❤️

The expression −8x3(7x6 − 3x5) is equivalent to?

Answers

Answer:

-56x^9+24x^8

Step-by-step explanation:

3 1/3x(-3 3/5) divided by (-1/3)

Answers

Answer:

31/3

Step-by-step explanation:

31/3*(-33/5) divided by (-1/3)

10/3*(-15/5) *(-1/3)

2/3*(-5)*(-1)

= 10/3

What is a polynomial in factored form and standard form of degree 7 or greater with at least two positive zeros, at least one negative zero. ​

Answers

Answer:

if a polynomial of lowest degree p has zeros at x=x1,x2,…,xn x = x 1 , x 2 , … , x n , then the polynomial can be written in the factored form: f(x)=a(x−x1)p1(x−x2)p2⋯(x−xn)pn f ( x ) = a ( x − x 1 ) p 1 ( x − x 2 ) p 2 ⋯ ( x − x n ) p n where the powers pi on each factor can be determined by the behavior of the graph ...

Round 125000 to 3 decimals!

Answers

12.50000 is the answer

Answer:

1.25 is your answer, Since the zeros are just place holders

Step-by-step explanation:

hope this helps!

Have a good day.

Find each product mentally using the distributive property show the steps that you used.
1.8x71

2.6x83

Answers

Answer:

1.8 x 71 = 127.8, just 18 x 71 = 1278, then count the the decimal place from the right, i.e 127.8

same for 2.6 x 83 = 215.8

.......ok try 5.52 x 5

Answer:

27,579.24 OR 343.6

Step-by-step explanation:

I don't know if this is what you mean but 1.8x71x2.6x83=27,579.24

Also, (if that was not what you meant), 1.8x71=127.8, 2.6x83=215.8, 127.8+215.8=343.6

Guided practice (page 3)
A photo studio charges a $50 sitting fee and $25 per photo sheet purchased. The number of photo sheets purchased equals
spent plus
times the number of dollars
Hint
Submit Answer

Answers

Answer:0

Step-by-step explanation:0=0 he has no money

The equation that represents the total amount will be y = 25x + 50.

What is a linear equation?

A connection between two or more factors results in a linear model when displayed on a graph. The parameter will have a degree of one.

The linear equation is given as,

y = mx + c

Where c is the line's y-intercept and m is the line's y-slope.

A photo studio charges a $50 sitting fee and $25 per photo sheet purchased.

Let x be the number of photo sheets purchased and y be the total amount.

The slope of the equation is 25 and the y-intercept will be 50.

y = 25x + 50

The equation that represents the total amount will be y = 25x + 50.

More about the linear equation link is given below.

https://brainly.com/question/11897796

#SPJ2

Choose all the numbers divisible by 3

Answers

The choices are B and E

A dolphin jumped from 5 feet below the surface of the sea. Jane wrote −5 to represent the depth from which the dolphin jumped. What does 0 represent in this situation?

Answers

0 represent the sea surface since the dolphin jumped from five feet below the sea surface and since Jane put that the dolphin jumped at -5 below the sea surface that would mean that 0 must represent the sea surface.

Answer:

it represents the surface of the sea

Step-by-step explanation:

i did the test lol :)

A school dance sold 2 types of tickets: one for freshman/sophomores (9th/10th graders), and one for juniors/seniors (11th/12th graders). The freshman/sophomore tickets cost $8, and the junior/senior tickets cost $6. They sold a combined total of 525 tickets, and earned a total of $4600. How many tickets of each type did the school sell? Write and show all algebra work; you cannot use graphing to solve this.

Answers

9514 1404 393

Answer:

freshman/sophomore: 725junior/senior: -200

Step-by-step explanation:

It is convenient to let a variable represent the number of the highest-price tickets sold. We'll use x for that. Then the number of lower-price tickets sold is 525 -x and the total revenue is ...

  8x +6(525 -x) = 4600

  2x = 1450 . . . . . . . . . . . . subtract 3150

  x = 725

  525 -x = -200

__

The school sold 725 freshman/sophomore tickets and -200 junior/senior tickets.

_____

Comment on the question

There is apparently an error in the question somewhere. The average revenue per ticket is ...

  $4600/525 ≈ $8.76

You cannot sell $6 and $8 tickets and earn more than $8 per ticket sold. (Average revenue must be between $6 and $8 per ticket.)

__

If we assume a typo in revenue, and that it should be $3600, then the sales will be 225 freshman tickets and 300 senior tickets.

Are these ratios equivalent?
16 shipments every 6 years
8 shipments every 3 years
yes
no

Answers

Answer: Yes

Step-by-step explanation:

If you do 16/6 you get 2.667 and if you do 8/3 you also will get 2.667

5.) You are working for a car dealership. You are doing an inventory of the lot. You counted 34
trucks. 15 cars, and 8 SUVs. What is the ratio of SUVs to total vehicles on the lot?

Answers

8:49 should be the answer

Answer: 8 SUVs to 57 total

Step-by-step explanation: You add 34 + 15 + 8 which gives you 57 Then you see there are 8 SUVs. So the answer is 8 to 57.

Sorry. I kinda got ahead of myself and excited. Guess mistakes happen. I was right the first time. I'm just so tired right now

PLEASEEE HELP FAST! One solution each is given on four quadratic equations. Assuming that each quadratic equation has two solutions, what is the second solution for each equation?

Answers

Answer:

Step-by-step explanation:

If one solution of a quadratic equation is a complex number (a + bi),

Other solution of the equation will be the conjugate of the first solution.

So the other solution will be in the form of (a - bi)

If one solution is, x = -4 - 5i

Other solution will be, x = -4 + 5i

If one solution is, x = 4 + 5i

Other solution will be, x = 4 - 5i

If one solution is, x = 5 - 4i

Other solution will be, x = 5 - 4i

1. Given that f(x) = 4x2 + 10x + k ,find the value
of k when f(-1).
A. 6
c. 4
B. 3
D.2​

Answers

when it shows f(x) then f(-1) it just means substitute

0 = 4(-1)^2 + 10(-1) + k

-6 + k = 0

k = 6

it should be a


The ratio of students that wear glasses to students that do not wear glasses is 1:4. If there were a total of 30 students. How many students wear glasses?

Answers

Answer:

34

Step-by-step explanation:

Answer:

6

Step-by-step explanation:

For every 5 students, 1 wears glasses and 4 don't.  If you take the total number of students, 30, and divide it by 5, you have 6 groups of 5 students.  There will be 1 student for each of those groups wearing glasses, or 6 students.

HELP ASAPPPP
What is the equation of this graphed line? If the line passes through (-6,-3) and (6,-7)

Answers

Answer:

y= -1/3x - 5

Step-by-step explanation:

Answer:

The answer would be y=-1/3x-7.

Step-by-step explanation:

Using the formula y2-y1/x2-x1 to find the slope of the line, plug in the numbers and you will get a slope of -1/3. Now using the the formula y-y1=m(x-x1) and plug in the numbers, and you will get the equation y=-1/3x-7.

Sienna earns $108 for working 9 hours. Ted earns $77 for working 7 hours. Colby earns $92
for working 8 hours. Jennifer earns $115 for working 10 hours. Who has the highest hourly
wage?

Answers

Answer:

Sienna

Step-by-step explanation:

Red has already lost three iphones.He didn't spend a lot of money on his newest cellphone because he knows he will eventually lose that one too.Is it deductive or inductive reasoning?​

Answers

Answer:

deductive reasoning

Step-by-step explanation:

What is 5 tens + 5 tens in standard form

Answers

Answer:

19531250

Step-by-step explanation:

I hope it helps

IM A DUMB BLONDE AND I NEED YOUR HELP RIGHT NOW

Answers

Answer:

- 1 and 3

Step-by-step explanation:

All six digits in the number are even one digit is used twice all other digits are used once the ones digit is 4 times the hundreds digit the tens digit id the difference of the ones digit and hundreds digit the same digit is in the tens place and hundreds thousands place the thousands digit is not zero

Answers

Answer:

rffrefr

Step-by-step explanation:

write 9.57 in word from

Answers

Answer:

Nine Point Five Seven....

Step-by-step explanation:

Answer: nine and fifty-seven hundredths

Step-by-step explanation: the numbers to the right of the decimal ends with a ths.

Question 1
Manipulate the base area and the height of the pyramid, setting different values for each. Record the base dimensions, height, and the exact
volume for two kinds of pyramids: one with a square base and one with a rectangular base. Also calculate the decimal value of each volume, and
verify that it matches the volume displayed by the tool. (You might see some discrepancies in the tool due to rounding of decimals.)

Answers

Answer:

Step-by-step explanation:

what is 0945987123987634678 divided by 67899876543444445678

Answers

Step-by-step explanation:

1.23

hdhdjdjdhdhdhjdjdjdjd

Answer:35652846437467

Step-by-step explanation:simple math

Which diagram best shows how fraction bars can be used to evaluate 1/2 divided by 1/4

Answers

Answer:

first one

Step-by-step explanation:

it is what is it

help me please I will mark brainlyest​

Answers

Answer:

B

Step-by-step explanation:

The equation is y=mx+b

Where m is the slope

And B is the y point of the y-intercept

Answer:

The answer is b to this question

Other Questions
what came in the mail today The amount of money an employee earns is proportional to the number of hours worked. If the employee works 40 hours and earns $1,100, how much does the employee earn per hour? Read the passage from Beowulf.Beowulf, Ecgtheows son, replied:"Well, friend Unferth, you have had your sayabout Breca and me. But it was mostly beerthat was doing the talking. The truth is this:when the going was heavy in those high waves,I was the strongest swimmer of all.Wed been children together and we grew updaring ourselves to outdo each other,boasting and urging each other to riskour lives on the sea.Now read Lukas inference.Even though hes a warrior, Beowulf often uses diplomacy instead of rushing to anger and aggression.Which sentence from the passage best supports his inference? Which ordered pair is a solution for both ? write your favourite comic character Match the image with the correct adjective.Image 01, Red-haired man Image 02, Pretty panda mom and a monster; arrow pointing to panda Image 03, Blond girl, acting silly Image 04, Man and woman; arrow pointing to woman who is much shorter Match Term DefinitionImage 01 A) BonitaImage 02 B) BajaImage 03 C) PelirrojoImage 04 D) Rubia 9. Qu helado _____ gusta a ti? Please help! I'm very behind on work and just don't have the time to catch up without costing the grade of another class. Thank you so much! easy math - lots of points y=8x-18 find the intercepts of the line Sue graphed the formula for converting temperatures from Fahrenheit to Celsius.If the temperature is 50 degrees Fahrenheit, what is the temperature in Celsius? 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Obscuring the recognizable school-bus yellow exterior with ads and distracting other drivers present a safety problem around school buses that cannot be ignored, the group said in its position paper.9 National studies have shown that anything that diverts a drivers attention from the road can pose a hazard, opponents say.10 And a big yellow bus doesnt distract? Berry asked.11 Several years ago in Indiana, a Department of Education lawyer suggested that allowing advertisements on buses would invite First Amendment challenges by spurned advertisers, the group noted.12 Berry said the bill would limit the types of advertising that would be allowed.13 The regulations would prohibit ads that contain obscene or sexual material, or messages associated with gambling, tobacco products, alcohol, and political campaigns or causes. The ads could not promote drug use or any illegal activity or antisocial behavior. 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