Consider the function f(x)=x^3-4x^2+2. Calculate the limit of the difference quotient at x0=3 for f(x).

Consider The Function F(x)=x^3-4x^2+2. Calculate The Limit Of The Difference Quotient At X0=3 For F(x).

Answers

Answer 1

The limit of the difference quotient of the above function [tex]f(x)[/tex] at [tex]x=3[/tex] is [tex]3[/tex] such that [tex]f(x)=x^{3} - 4x^{2} + 2[/tex].

Difference of quotient

The difference quotient of a function [tex]f(x)[/tex] is [tex]\frac{f(x+h)-f(x)}{h}[/tex].

How to evaluate the limit of the function?

The given equation is, [tex]f(x)=x^{3} -4x^{2} +2[/tex]

So, [tex]f(x+h)=(x+h)^{3} -4(x+h)^{2} +2= x^{3} +h^{3}+3x^{2} h+3xh^{2} -4x^{2} -4h^{2} -8xh+2[/tex]

Now, [tex]f(x+h)-f(x)[/tex]

[tex]=x^{3}+h^{3}+3x^{2}h+3xh^{2}-4x^{2}-4h^{2}-8xh+2-x^{3}+4x^{2}-2[/tex]

[tex]=h^{3}+3x^{2}h+3xh^{2}-4h^{2}-8xh[/tex]

So, [tex]\frac{f(x+h)-f(x)}{h} =\frac{h^{3}+3x^{2}h+3xh^{2} -4h^{2}-8xh }{h}[/tex]

[tex]=h^{2}+3x^{2}+3xh-4h-8x[/tex]

Now, at [tex]x=3[/tex],

[tex]h^{2}+3x^{2}+3xh-4h-8x=h^{2}+27+9h-4h-24=h^{2}+5h+3[/tex]

If   [tex]h[/tex]→[tex]0[/tex], the value of [tex]h^{2}+5h+3=3[/tex]

Thus, the limit of the difference quotient of the above function [tex]f(x)[/tex] at [tex]x=3[/tex] is [tex]3[/tex].

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Related Questions

A survey of high schools within a district revealed that for ninth graders, 38% offer no honors classes, 12% offer one
honors class, 25% offer two honors classes, 20% offer three honors classes, and 5% offer four honors classes. A
high school is selected at random. What is the probability that it offers an even number of honors classes?

0.30
O 0.32
O 0.62
O 0.68

Answers

Answer:

0.30

Step-by-step explanation:

Find the probability by adding the probabilities together for having two and four honors classes.

25% offer two honors classes and 5% offer four honors classes. Add these together:

25 + 5

= 30

So, there is a 30% probability that the school offers an even number of honors classes.

The correct answer is 0.30.

1) Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of -3. Then graph the line. 2) Write an equation in point-slope form of the line with slope -3/5 that contains(-10 ,8). Then graph the line.

Answers

Answer:

An equation in the slope-intercept form is:

y = a*x + b

Where a is the slope, and b is the y-intercept.

a)

Here we have a slope of 6 and a y-intercept of -3

Then the equation is:

y = 6*x - 3

Now we want to graph this.

To graph it, we first need to find two points (x, y) that belong to this equation, then we can graph the points, and connect them with a line.

To find the points, we evaluate in two different values of x.

x = 0

y = 6*0 - 3 = -3

Then we have the point (0, -3)

x = 1

y = 6*1 - 3 = 3

Then we have the point (1, 3)

The graph of this line can be seen in the image below (the red one)

b) Similar to before, here the slope is -3/5, then the equation is something like:

y = (-3/5)*x + b

Now we also know that the line passes through the point (-10, 8)

This means that when x = -10, we must have y = 8

Replacing these two in the equation we get:

8 = (-3/5)*-10 + b

8 = 6 + b

8 - 6 = 2 = b

Then this equation is:

y = (-3/5)*X + 2

The graph can be found in the same way as before, the graph of this function can also be seen in the image below (the green one)


74. A portion of a board has length x feet. The other part has
length (7x – 9) feet. Express the total length of the board
as a simplified expression in x.

Answers

Step-by-step explanation:

Given that,

Length of the one portion of the board = x feetLength of the another portion = (7x – 9) feet

According to the question,

[tex]\longrightarrow[/tex] Total length = Sum of the length of the two pieces

[tex]\longrightarrow[/tex] Total length = {x + (7x – 9)} feet

[tex]\longrightarrow[/tex] Total length = {x + 7x – 9} feet

[tex]\longrightarrow[/tex] Total length = (8x – 9) feet

Therefore, the total length of the board as a simplified expression in x is (8x 9) feet.

The diagram below is divided into equal parts. Which fraction of the parts is white?

A diagram is divided into 4 blue parts and 3 white parts.
Three-sevenths
Four-sevenths
Three-fourths
Four-thirds

Answers

Answer: This problem is a fraction since we have several equal parts that make up one whole. The problem asks us to talk about the relationship of white pieces to the whole. Since we know the whole is made up of 7 pieces (4 blue parts and 3 white parts = 7 total parts), then 7 will be our denominator (number on the bottom of the fraction).

Now that we have our number on the bottom, we need to look back at the question to carefully decide what parts of the whole we are looking at. The question wants to know how many of the parts are white. We know that 3 of the parts are white, so that is our numerator (number of the top of the fraction).

Our final answer is 3/7 or "three-sevenths." Said another way, three of the seven pieces are white.

Step-by-step explanation:

Cole biked at 5 mph for 1 1/2 hours. Which of the following choices show how far he biked?

Answers

Answer:

Should be 5 1/2 if thats on there

Step-by-step explanation:

u take 11/2 and take out the 1 u get 10/2 so u cut 10 in half get 5 then add the one and make it 5 1/2

5 1/2 jus took that question got it right.


An environmentalist would like to estimate the true mean weight of all cars. To do so, she selects a random sample of
30 cars and determines that the 90% confidence interval for the true mean weight to be 2.8 to 3.4 tons. Which of the
following would increase the margin of error for this confidence interval?
O selecting another sample
O increasing the sample size
O increasing the confidence level
O decreasing the confidence level

Answers

decreasing the confidence level

If the confidence level will increase, the margin of error will also increases.

What is margin of error?

The margin of error is defined a range of values below and above the sample statistic in a confidence interval.

What is confidence interval?

The confidence interval is a way to show what is uncertainty is with a certain statistic.

According to the given question

Environmentalist estimating true mean weight or all cars.

For the true mean weights of 2.8 to 3.4 tons the confidence level is 90%.

Since, the confidence level increases, the critical value increases and hence the margin of error increases.

Therefore, if the confidence level will increase, the margin of error will also increases.

Find out more information about confidence interval and margin error here:

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How many fourths are in 3 4 ?
O 4 Over 16
O One-half
O 3
O 4

Answers

Answer:

⭕ 3

hope it helps much

which is the best estimate of (-3/5) (17 5/6)?​

Answers

Answer:

-170 / 10

Step-by-step explanation:

Suppose that all of the points on the circular dartboard shown are equally likely to be hit by a dart. If the radius of the shaded center circle is 1 and the radius of the entire dartboard is 4, what is the probability of throwing a dart and hitting the white part of the board? Round your answer to the nearest whole number.


6%


34%


67%


94%

Answers

Answer:

94%

Step-by-step explanation:

There are 9 people in an office with 4 different phone lines. If all the lines begin to ring at once, how many groups of 4 people can answer these lines?

Answers

Two groups of four people can answer the lines

The quadratic function f(x) = -x2 - 6x - 8 is graphed.
What are the solutions of the quadratic equation 0 =
x2 - 6x - 8?
Ту
2
O 2 and 4
0-2 and 4
0-2 and 4
O: 2 and 4
6 5 A
-3
-1
X
-2
ون

Answers

Answer:

-2 and -4

Step-by-step explanation:

When the quadratic equation equals 0, that means when y = 0.

By looking at the graph, whenever y = 0 is when the graph crosses the x-axis.

The graph crosses the x-axis at only two points:

(-4, 0) and (-2, 0)

So the solutions to the graph when  -x^2 - 6x - 8 equals 0 is -2 & -4.

Hope it helps (●'◡'●)

Answer:

-2 and -4

Step-by-step explanation:

if you'll notice the function goes through the x axis at -2 and -4

sum of first n terms of an ap sequance is 3n2+5n.what is the sum of its first n+1 terms.find the sum of its first 10 terms.



Answers

Answer:

a) 3n^2 + 11n + 8

b) 350

Step-by-step explanation:

10) 3(10^2)+ 5*10 = 350

n+1) 3(n+1)^2 + 5(n+1)

        3(n^2 + 2n + 1) + 5n+5

         3n^2 + 6n+3 + 5n + 5

           3n^2 + 11n + 8

A meteorologist preparing a talk about global warming compiled a list of weekly low temperatures (in degrees Fahrenheit) he observed at his southern Florida home last year. The coldest temperature for any week was 36°F, but he inadvertently recorded the Celsius value of 2°. Assuming that he correctly listed all the other temperatures, explain how this error will affect these summary statistics:a) measures of center: mean and median.b) measures of spread: range, $IQR,$ and standard deviation.

Answers

Answer:

nr.herkyrsfdlufshfsyfs

Step-by-step explanation:

dsfsyfksutryrysyrslufzmfyzydzufmzmhfzl

hdhfuthfzhkrskyrsgj

CAN SOMEONE HELP ME ON ANALYZING DOT PLOTS!!!

Answers

Answer:

yes

Step-by-step explanation:

but I can't see them here

suppose △abc≅△xyz. what is the corresponding congruent part for each segment or angle?​

Answers

Answer:

Step-by-step explanation:

log2(6x) – log2 (x)-2

Answers

Answer:

xlog(64)−xlog(2)−2

Step-by-step explanation:

Simplify 6log(2) by moving 6 inside the logarithm.

log(2^6)x − log(2)x − 2

Raise 2 to the power of 6.

log(64)x − log(2)x − 2

Reorder factors in log(64)x − log(2)x −2.

The probability that a certain movie will win an award for acting is 0.15, the probability that it will win an award for direcing is 0.23, and the probability that it will win both is 0.09. Find the probabilities of the following.

a. The movie wins an award for acting, given that it wins both awards.
b. The movie wins an award for acting, given that it wins exactly one award.
c. The movie wins an award for acting, given that it wins at least one award.

Answers

Answer:

a) 0.15 / 0.09

b) 0.15 / 1

c) 0.15 / 0.23

i’ll give brainliest to the right answer

Answers

the answer is A. 0.0000805

Answer:

First one , 0.0000805

Step-by-step explanation:

With negative exponents the decimal is moved to the left the amount of the exponent. The spaces are filled with zeros.

With positive exponents the opposite occurs.  The decimal moves to the right.

Find the measure of angle C of a triangle ABC, if angle A=a and angle B= 2a.
*The answer is not 180-3a

Answers

3.14 - 3a Radians

Because
180o = 3.14 radian

The angle C of the triangle ABC is ( π - 3a ).

What is an angle?

The angle is defined as the span between two intersecting lines or surfaces at or close to the point where they meet.

The angle will be calculated as follows:-

We know that the sum of the angles of the triangle is 180 degrees or π in radians.

∠A + ∠B + ∠C = π

a + 2a + ∠C = π

∠C = π - a - 2a

∠C = π - 3a

Therefore angle C of the triangle ABC is ( π - 3a ).

To know more about an angle follow

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An automatic machine inserts mixed vegetables into a plastic bag. Past experience revealed that some packages were underweight and some were overweight, but most of them had satisfactory weight.
Weight % of Total Underweight 2.5 Satisfactory 90.0 Overweight 7.5a) What is the probability of selecting and finding that all three bags are overweight?b) What is the probability of selecting and finding that all three bags are satisfactory?

Answers

Answer:

a) 0.000016 = 0.0016% probability of selecting and finding that all three bags are overweight.

b) 0.729 = 72.9% probability of selecting and finding that all three bags are satisfactory

Step-by-step explanation:

The condition of the bags in the sample is independent of the other bags, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

a) What is the probability of selecting and finding that all three bags are overweight?

2.5% are overweight, which means that [tex]p = 0.025[/tex]

3 bags means that [tex]n = 3[/tex]

This probability is P(X = 3). So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 3) = C_{3,3}.(0.025)^{3}.(0.975)^{0} = 0.000016[/tex]

0.000016 = 0.0016% probability of selecting and finding that all three bags are overweight.

b) What is the probability of selecting and finding that all three bags are satisfactory?

90% are satisfactory, which means that [tex]p = 0.9[/tex]

3 bags means that [tex]n = 3[/tex]

This probability is P(X = 3). So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 3) = C_{3,3}.(0.9)^{3}.(0.1)^{0} = 0.729[/tex]

0.729 = 72.9% probability of selecting and finding that all three bags are satisfactory

If the measure of angle 2 is 920 and the measure of angle 4 is 2
what is the value of x?
14.
23

Answers

Step-by-step explanation:

the value of x will be 920

!!!!Plzzz help!!!!

For your initial post: Discuss your strategy for
establishing identities. Why do you think it is
usually preferable to start with the side
containing the more complicated expression
when establishing an identity?

Answers

Answer:

Yes it is very good for establishing identities.

Step-by-step explanation:

Since its a very preferable start it is a very good way to establish identity.

A boy leaves station X on a bearing of 035' to station Y. which is 21km away. He then travels to another station Z on a bearing of 125 degrees . If Z is directly East of X, what is the distance from X to his present position?​

Answers

9514 1404 393

Answer:

  36.6 km

Step-by-step explanation:

We assume the initial bearing of the boy is 35°. Then he will make a 90° turn to a heading of 125°. A diagram shows the distance of interest is the hypotenuse of a right triangle in which 35° is the angle opposite the side of length 21 km.

The relevant trig relation is ...

  Sin = Opposite/Hypotenuse

  sin(35°) = (21 km)/XZ

  XZ = (21 km)/sin(35°) ≈ 36.61 km

The distance from X to Z is about 36.61 km.

_____

The attached diagram has the angles measured in the usual way for a Cartesian plane: CCW from the +x axis. This will correspond to bearing measures if we relabel the axes so that +x is North, and +y is East.

Find two positive numbers whose product is 64 and whose sum is a minimum. (If both values are the same number, enter it into both blanks.) (smaller number) (larger number)

Answers

Answer:

Both the numbers are 8.

Step-by-step explanation:

Let the two numbers are p and 64/p.

The sum is given by

[tex]S = p +\frac{64}{p}\\\\\frac{dS}{dp}= 1 - \frac{64}{p^2}\\\\\frac{dS}{dp}=0\\\\\frac{64}{p^2}=1\\\\p= \pm 8[/tex]

So, the sum is minimum for p = 8 0r - 8, so the two numbers 8.

Determine the volume and the surface area of the three dimension figure

Answers

Answer:

Volume = 18 cm^3

Surface Area = 58 cm^2

Step-by-step explanation:

Find the volume with the formula V=w*h*l

W= width

H = height

L = length

W= 2cm

H= 1 cm

L= 9 cm

V= w*h*l

V= 2cm * 1 cm * 9cm

V= 18 cm^3

Find the surface area with the formula A= 2(w*l + h*l + h* w)

W= width

H = height

L = length

W= 2cm

H= 1 cm

L= 9 cm

A= 2(w*l + h*l + h* w)

A= 2(2cm*9cm + 1cm*9cm + 1cm* 2cm)

A= 2(29cm)

A= 58cm^2

Bindi is buying pet food with a coupon for 20% off. The original price of the bag of food is $39.
Which answer gives the best estimate for the price after the discount?

$48
$19
$34
$32

Answers

I think its $32.
.20 x 39 = 7.8
39 - 7.8 = $31.2

A water reservoir is shaped like a rectangular solid with a base that is 60 yards by 30 yards, and a vertical height of 30 yards. At the start of a three-month period of
no rain, the reservoir was completely full. At the end of this period, the height of the water was down to 6 yards. How much water was used in the three-month period?
How much water was used in the three-month period?


Please help :)

Answers

Answer:

43200 yd³

Step-by-step explanation:

The water reservoir is a rectangular solid that is a three dimensional shape with six quadrilateral faces (cuboid).

This reservoir has a base of 60 yards by 30 yards, and a vertical height of 30 yards. Therefore:

Volume of the reservoir = area of base * vertical height = 60 * 30 * 30 = 54000 yd³

This reservoir hence have a volume of 54000 yd³ when filled up with water.

After 3 months, the height of the water was down to 6 yards therefore the the volume is:

Volume after 3 months = area of base * vertical height = 60 * 30 * 6 = 10800 yd³

The amount of water used after 3 months = volume of water at beginning - volume of water after 3 months

The amount of water used after 3 months = 54000 - 10800 = 43200 yd³

Can you help me figure out this question I’ve been stuck on this for 20 minutes

Answers

Step-by-step explanation:

[tex]\dfrac{2x^2+x-6}{x+x-6} = \dfrac{(2x-3)(x+2)}{2(x-3)}[/tex]

what is the LCM of 2 Express on if there is no common factor​

Answers

Answer:

I started by dividing 2940 by the smallest prime that would fit into it, being 2. This left me with another even number, 1470, so I divided by 2 again. The result, 735, is divisible by 5, but 3 divides in also, and it's smaller, so I divided by 3 to get 245. This is not divisible by 3 but is divisible by 5, so I divided by 5 and got 49, which is divisible by 7.

​ solve it

[tex]4 \frac{1}{3} \times (3 \frac{1}{3} \times 3 \frac{1}{2} ) \ {}^{7} \div 9 \frac{3}{4} [/tex]
solve​ it fast

Answers

Answer:

[tex]\frac{257357187500}{19683}[/tex]

Step-by-step explanation:

We can convert these mixed fractions to ordinary fractions.

[tex]4(1/3)=\frac{(4*3)+1}{3}=\frac{13}{3}[/tex]

[tex]3(1/3)=\frac{10}{3}[/tex]

[tex]3(1/2)=\frac{7}{2}[/tex]

[tex]9(3/4)=\frac{39}{4}[/tex]

Then we have:

[tex]\frac{13}{3}*(\frac{10}{3}*\frac{7}{2})^{7}*\frac{4}{39}[/tex]

[tex]\frac{257357187500}{19683}[/tex]

I hope it helps you!

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Circle the letter in front of the correct meaning for each root word. Then, write two words that contain the root word. Bio A. Sea B. farC. life Pend A.one B. before C. hang path A. feelingB. fear C. allchron A.time B. fear C. study of portA. Carry B. out C. in Find the center and radius of the circle (x + 1)^2 + y^2 = 4 Of the travelers arriving at a small airport, 60% fly on major airlines, 20% fly on privately owned planes, and the remainder fly on commercially owned planes not belonging to a major airline. Of those traveling on major airlines, 50% are traveling for business reasons, whereas 70% of those arriving on private planes and 80% of those arriving on other commercially owned planes are traveling for business reasons. Suppose that we randomly select one person arriving at this airport. What is the probability that the persona. is traveling on business?b. is traveling for business on a privately owned plane?c. arrived on a privately owned plane, given that the person is traveling for business reasons?d. is traveling on business, given that the person is flying on a commercially owned plane? As a result of Stalin's Five-Year Plans, production skyrocketed in the Soviet Union. How much did the production of heavy machinery increase in the first Five Year Plan? A. Production quadrupled B. Production tripled C. Production increased by 75% D. Production doubled Indentify the structures A car travels 25 m/s for 30 seconds. What distance did the car travel ? A trapezoid has two basses that measure 11 cm and 8 cm. The height of the figure is 5 cm. What is the area of the trapezoid? A) 95 cm2. B) 64 cm2. C) 47.5 cm2. D) 24 cm2 If the function f is given by f(x)= 4x -3, find the value of f(2+h) list from soft to hardList the following minerals in order from softest to hardest:apatite, calcite, talc, quartz, diamond. Below, create a post, and in 3-5 complete sentences, explain how you would use a number line to solve the problem (-6) + 15. I need an essay in French of a typical school day Hey guys,I hope u r gonna answer this question fast,SI system is extended from of MKS system.Why? I will be waiting for the answer. Good luck thank u In Colorado Springs, which type of student/teenager is more likely to work at a fast food restaurant? Your friend Reggie needs to write a resume and asked for your help. In at least 150 words, create a guide for Reggie. What are the essential parts of the resume? What will he need to include? What will he need to avoid? Include information about how he should organize. Short story on helping old and disabled people Hey guys not good at math please help It is a letter in a form of picture writing.a. cuneiformc. bousterophedonb. hieroglyphics d. old roman Can someone help me with this 1. I . like tea.2. He . play football in the afternoon.3. You . go to bed at midnight.4. They.do the homework on weekends.5. The bus .arrive at 8.30 a.m.6. My brother .finish work at 8 p.m.7. Our friends . live in a big house.8. The cat . like me. Which decimal is equivalent to 48 over 100a. 0.048b. 0.48c. 4.08d. 4.8