Use the definition of the derivative to find the derivative of the function. Show your work by completing the four-step process. (Simplify your answers completely for each step.) f(x) = Step 1: Step 2: Step 3: Step 4: f'(x) = lim h→0 Step 1: Step 2: X + 9 Step 3: Step 4: [-/0.2 Points] Use the definition of the derivative to find the derivative of the function. Show your work by completing the four-step process. (Simplify your answers completely for each step.) f(x)=√x + 8 f(x + h) = f(x +h)-f(x) = f(x +h)-f(x) h DETAILS f'(x) = lim h→0 f(x +h)-f(x) = h f(x + h) = f(x +h)-f(x) = f(x+h)-f(x) h (Express your answer as a single fraction.) f(x+h)-f(x) h (Rationalize the numerator.)

Answers

Answer 1

The derivative of the function f(x) = √x + 8 is f'(x) = 1 / (2√x).

To find the derivative of the given function using the definition of the derivative, we follow the four-step process:

Step 1: Set up the difference quotient:

f'(x) = lim h→0 [f(x + h) - f(x)] / h

Step 2: Substitute the function into the expression:

f'(x) = lim h→0 [√(x + h) + 8 - (√x + 8)] / h

Step 3: Simplify the numerator:

f'(x) = lim h→0 [√(x + h) - √x] / h

Step 4: Rationalize the numerator by multiplying the numerator and denominator by the conjugate of the numerator:

f'(x) = lim h→0 [√(x + h) - √x] / h * [√(x + h) + √x] / [√(x + h) + √x]

Simplifying further:

f'(x) = lim h→0 [(x + h) - x] / [h(√(x + h) + √x)]

f'(x) = lim h→0 h / [h(√(x + h) + √x)]

f'(x) = lim h→0 1 / (√(x + h) + √x)

Taking the limit as h approaches 0, we find:

f'(x) = 1 / (√x + √x) = 1 / (2√x)

Therefore, the derivative of the function f(x) = √x + 8 is f'(x) = 1 / (2√x).

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Answer 2

The derivative of the function f(x) = √x + 8 is f'(x) = 1 / (2√x).

To find the derivative of the given function using the definition of the derivative, we follow the four-step process:

Step 1: Set up the difference quotient:

f'(x) = lim h→0 [f(x + h) - f(x)] / h

Step 2: Substitute the function into the expression:

f'(x) = lim h→0 [√(x + h) + 8 - (√x + 8)] / h

Step 3: Simplify the numerator:

f'(x) = lim h→0 [√(x + h) - √x] / h

Step 4: Rationalize the numerator by multiplying the numerator and denominator by the conjugate of the numerator:

f'(x) = lim h→0 [√(x + h) - √x] / h * [√(x + h) + √x] / [√(x + h) + √x]

Simplifying further:

f'(x) = lim h→0 [(x + h) - x] / [h(√(x + h) + √x)]

f'(x) = lim h→0 h / [h(√(x + h) + √x)]

f'(x) = lim h→0 1 / (√(x + h) + √x)

Taking the limit as h approaches 0, we find:

f'(x) = 1 / (√x + √x) = 1 / (2√x)

Therefore, the derivative of the function f(x) = √x + 8 is f'(x) = 1 / (2√x).

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Related Questions

Better taxi service charges $3.00 for the first mile plus $0.20 for each additional mile. Best
taxi service charges $6.00 for the first two miles plus $0.10 for each additional mile. How
many miles will a person travel in order to spend an equal amount of money using either
service?
PLEASE AN EXPERT THIS IS WORTH 50 POINTS!!!!!!!!!

Answers

8 miles i think because this and that are $3.00 plus $0.20 so it will be 8 mile to equal the other taxi service

Equations
6) 2x + 3y = -1
3x + 4y = -4

Answers

Answer:

x = -8

y = 5

Step-by-step explanation:

2x = -3y-1

x = (-3y-1)/2

3x + 4y = -4

3[(-3y-1)/2]+4y = -4

(-9y-3/2)+4y = -4 (x2 all)

-9y-3+8y = -8

-9y+8y = -8+3

-y = -5

y = 5

3x+4.5 = -4

3x+20 = -4

3x = -24

x = -8

checking:

2.(-8)+3.5 = -1

-16+15 = -1 (checked)

A student is evaluating the expression below for n=3.7.12.8+6(n-0.5) in what order should the student do the steps?

Answers

Answer:

Step-by-step explanation:

The given expression is :

12.8+6(n-0.5)

We need to evaluate the expression for n = 3.7

Put n = 3.7 in the above expression.

12.8+6(3.7-0.5)

= 12.8+6(3.2)

= 12.8 + 19.2

= 32

So, the value of the given expression is equal to 32.

Answer all the questions Question One a. Show the equations for calculating 1. Bulk Volume of a reservoir in ft 3 and barrels 2 . Pore Volume of a reservoir in ft 3 and barrel 3 . Hydrocarbon Pore Volume in ft 3 and in barrel.

Answers

The equations for Bulk Volume of a reservoir in ft³ is VB = A*h and in barrels is VB = (A*h) / 5.615. The equations for Pore Volume of a reservoir in ft³ is VP = φ*VB and in barrels is VP = (φ*VB)/5.615. The equations for Hydrocarbon Pore Volume in ft³ is VHC = φ*S*VB and in barrels is VHC = (φ*S*VB)/5.615.

The equations for calculating the bulk volume, pore volume, and hydrocarbon pore volume of a reservoir are as follows:

1. Bulk Volume (VB):

In cubic feet (ft³):VB = A * hIn barrels (bbl):VB = (A * h) / 5.615

Where:

VB = Bulk Volume

A = Cross-sectional area of the reservoir in square feet (ft²)

h = Thickness of the reservoir in feet (ft)

2. Pore Volume (VP):

In cubic feet (ft³):VP = φ * VBIn barrels (bbl):VP = (φ * VB) / 5.615

Where:

VP = Pore Volume

φ = Porosity of the reservoir (dimensionless)

VB = Bulk Volume

3. Hydrocarbon Pore Volume (VHC):

In cubic feet (ft³):VHC = φ * S * VBIn barrels (bbl):VHC = (φ * S * VB) / 5.615

Where:

VHC = Hydrocarbon Pore Volume

φ = Porosity of the reservoir (dimensionless)

S = Saturation of hydrocarbons in the reservoir (dimensionless)

VB = Bulk Volume

The conversion factor from cubic feet (ft³) to barrels (bbl) is 5.615.

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Emmanuel sections off a part of the backyard of his house to build the rectangular stone patio for his grill. The length of the base will be 3 feet more than the width. Let the width be defined as x. Write an expression for the length in terms of the width.

Answers

Answer: x + 3feet

Step-by-step explanation:

From the question, Emmanuel sections a part of his house backyard in order to build the rectangular stone patio for his grill. We are told that the length of the base is going to be 3 feet more than the width and the width has been defined as x.

Since the length of the base will be 3 feet more than the width and the width is x. Therefore, the length will be:

Length = x + 3feet

HELPPPPP
Find the perimeter. Simplify your answer.
40 +4cd+12
-502-5cd+2
c-d2+1
3cd+4d2-30

Answers

Answer:

Step-by-step explanation:

−

500

−

What percent of 24 seconds to 4 hours?

Also can you add them label it would help a lot.

:)

Answers

24 seconds is approximately 0.17% of 4 hours. The percentage of 24 seconds to 4 hours, we can start by converting both values to the same units.

To find the percentage of 24 seconds to 4 hours, we can start by converting both values to the same units. One hour is equal to 3600 seconds, so 4 hours is equal to:

4 hours * 3600 seconds/hour = 14,400 seconds

Now we can express the ratio of 24 seconds to 14,400 seconds as a fraction and convert it to a percentage:

(24 seconds / 14,400 seconds) * 100% ≈ 0.17%

Therefore, 24 seconds is approximately 0.17% of 4 hours.

Here's how you could label the calculation:

24 seconds is what percent of 4 hours?

24 seconds / 14,400 seconds = 0.00167 or 0.167%

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please help me answer this question asap

Answers

Answer:

It's quite easy

Step-by-step explanation:

people less than 30 years = frequency of people 0 to 15 + 15 to 30 = 8+15 =23

Therefore there are 23 people less than 30 years old.

pls mark me as brainliest pls.

Consider points R, S, and T. Point S is in the middle. Point R is below and to the left of point S. Point T is below and to the right of point S. Which statement is true about the geometric figure that can contain these points? No line can be drawn through any pair of the points. One line can be drawn through all three points. One plane can be drawn so it contains all three points. Two planes can be drawn so that each one contains all three points.

Answers

Answer:

Answer:One plane can be drawn so it contains all three points.

Step-by-step explanation:

Took the test :P

One plane can be drawn, so it contains all three points. Then the correct option is C.

What is coordinate geometry?

Coordinate geometry is the study of geometry using the points in space.

Consider points R, S, and T. Point S is in the middle.

Point R is below and to the left of point S. Point T is below and to the right of point S.

Let the coordinate of the S is at origin that is (0, 0)

Then the coordinate of R and T with respect to S will be (-x, -y) and (x, -y).

One plane can be drawn, so it contains all three points.

Then the correct option is C.

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Consider points R, S, and T. Point S is in the middle. Point R is below and to the left of point S. Point

Combine like terms 3x-7x=

Answers

Answer:

-4x

Step-by-step explanation:

3-7=-4 just add a x

Alexa is ordering a taxi from an online taxi service. The taxi charges $4 just for the pickup and then an additional $1.25 per mile driven. How much would a taxi ride cost if Alexa is riding for 9 miles? How much would a taxi ride cost that is mm miles long? what is the eaquation for m miles

Answers

Answer:

Alex's ride is $15.25

The equation to find this is

1.25m + 4 = cost

m = miles traveled

Subtract 8x-6 from 6x-9 as a algebraic expression

Answers

Answer:

2x + 3

Step-by-step explanation:

Step 1: Write expression

8x - 6 - (6x - 9)

Step 2: Distribute negative

8x - 6 - 6x + 9

Step 3: Combine like terms (x)

2x - 6 + 9

Step 4: Combine like terms (constants)

2x + 3

Review Questions
1. Cindy is a baker and runs a large cupcake shop. She has already

a. How many workers will the firm hire if the market wage rate is

hired 11 employees and is thinking of hiring a 12th. Cindy esti- $27.95 ? \$19.95? Explain why the firm will not hire a larger or mates that a 12 th worker would cost her $100 per day in wages $ smaller number of units of labor at each of these wage rates. and benefits while increasing her total revenue from $2,600per. day to $2,750 per day. Should Cindy hire a 12 th worker? b. Show this firm Explain. L016.2 c. Now again determine the firm's demand curve for labor. Complete the following labor demand table for a firm that is assuming that it is selling in an imperfectly competitive marhiring labor competitively and selling its product in a competiket and that, although it can sell 17 units at $2.20 per unit, it tive market. L016.2 ginal product of each successive labor unit. Compare this demand curve with that derived in part b. Which curve is more elastic? Explain. 3. Alice runs a shoemaking factory that uses both labor and capital to make shoes. Which of the following would shift the factory's demand for capital? You can select one or more correct answers from the choices shown. LO16.3 a. Many consumers decide to walk barefoot all the time. b. New shoemaking machines are twice as efficient as older machines. c. The wages that the factory has to pay its workers rise due to an economywide labor shortage.

Answers

Cindy should hire the 12th worker as it would result in a net increase in profit, with additional revenue exceeding the cost of hiring. Insufficient information is provided to determine the demand curve for labor or compare its elasticity. Events that would shift the factory's demand for capital include new, more efficient machines and rising wages due to a labor shortage.

a. To determine whether Cindy should hire a 12th worker, we need to compare the additional revenue generated with the additional cost incurred. Hiring the 12th worker would increase total revenue by $150 ($2,750 - $2,600) per day, but it would also increase costs by $100. Therefore, the net increase in total profit would be $50 ($150 - $100). Since the net increase in profit is positive, Cindy should hire the 12th worker.

b. By hiring the 12th worker, Cindy can increase her total revenue from $2,600 per day to $2,750 per day. The additional revenue generated by the 12th worker exceeds the cost of hiring that worker, resulting in a net increase in profit.

c. To determine the firm's demand curve for labor, we need information about the marginal product of labor (MPL) and the wage rates. Unfortunately, this information is not provided, so we cannot complete the labor demand table or derive the demand curve for labor.

Without specific data or information about changes in the quantity of labor demanded and wage rates, we cannot determine which demand curve (from part b or c) is more elastic. The elasticity of the demand curve depends on the responsiveness of the quantity of labor demanded to changes in the wage rate.

The events that would shift the factory's demand for capital are:

a. New shoemaking machines being twice as efficient as older machines would increase the productivity of capital. This would lead to an increase in the demand for capital as the factory would require more capital to produce the same quantity of shoes.

b. The wages that the factory has to pay its workers rising due to an economy-wide labor shortage would increase the cost of labor relative to capital. This would make capital relatively more attractive and lead to an increase in the demand for capital as the factory may substitute capital for labor to maintain production efficiency.

The event "Many consumers decide to walk barefoot all the time" would not directly impact the demand for capital as it is related to changes in consumer behavior rather than the production process of the shoemaking factory.

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Use the Order of Operations to evaluate this expression
(-2+1square2) +5 (12/3) -9
Type the correct answer in the box. Use numerals instead of words.

Answers

(-2+1^2)+5(12/3)-9
(1)+5(4)-9
1+20-9
21-9
12

The answer when simplified gives the expression as 11.

What is the BODMAS rule?

According to the BODMAS rule, the brackets have to be solved first followed by powers or roots (i.e. of), then Division, Multiplication, Addition, and at the end Subtraction. Solving any expression is considered correct only if the BODMAS rule or the PEMDAS rule is followed to solve it.

Given here the expression as -2+1²×2+5×12/3 -9=0+4×5-11

= 11

Hence, The answer when simplified gives the expression as 11.

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A triangle has two sides of length 9 and 4. What is the largest possible whole-number length
for the third side?

Answers

Answer:

Step-by-step explanation:

What is the importance of polynomial functions and how this real life applications improve or contribute to the value of life

Answers

Answer:

What is the importance of polynomial functions?

Polynomials are an important part of the "language" of mathematics and algebra. They are used in nearly every field of mathematics to express numbers as a result of mathematical operations. Polynomials are also "building blocks" in other types of mathematical expressions, such as rational expressions.

How these real-life applications improve or contribute to the value of life?

Since polynomials are used to describe curves of various types, people use them in the real world to graph curves. Combinations of polynomial functions are sometimes used in economics to do cost analyses, for example. Engineers use polynomials to graph the curves of roller coasters and bridges.

If anyone can solve all 3 questions that would be amazing thanks

Answers

Answer:

Step-by-step explanation:

a) y=x²-4x+10

$x^2-4x+10=\\\\x^2-2(x)(2)+2^2-2^2+10=\\\\(x^2-2(x)(2)+2^2)-2(2)+10=\\\\(x-2)^2-4+10=\\\\(x-2)^2+6\\\\Hence \ y=(x-2)^2+6$

b) y=x²+8x+9

$x^2+8x+9=\\\\x^2+2(x)(4)+4^2-4^2+9=\\\\(x^2+2(x)(4)+4^2)-4^2+9=\\\\(x+4)^2-16+9=\\\\(x+4)^2-7\\\\Hence \ y=(x+4)^2-7$

c) y=x^2-22y+124

$x^2-22x+124=\\\\x^2-2(x)(12)+12^2-12^2+12^2=\\\\x^2-2(x)(12)+12^2=\\\\(x-12)^2\\\\Hence \ y=(x-12)^2$

A rectangular block of candy 10 inches by 10 inches by 5 inches is coated on all faces with a very thin layer of chocolate. The block of candy is then cut into cubes measuring 1 inch by 1 inch by 1 inch. What percent of the cubes have no chocolate on them

Answers

38.2% of the cubes are without chocolate coated.

Calculation

Given that,

The length ,l= 10 inch

The width ,b=10 inch

The height, h=5 inch

We know that the lateral surface area of the cuboid or rectangular block,

SA= 2(lh+lb+hb)

⇒ SA = 2( 10*5 +10*10+5*10)

⇒ SA =2*200=400 $inch^{3}$

The volume of the cuboid,V= 10*10*5 =500 $inch^{3}$

The block of candy is then cut into another small cube 1 inch per side.

length,l=1 inch

width, b=1 inch

height,h=1 inch

So, the surface area of each small cube ,sa= 2(1+1+1) = 6 $inch^{3}$

Volume of each candy,v= 1*1*1=1 $inch^{3}$

So, the total number of candy= 500/1 =500 nos.

There is chocolate on each and every piece on the exterior inch. But those who are totally within do not.

A cube measuring 8 inches by 8 inches by 3 inches makes up the inside.

So, the total no of boxes that have no chocolate =8*8*3=192 boxes

So, the percentage = 192*100/500= 38.2 %

Therefore, it is concluded that 38.2% of cubes are without chocolate.

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what does it mean by give your answer in 'terms of pi' ??
please help me and thanks

Answers

Therefore, the area of the quarter circle of radius 8 cm is 16π square centimeters (or approximately 50.2655 square centimeters if we use a decimal approximation for π).

What is circle?

A circle is a two-dimensional shape that is defined as the set of all points that are a fixed distance (called the radius) from a central point (called the center) in a plane. It is one of the most basic geometric shapes and is often used in a variety of mathematical and scientific contexts. One important thing to note about circles is that they are symmetrical: any line passing through the center of a circle divides the circle into two halves that are mirror images of each other. Circles are commonly used in a variety of contexts, including geometry, trigonometry, physics, engineering, and many other fields. They are also used in everyday life, such as in the design of wheels and other circular objects, the calculation of the areas of circular fields or gardens, and the measurement of circular objects like plates and bowls.

Here,

The formula for the area of a quarter circle is given by:

A = (1/4)πr²

where A is the area of the quarter circle, r is the radius of the circle, and π is the mathematical constant pi (approximately 3.14159).

In this case, the radius of the quarter circle is 8 cm, so we can substitute this value into the formula and simplify as follows:

A = (1/4)π(8 cm)²

A = (1/4)π(64 cm²)

A = 16π cm²

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19 Translate into an equation. (TEKS: A5A)
Five times a number decreased by 2 is 28.
A 5n+ 2 = 28
B 5n = 28
C 5n -2=28
D 5n + 2 = 2

Answers

Answer:

your answer is C. 5n-2=28

Step-by-step explanation:

hope this helps you ☺️

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A bookmark has a perimeter of 50 centimeters. its area is 136 square centimeters. what are the dimensions of the bookmark?

Answers

Answer:

l + w = 25 and lw = 136

w(25 - w) = 136

25w - w² = 36

w² - 25w - 136 = 0

(w - 17)(w - 8) = 0

w = 8, so l = 17

The dimensions of the bookmark are 8 cm by 17 cm.

Bonjour pouvez vous m'aider svp c'est urgent !! Calculer la valeur de chaque expression pour a=3 et b=10: a+b= a-b= -7 ab= a/b= a²= 2b

Answers

Bonjour!

a+b=13
a-b=-7
ab= 30
a/b=0.3
a²=9
2b= 20

Je ne suis pas tout à fait sûr, mais j'espère que cela vous aidera.

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
The slope of the line shown in the graph is _____
and the y-intercept of the line is _____ .

$Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction$

Answers

The slope of the line shown in the graph is __2/3__

and the y-intercept of the line is __6___

How to find the slope and the y-intercept?

The general linear equation is written as follows:

y = ax + b

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

On the graph we can see that the y-intercept is y = 6, then we can write the line as:

y = ax + 6

The line also passes through the point (-9, 0), replacing these values in the line we will get:

0 = a*-9 + 6

9a = 6

a = 6/9

a = 2/3

That is the slope.

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PLEASE HELP!!! I’ll give brainlest

Answers

The answer to your question is g(x)= -3f(x).

Use the figure to find the exact value of the trigonometric function

Answers

In given figure we have right angled triangle,

with

Hypotenuse=5

perpendicular=3

Base=4

Apply the trignometric expression for tanΘ

$\tan \emptyset=\frac{Perpendicular}{\text{base}}$

Substitute the value of Perendicular and base,

$\tan \emptyset=\frac{3}{4}$

Multiply bothe side by arc(tan)

$\begin{gathered} \arctan (\tan \emptyset)=\arctan \frac{3}{4} \\ \emptyset=\arctan \frac{3}{4} \end{gathered}$

The trignometric expression for tan2Θ is:

$\tan 2\emptyset=\frac{2\tan \emptyset}{1-\tan ^2\emptyset}$

Put the value of Θ in the tan2Θ

$\begin{gathered} \tan 2\emptyset=\frac{2\tan (\arctan \frac{3}{4)}}{1-\tan ^2(\arctan \frac{3}{4})} \\ \tan 2\emptyset=\frac{2\frac{3}{4}}{1-(\frac{3}{4})^2} \\ \tan 2\emptyset=\frac{\frac{6}{4}}{1-\frac{9}{16}} \\ \tan 2\emptyset=\frac{\frac{6}{4}}{\frac{16-9}{16}} \\ \tan 2\emptyset=\frac{\frac{6}{4}}{\frac{7}{16}} \\ \tan 2\emptyset=\frac{6}{4}\times\frac{16}{7} \\ \tan 2\emptyset=\frac{6\times4}{7} \\ \tan 2\emptyset=\frac{24}{7} \end{gathered}$

ANSWER : (D). 24/7

The exact value of the trigonometric function is tan2θ = 15/8.

What are Trigonometric functions?

Trigonometric functions are defined as the functions which show the relationship between the angle and sides of a right-angled triangle.

The figure given in the question has a right-angled triangle,

As per the given triangle,

Hypotenuse = 5

Perpendicular = 3

Base = 4

Apply the trignometric expression for tanθ

⇒ tanθ = Perpendicular / Base

⇒ tanθ = 3/5 ....(i)

We know that the identity of tangent functions as

⇒ tan2θ = 2tanθ /(1 - tan²θ)

Substitute the value of tanθ = 3/5 in the above identity

⇒ tan2θ = 2(3/5)/(1 - (3/5)²)

⇒ tan2θ = (6/5)/(1 - 9/25)

⇒ tan2θ = (6/5)/(16/25)

⇒ tan2θ = (6/5) × (25/16)

⇒ tan2θ = 15/8

Therefore, the exact value of the trigonometric function is tan2θ = 15/8.

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Simplify (write without the absolute value sign):

|x÷3|, if x < 0

Answers

Answer:

Step-by-step explanation:

zero divided by anything that is integral is zero and 3 in a integer

Answer:

Step-by-step explanation:

PLSS HELP ASAP!! WILL GIVE YOU BRAINLYEST!!

A spinner is divided into 9 equal parts, numbered 1-9. You spin the spinner 15 times and even numbers come up six times. What is the difference between the experimental
probability of spinning an even number and the theoretical probability of spinning an
even number?

Experimental probability:

Theoretical probability:

Answers

Answer:

5/9

Step-by-step explanation:

Experimental was what actually happened.

Theoretical was what should happen in a perfect, equal world.

E: 6/6 = 1

T: 4/9

E-T= 1-4/9=5/9

What is the volume of this rectangular prism?

Answers

Answer:

volume = l×b×h

5/4 ×4/3 × 3/5 = 1 cubic centimeter

Volume = length x width x height

5/4 x 4/3 = 20/12

20/12 x 3/5 = 60/60 = 1

Volume = 1 cm^3

on ran around a track that was 1/8 of a mile long.He ran around the track 24 times.how mant miles did jon run in all?

Answers

Answer:

3 miles

Step-by-step explanation:

1/8*24=3

Answer:

Step-by-step explanation:

He ran 24 times around the track1

Circumfrence of track=1/8 Mile

Total distance he covered=1/8*24

Distance = 3 miles

1 Determine the average value of f(x) over the interval from x = a to x = b, where f(x) = – 1 a= 3 and b = 3. X The average value is (Type an exact answer.) 1 During a certain 18-hour period, the average value is

Answers

The average value of f(x) over the interval from x = 3 to x = 3, where f(x) = -1, is -1.

To find the average value of a function over an interval, we need to calculate the definite integral of the function over that interval and then divide it by the length of the interval.

In this case, we are given that the function is f(x) = -1, and the interval is from x = 3 to x = 3. Since the interval has no length (b - a = 3 - 3 = 0), the average value of the function over this interval would simply be the value of the function at any point within the interval.

As per the given function, f(x) = -1 for all values of x, including x = 3. Therefore, the average value of f(x) over the interval from x = 3 to x = 3 is -1.

Therefore, the average value of f(x) over the interval from x = 3 to x = 3, where f(x) = -1, is -1.

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