What is the equation of the line that passes through the point (6, -3) and has a
slope of -?

The standard deviation is 19.21 while the mean is 20.785.

a) Because E(Y) = 12, var(Y) = 9.

then, for 2Y+20

E(2Y+20)=2E(Y)+20=24+20=44

Var(2Y+20)=4(9)=36.

The average deviation is six.

b)E(X) = 80 var(X) = 144

so, for 3X

E(3X)=3(80)=240

Var(3X)=9Var(X)=9(144)=1296

The average deviation is 36.

c)0.25x+y

E(0.25x+y)=0.25E(x)+E(y)=0.25(80)+12=32

Var(0.25x+y)=0.0625 var(x)+var(y) +2(0.25)Cov(x,y)

Cov(x,y)=0 since x and y are independent.

Var(0.25x+y)=18

Standard deviation = 4.24 d) x – 5 y

E(x-5y)=E(x)-5E(y)=80-5(12)=20

Var(x-5y)=var(x)+25var(y)-2(5)Cov(x,y)

Because X and Y are unrelated, Cov(x,y) = 0

Var(x-5y)=369

The standard deviation is 19.21, while the mean is 20.785

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The equation below that does not show equivalent expressions when y = 2, is  5y + 25  = 5 ( y + 4 ).

option C.

The equation below that does not show equivalent expressions when y = 2, is determined by solving for y in all the given equation.

y + 4y  = y ( 1 + 4 )

y + 4y = y + 4y

5y = 5y

y = y

y + y + y = 3y

3y = 3y

y = y

5y + 25  = 5 ( y + 4 )

5y + 25 = 5y + 20

5y + 5 = 5y

5 ( y + 1 ) = 5y

y + 1 = y

2 (y + 2 ) = 2y + 4

2y + 4 = 2y + 4

2y = 2y

y = y

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Answer:

Given Figure has set of Stacked cubes.

To Draw the Base Plan, lets see its right side, Left Side and Front side view.

Right side:

It has 3 cubes in base.

2 cubes in 2nd layer clearly seen in figure and only 1 cube in 3rd layer.

Left Side:

It has 2 Cubes in base.

Both Cubes attached to 1st and 2nd cube of right side cubes.

Front Side:

It can be seen that, here base only has 2 rows of cubes.

Therefore,  1st Figure is correct Base Plan (or enclosed figure)

Note: Numbers in enclosed figure just shows the total no. of cubes in that stack.

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Answer:

-0.4444444444 or just -0.4

Answer: 5 (v+1)

Step-by-step explanation:

( i_i )

The value of the limit that exists at the given numbers is approximately 0.5.

First of all at x = 3.1, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 3.1} \frac{x^2-3 x}{x^2-9} $$[/tex] = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] = [tex]\frac{(3.1)^2-3(3.1)}{(3.1)^2-9}[/tex] = [tex]\frac{0.31}{0.61}[/tex] ≅ 0.508197

At x = 3.05, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 3.05} \frac{x^2-3 x}{x^2-9} $$[/tex] = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex]  =  [tex]$\frac{(3.05)^2-3(3.05)}{(3.05)^2-9}=\frac{0.1525}{0.3025}$[/tex] ≅  0.504132

At x = 3.01, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 3.01} \frac{x^2-3 x}{x^2-9} $$[/tex] = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex]  = [tex]$\frac{(3.01)^2-3(3.01)}{(3.01)^2-9}=\frac{0.0301}{0.0601}$[/tex] ≅  0.500832

At x =  3.001, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 3.001} \frac{x^2-3 x}{x^2-9} $$[/tex] = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] =  [tex]$\frac{(3.001)^2-3(3.001)}{(3.001)^2-9} $[/tex]=  [tex]$\frac{3.001 \times 10^{-3}}{6.001 \times 10^{-3}}$[/tex] ≅0.500083

At x =  3.0001, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 3.0001} \frac{x^2-3 x}{x^2-9} $$[/tex] = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] = [tex]$\frac{(3.0001)^2-3(3.0001)}{(3.0001)^2-9} $[/tex]=  [tex]$\frac{3.001 \times 10^{-4}}{6.001 \times 10^{-4}}$[/tex] ≅ 0.500008

At x = 2.9, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 2.9} \frac{x^2-3 x}{x^2-9} $$[/tex]  = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] =  [tex]$\frac{(2.9)^2-3(2.9)}{(2.9)^2-9} $[/tex]=  [tex]$\frac{-0.29}{-0.59}$[/tex] = 0.491525

At x = 2.95, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 2.95} \frac{x^2-3 x}{x^2-9} $$[/tex]  = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] =  [tex]$\frac{(2.95)^2-3(2.95)}{(2.95)^2-9} $[/tex]=  [tex]$\frac{-0.1475}{-0.2975}$[/tex] = 0.495798

At x = 2.99, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 2.99} \frac{x^2-3 x}{x^2-9} $$[/tex]  = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] =  [tex]$\frac{(2.99)^2-3(2.99)}{(2.99)^2-9} $[/tex]=  [tex]$\frac{-0.0299}{-0.0599}$[/tex] = 0.499165

At x = 2.999, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 2.999} \frac{x^2-3 x}{x^2-9} $$[/tex]  = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] =  [tex]$\frac{(2.999)^2-3(2.999)}{(2.999)^2-9} $[/tex]=  [tex]$\frac{-2.999 \times 10^{-3}}{-5.999 \times 10^{-3}}$[/tex] = 0.499917

At x = 2.9999, the value of [tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9} $$[/tex] will be

[tex]$$\lim _{x \rightarrow 2.9999} \frac{x^2-3 x}{x^2-9} $$[/tex]  = [tex]\frac{x^2-3 (x) }{x^2-9}$$[/tex] =  [tex]$\frac{(2.9999)^2-3(2.9999)}{(2.9999)^2-9} $[/tex]=  [tex]$\frac{-2.999 \times 10^{-4}}{-5.999 \times 10^{-4}}$[/tex] = 0.499992

From above all, we can see that the limit is approximately equal to and tends towards 0.5. So 0.5 will be our answer.

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Your question was incomplete, your most probable question was:
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

[tex]$$\lim _{x \rightarrow 3} \frac{x^2-3 x}{x^2-9}[/tex],  x = 3.1, 3.05, 3.01, 3.001, 3.0001, 2.9, 2.95, 2.99, 2.999, 2.9999

The diameter of the back wheel is 18 in.

A diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle.

Given that, a model had a front wheel with a 52 in. diameter and one spoke on the back wheel measuring 9 in, we need to find the diameter of the back wheel.

Since, the spoke in a wheel will act as a radius of the wheel,

Radius = diameter / 2

Also, it is given that, the one spoke on the back wheel measuring 9 in,

Therefore, the diameter of the back wheel = 9 × 2

= 18 in

Hence, the diameter of the back wheel is 18 in.

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The standard deviation of the variable "Defects in batch" is approximately 0.47.

The closest answer in the options is (C) 1.51.

The standard deviation is described as a measure of the amount of variation or dispersion of a set of values.

Standard Deviation is given as  = √(sum(p(x - μ)^2) / n)

Where μ = sum(xp)

= 2 * 0.35 + 3 * 0.23 + 4 * 0.20 + 5 * 0.09 + 6 * 0.07 + 7 * 0.06

= 2.49

substituting the values into the standard deviation formula, we have :

SD = √(sum(p(x - μ)^2) / n)

= √((0.35 * (2 - 2.49)^2 + 0.23 * (3 - 2.49)^2 + 0.20 * (4 - 2.49)^2 + 0.09 * (5 - 2.49)^2 + 0.07 * (6 - 2.49)^2 + 0.06 * (7 - 2.49)^2) / 6)

= √(1.32 / 6)

= √(0.22)

= 0.47

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The standard deviation of this variable is given as follows:

1.51.

Before obtaining the standard deviation, we must obtain the expected value, which is given by the sum of each value multiplied by it's respective probability, hence:

E(X) = 2 x 0.35 + 3 x 0.23 + 4 x 0.20 + 5 x 0.09 + 6 x 0.07 + 7 x 0.06

E(X) = 3.48.

Then we obtain the sum of the differences squared between each observation and the mean, multiplied by it's respective probabilities, hence:

(2-3.48)² x 0.35 + (3-3.48)² x 0.23 + (4-3.48)² x 0.20 + (5-3.48)² x 0.09 + (6-3.48)² x 0.07 + (7-3.48)² x 0.06 = 2.2696.

The standard deviation is the square root of this sum, hence:

S(X) = sqrt(2.2696)

S(X) = 1.51.

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In order to determine cash flows from operating activities, firms may use a method in which every item on the income statement is adjusted from accrual accounting to cash accounting which is called Modified cash basis.

This is referred to as an accounting method that combines elements of the two primary bookkeeping practices known as  cash and accrual accounting.

In this scenario, different adjustments are made so as to get the accurate value of the income generated by the company or organization.

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Solving a system of equations we wll see that there are  are 24 women, and the other 15 students are men in foruth grade.

We know that there are 39 students in fourth grade, we know that:

3/4 of the women went

2/3 of the men went

And this leaves 11 students at the school, then we can write the equation:

(3/4)*W + (2/3)*M + 11 = 39

Where W and M are the number of women and men respectively.

(3/4)*W + (2/3)*M = 39 - 11  = 28

(3/4)*W + (2/3)*M = 28

And we also know that the total number is 39, then:

M + W = 39

So we can write the system of equations:

(3/4)*W + (2/3)*M = 28

M + W = 39

M = 39 - W

REplacing that on the other equation we get:

(3/4)*W + (2/3)*(39 - W) = 28

(3/4)*W - (2/3)*w + 26 = 28

(3/4)*W - (2/3)*W = 28 - 26 = 2

(9/12)*W - (8/12)*W = 2

(1/12)*W = 2

W = 2*12 = 24

There are 24 women, and the other 15 students are men.

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Answer:0.3

Step-by-step explanation:

Therefore , the solution of the given problem of rectangle comes out to be  A= b* √6²-b².

In the Euclidean plane, a rectangle is merely a cube with four right angles. It is sometimes referred to as an equiangular quadrilateral or a quadrant that seems to have equal angles. The parallelogram also has the option of a straight angle. Four of a square's sides are all the same length. Rectangular quadrilaterals have four equal sides and four 90° vertices. It is sometimes referred as an "equirectangular square" as a result. A rectangle is also referred to as a parallelogram due to the identical and parallel lengths of its opposite sides.

Here,

radio= 3 cm

Al estar un rectángulo inscrito en un círculo, el diámetro del círculo corresponderá a la diagonal del rectángulo

Siendo b la base del rectángulo y a la altura, se tiene que:

6²= a²+b²

a²= √6²-b²

Nota: Se empleó el teorema de Pitágoras

El área del rectángulo está dado por:

A= base* altura

A= b*a

Reemplazando a:

A= b* √6²-b²

Therefore , the solution of the given problem of rectangle comes out to be  A= b* √6²-b².

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Answer:$69.08

Step-by-step explanation:

Yes, By SAS congruency criteria both ΔDEF and ΔHIG are congruent.

A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

We have to given that;

In the given figure,

⇒ DE = IH

⇒ EG = FI

⇒ ∠E = ∠I

Now, In ΔDEF and ΔHIG;

⇒ DE = IH   (Given)

⇒ ∠E = ∠I    (Given)

Since, EG = FI

Hence, We get;

⇒ GF + FI = FG + EG

⇒ GI = FE

So, In ΔDEF and ΔHIG;

⇒ EF = GI   (Common side)

Thus, By SAS congruency criteria both ΔDEF and ΔHIG are congruent.

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The expression for the number of butterflies after 4 minutes 3([tex]2^4[/tex]) - 7.

A grouping of numbers in a specific order is known as a sequence. On the other hand, a series is described as the accumulation of a sequence's constituent parts.

An ordered list of numbers is a sequence. The three dots indicate that the established pattern should move forward.

Given:

Initially we have 3 butterflies

After one minute =  3(2)

                             = 6 butterflies

After two minutes = 3(2)(2)

= 3(2²) butterflies

After three minutes = 3(2³) butterflies

After four minutes = 3([tex]2^4[/tex]) butterflies

as, she captured 7 butterflies

Then, after 4 min the number of butterflies on screen

= 3([tex]2^4[/tex]) - 7

= 3(16) - 7

= 48-7

= 41

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The solution is Marginal cost is  (1200 - 1000) / (30 - 20) = 12.

Subtraction is the process of taking away a number from another. It is a primary arithmetic operation that is denoted by a subtraction symbol (-) and is the method of calculating the difference between two numbers.

here, we have,

In order to calculate marginal cost, producers must compare the difference of producing one unit to the cost of producing the next unit

Marginal cost is the change in total cost as a result of increasing output produced by one unit.  

For example, if the total cost of producing 20 units of a good is 1000 and the total cost of producing 30units is 1200.

Marginal cost is  (1200 - 1000) / (30 - 20) = 12

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Answer:

18.2

Step-by-step explanation:

replace where x is with 3

[tex]3 + \frac{1}{2} + (5 \times 3) - \frac{3}{10} [/tex]

[tex]3 \frac{1}{2} + 15 - \frac{3}{10} [/tex]

[tex]15 - \frac{3}{10} = 14 \frac{7}{10} [/tex]

[tex]3 \frac{1}{2} + 14 \frac{7}{10} = 18 \frac{1}{5} [/tex]

[tex]3.5 + 15 - 0.3 = 18.2[/tex]

Step-by-step explanation:

Since force doesn't depend on mass and varies directly, you can set up a proportion. group the F's together and the acceleration together, as so

[tex] \frac{20}{16} = \frac{5}{x} [/tex]

Next, cross m multiply

[tex]20x = (16)(5)[/tex]

[tex]20x = 80[/tex]

Lastly, divide by 20

[tex]x = 4m {s}^{2} [/tex]

The acceleration of the object when force is 16N is 4m/s².

A ratio is a comparison between two similar quantities in simplest form.

Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.

In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.

Given, The force acting on the object varies directly with the object's acceleration.

Therefore, F ∝ a.

F = ka.

20 = 5k.

k = 4.

Now,

16 = 4a.

a = 4.

So, The acceleration is 4m/s²

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Answer: $25,000

Step-by-step explanation:

You invest $10,000 annually. If you divide 10,000 by 4, you get 25,000. It's $25,000 quarterly.

Answer: Find the length of one side and square it

Example: Lets say one side of the square has a length of 7. The area would be 49 because 7x7=49

The total cost of the materials used to construct the wheel is $51.57

The total cost of the materials used to construct the wheel is the sum of the total cost of the wood and the rubber.

Total cost = total cost of the wood + total cost of the rubber

Total cost of the wood = (area of the wood used x cost per square foot) + (circumference of the wheel x cost of the rubber)

Area of the wood = πr²

Where :

π = pi = 3.14

R = radius

3.14 x 1.5² = 7.065 ft²

Total cost of the wood = $17.66

Circumference of the wood = 2πr

2 x 3.14 x 1.5 = 9.42 feet

Total cost of the rubber = 9.42 x 3.60 = $33.91

Total cost of the materials = $33.91 + $17.66 = $51.57

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there is no difference in the values of Time for Idil or Sophie in that their amount will be tripled.

The interest earned on savings that are computed using both the original principal and the interest accrued over time is known as compound interest. A = P(1 + r/n)^nt, where P is the principal balance, r is the interest rate, n is the number of times interest is compounded annually, and t is the number of years, which is the formula for compound interest.

Given, Idil invested $150 in an account paying an interest rate of 5.785% compounded annually. Sophie invested $150 in an account paying an interest rate of 5.625% compounded continuously.

From the general formula of compound interest:

Final value A = (1 + r/n)^nt,

where P is the principal balance

r is the interest rate

n is the number of times interest is compounded annually

t is the number of years,

For Idil,

P = 150, r = 5.782, A = 3*p = 450, and n = 1

Thus,

450 = 150 (1 + 5.785%)^t

3 = (1.05785)^t

t = 19.535 years

The time required to get a total amount of $450.00 with compounded interest on a principal of $150.00 at an interest rate of 5.785% per year and compounded 1 time per year is 19.535 years.

(about 19 years 6 months)

For Sophie,

P = 150, r = 5.625, A = 3*p = 450

First, convert R as a percent to r as a decimal

r = R/100

r = 5.625/100

r = 0.05625 per year,

Then, solve the equation for t

t = ln(A/P) / r

t = ln(450.00/150.00) / 0.05625

t = 19.531 years

The time required to get a total amount of $450.00 with compounded interest on a principal of $150.00 at an interest rate of 5.625% per year and compounded continuously is 19.531 years.

(about 19 years 6 months)

therefore, there is no difference in the values of Time for Idil or Sophie.

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The sales representative has 24 possible sequences of cities to visit.

The multiplication rule of counting states that if there are n choices for the first event, m choices for the second event, p choices for the third event, and q choices for the fourth event, then there are a total of n x m x p x q possible sequences of choices. Applying this to the situation of the sales representative visiting four cities, we can say that there are 4 x 3 x 2 x 1 = 24 possible sequences of the cities.

To break this down further, the sales representative has 4 choices for the first city, Omaha, Dallas, Wichita, and Oklahoma City. Once the first city has been chosen, there are then 3 choices for the second city, Omaha, Dallas, and Wichita. From these 3 choices, there are then 2 choices for the third city, Omaha and Dallas. And then finally, there is 1 choice for the fourth city, Omaha. Multiplying these numbers together gives us 4 x 3 x 2 x 1 = 24 possible sequences of cities.

Therefore, the sales representative has 24 possible sequences of cities to visit.

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Answer:

-6, -1

Step-by-step explanation:

midpoint M of the line segment

M = (x1 + x2) / 2 , (y1 + y2) / 2

   =  (-5-7) / 2, (5-7) / 2

   =  -12 / 2, -2 / 2

   =  -6 , -1

Answer:

1.a)  x = g(2)

1.b)  See below

2.a)  See below

2.b) α = 60°

3.a)  2 + h

3.b)  2

4.a)  Interval [3, 4] = 3 m/s

       Interval [3, 3.1] = 2.1 m/s

4.b)  2 m/s

Step-by-step explanation:

Part (a)

Given function f(x):

[tex]f(x)=\dfrac{x}{x-2}[/tex]

If the range of f is (-∞, 1) U (1, ∞), then a value of y in the range of f is y=2.

Set the function equal to the value of y and solve for x:

[tex]\begin{aligned}y=2 \implies \dfrac{x}{x-2}&=2\\x&=2(x-2)\\x&=2x-4\\-x&=-4\\x&=4\end{aligned}[/tex]

The solution written as a function x = g(y) is:

Part (b)

Given function g(y):

[tex]g(y)=\dfrac{2y}{y-1}[/tex]

Verify that g is the inverse function of f by calculating g(f(x)):

[tex]\begin{aligned}\implies g(f(x)) &=\dfrac{2f(x)}{f(x)-1}\\\\&=\dfrac{2\left(\frac{x}{x-2}\right)}{\left(\frac{x}{x-2}\right)-1}\\\\&=\dfrac{2\left(\frac{x}{x-2}\right)}{\left(\frac{x-(x-2)}{x-2}\right)}\\\\&=\dfrac{\left(\dfrac{2x}{x-2}\right)}{\left(\dfrac{2}{x-2}\right)}\\\\&=\dfrac{2x}{2}\\\\&=x \quad (\text{for all $x\neq 2$})\end{aligned}[/tex]

Similarly, calculate f(g(y)):

[tex]\begin{aligned}\implies f(g(y))&=\dfrac{g(y)}{g(y)-2}\\\\&=\dfrac{\left(\frac{2y}{y-1}\right)}{\left(\frac{2y}{y-1}\right)-2}\\\\&=\dfrac{\left(\frac{2y}{y-1}\right)}{\left(\frac{2y-2(y-1)}{y-1}\right)}\\\\&=\dfrac{\left(\dfrac{2y}{y-1}\right)}{\left(\dfrac{2}{y-1}\right)}\\\\&=\dfrac{2y}{2}\\\\&=y \quad \text{(for all $y \neq 1$)}\end{aligned}[/tex]

Hence verifying that g is the inverse function of f.

[tex]\boxed{\begin{minipage}{5 cm}\underline{Trigonometric Identities}\\\\$\tan x=\dfrac{\sin x}{\cos x}$\\\\\\$\cos \left(\arctan \left(x\right)\right)=\dfrac{1}{\sqrt{1+x^2}}$\\\\\\$\sin\left(\arctan \left(x\right)\right)=\dfrac{x}{\sqrt{1+x^2}}$\\\end{minipage}}[/tex]

Part (a)

If  y = arctan(x)  then  tan(y) = x.

Use the trigonometric identities to express sin(y) and cos(y) in terms of x.

[tex]\boxed{\begin{aligned}\tan(y) &= x\\\implies \dfrac{\sin y}{\cos y}&=x\\\sin y&=x \cos y\\ \sin y&=x \cos (\arctan x)\\ \sin y&=\dfrac{x}{\sqrt{1+x^2}}\end{aligned}}[/tex]

[tex]\boxed{\begin{aligned}\tan(y) &= x\\\\\implies \dfrac{\sin y}{\cos y}&=x\\\\\dfrac{\sin y}{x}&= \cos y\\\\ \cos y&=\dfrac{\sin(\arctan x)}{x}\\\\ \cos y&=\dfrac{\dfrac{x}{\sqrt{1+x^2}}}{x}\\\\\cos y&=\dfrac{x}{\sqrt{x^2+1}x}\\\\\cos y&=\dfrac{1}{\sqrt{x^2+1}}\end{aligned}}[/tex]

Part (b)

Given linear equation:

[tex]y=\sqrt{3}x+1[/tex]

As the slope of a linear equation y = mx + b is m, then the slope (m) of the given line is √3.

If m = tan(α) then:

[tex]\begin{aligned}\implies m&=\sqrt{3}\\\tan (\alpha)&=\sqrt{3}\\\alpha&=\arctan \sqrt{3}\\\alpha&=60^{\circ}+180^{\circ}n\\\end{aligned}[/tex]

If α is the angle the line forms with the x-axis, then α = 60°.

Given function:

[tex]f(x)=x^2-1[/tex]

Part (a)

[tex]\begin{aligned}\dfrac{f(1+h)-f(1)}{h} &=\dfrac{((1+h)^2-1)-(1^2-1)}{h}\\\\&=\dfrac{(1+2h+h^2-1)-0}{h}\\\\&=\dfrac{2h+h^2}{h}\\\\&=2+h\end{aligned}[/tex]

Part (b)

Part a used differentiating from first principles to find the gradient of f(x) at x = 1.  As h gets smaller, the gradient of the straight line gets closer and closer to the gradient of the curve.  Therefore, as h gets close to zero, (2 + h) gets close to 2.  Therefore, the slope of the tangent line to the graph of f(x) at the point where x = 1 is 2.

A particle is moving on the x-axis and its position at time is given by

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

Evaluate the position of the particle at t = 3, t = 4 and t = 3.1:

[tex]x(3)=3^2-4(3)+3=0[/tex]

[tex]x(4)=4^2-4(4)+3=3[/tex]

[tex]x(3.1)=(3.1)^2-4(3.1)+3=0.21[/tex]

Average velocity formula

[tex]\overline{v}=\dfrac{s(t_2)-s(t_1)}{t_2-t_1}[/tex]

Therefore, the average velocity of this particle over the interval [3, 4]:

[tex]\overline{v}=\dfrac{x(4)-x(3)}{4-3}=\dfrac{3-0}{1}=3\;\; \rm m/s[/tex]

The average velocity over the interval [3, 3.1] is:

[tex]\overline{v}=\dfrac{x(3.1)-x(3)}{3.1-3}=\dfrac{0.21-0}{0.1}=2.1\;\; \rm m/s[/tex]

Part (b)

To find an equation for velocity, differentiate the given equation for displacement:

[tex]v(t)=\dfrac{\text{d}x}{\text{d}t}=2t-4[/tex]

To calculate the instantaneous velocity of the particle at time t = 3, substitute t = 3 into the equation for velocity:

[tex]v(3)=2(3)-4=2\;\; \rm m/s[/tex]

The numerical form of the statement is (5/2)n - 17 = 48.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

Let, 'n' be the unknown number,

So, the statement The difference between 5/2 of a number and 17 is 48

can be written as,

(5/2)n - 17 = 48.

5n - 34 = 96.

5n = 96 + 34.

5n = 130.

n = 130/5.

n = 26.

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The number 9,839,038 after indicated rounding off gives the number 9,839,000.

Rounding off of a number is defined as the simplification of the number by keeping the value of the number same but is made closer to the next number.

Rounding can be done for whole numbers as well as decimals.

Given is a whole number 9,839,038.

We have to round it to the place shown where the digit is 9.

Look at the next digit right to 9.

It is 0, which is less than 5.

So replace all the digits next to 9 by 0.

We get the number 9,839,000.

Hence the rounded number is 9,839,000.

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Survey was wrong: holiday, systematic sample, 6 similar flights. To improve: questionnaire, various times/locations, entrance, different days.

The problems with the survey conducted by the airline are:

1) Conducting the survey on a holiday weekend is not representative of the entire population of air travelers;

2) The survey uses systematic sampling which may not accurately reflect the entire population;

3) The survey was only conducted on 6 similar flights and does not represent the entire population of air travelers.

To improve the survey, the following steps can be taken:

1) Ask travelers to fill out a questionnaire;

2) Conduct the survey at the entrance to the airport;

3) Conduct the survey during different times of the year;

4) Use various days of the week to conduct the survey;

5) Conduct the survey using flights to and from various locations.

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