Please help!
Simplify:

Answer:

formula for the derivative of x*lnx is equal to 1 + lnx.

Step-by-step explanation:

we need to find derivative of f(x) = x*lnx,

so we take the limiting value as x approaches x + h.

To simplify this, we set x = x + h,

and we want to take the limiting value as h approaches 0.

Now according to first derivative formula we have;

f'(x) = [tex]\lim_{h \to \zer0}[/tex]f(x+h) - f(x)]/[(x+h) - x]

[tex]\lim_{h \to \zer0}[/tex][ ln (1 + x) ] / x = 1

ln a - ln b = ln(a/b)

Using the above formulas, we have

d(xlnx)/dx = [tex]\lim_{h \to \zer0}[/tex][(x+h) ln(x+h) - xlnx]/[(x+h) - x]

= [tex]\lim_{h \to \zer0}[/tex][x ln(x+h) + h ln(x+h) - xlnx]/h

= [tex]\lim_{h \to \zer0}[/tex][x ln(x+h) - x lnx + h ln(x+h)]/h

= [tex]\lim_{h \to \zer0}\\[/tex][x(ln(x+h) - lnx) + h ln(x+h)]/h

=[tex]\lim_{h \to \zer0}[/tex][x ln [(x+h)/x] + h ln(x+h)]/h

=[tex]\lim_{h \to \zer0}[/tex][x ln (1+h/x) + h ln(x+h)]/h

=[tex]\lim_{h \to \zer0}[/tex] [x ln (1+h/x)]/h + limh→0 ln(x+h)

= [tex]\lim_{h \to \zer0}[/tex][ ln (1 + h/x) ] / (h/x) + limh→0 ln(x+h)

= 1 + lnx --- [Using limit formula [tex]\lim_{h \to \zer0}[/tex] [ ln (1 + x) ] / x = 1]

Hence, we have proved that the formula for the derivative of x*lnx is equal to 1 + lnx.

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Answer:

Step-by-step explanation:

In this section, we will determine the differentiate of xlnx using the first principle of derivatives, that is, the definition of limits. To derive the derivative of f(x) = xlnx, we take the limiting value as x approaches x + h. To simplify this, we set x = x + h, and we want to take the limiting value as h approaches 0. We will use the following formulas to prove the result:

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

limx→0 [ ln (1 + x) ] / x = 1

ln a - ln b = ln(a/b)

Using the above formulas, we have

d(xlnx)/dx = limh→0[(x+h) ln(x+h) - xlnx]/[(x+h) - x]

= limh→0[x ln(x+h) + h ln(x+h) - xlnx]/h

= limh→0[x ln(x+h) - x lnx + h ln(x+h)]/h

= limh→0[x(ln(x+h) - lnx) + h ln(x+h)]/h

= limh→0[x ln [(x+h)/x] + h ln(x+h)]/h

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Answer:

631

Step-by-step explanation:

This problem can be expressed as:

0.05x + 15 = 21.55
0.05x + 15 - 15 = 21.55 - 15
0.05x = 6.55
0.05x/0.05 = 6.55/0.05
x = 131

We also know that:

0.05x + 15 = x + 500

If x = 131:

x + 500
131 + 500
631

Answer:

-7

Step-by-step explanation:

To add a positive number, 5, and a negative number, -12, take the absolute value of both numbers.

You get 5 and 12.

Subtract the smaller absolute value from the larger absolute value.

12 - 5 = 7

The sign of the answer is the same sign as the number that has the larger absolute value. The absolute of 5 is 5. The absolute value of -12 is 12. -12 has the larger absolute value, and -12 is negative, so the answer is negative.

5 + (-12) = -7

The values of x and y are 25° and 33° respectively.

It is important to note the following;

From the information given, we have that

Angles 5x + 4 and 3x - 24 are supplementary.

This is represented as;

5x + 5 + 3x - 24 = 180

collect like terms

8x = 180 + 20

add lik terms

8x = 200

x = 200/8

x = 25°

Also,

2y and 114 are on a straight line

This is represented as;

2y + 114 = 180

collect like terms

2y = 180 - 114

2y = 66

Make 'y' the subject

y = 66/2

y = 33°

Thus, the values of x and y are 25° and 33° respectively.

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She earned a total of:

[tex]5(9.50)+2(11.25)+2(8.75)=\$ 87.50[/tex]

If she pays 15% for the booth space, she keeps 85%.

[tex]87.50(0.85) \approx \boxed{\$ 74.38}[/tex]

The slope of the road as reduced fraction  [tex]\frac{1}{20}[/tex] for grade which is 5%.

The slope of a road is defined as the ratio of the vertical height that a road rises to the horizontal distance covered to rise to that height.

While slope is expressed as a fraction grade is the percentage form of that fraction. The slope can also be defined as [tex]tan[/tex] [tex]\theta[/tex] where [tex]\theta[/tex] is the angle of elevation of the road. For example a slope of [tex]\frac{1}{4}[/tex] has a grade of 25%.

Here the given grade is 5%.

Per-cent refers to per one-hundred. So a 5% grade is 5 per one -hundred.

That means if the road is climbing then for every 100 units of horizontal distance covered, the altitude increases by 5 units.

Now slope of the road=vertical distance ÷ horizontal distance

[tex]=\frac{5}{100}\\=\frac{1}{20}[/tex]

Therefore we can say that the slope of the road is [tex]\frac{1}{20}[/tex]

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Answer:14

Step-by-step explanation:

Step-by-step explanation:

B is the midpoint of AC, in other words it is the halfway point.

So A to B should be equal to B to C

Our expression is:

2x + 9 = 37

Subtract 9

2x = 28

Divide by 2

x = 14

Answer:

750

Step-by-step explanation:

3/4 is 75%

Juan did 75% of the 1000 meters meaning he did 750

To get the answer u just do (3/4)*1000 and you get 750

Group C ran more miles than Group D

the central value of a set of data is expressed by the average of a list of data. It is defined mathematically as the ratio of the total number of data points to the number of units in the list. In terms of statistics, the term "mean" also refers to the average of a certain set of numerical data. The average of 2, 3, and 4 is, for instance, (2+3+4)/3 = 9/3 = 3. The center value of 2, 3, and 4 in this instance is 3, thus. Finding the mean value of a bunch of numbers is the definition of average.

Average = Total Values / Total Values

Group C = 27/4 = 6.75

Group D = 39 /6 = 6.5

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Answer:

1/9

Step-by-step explanation:

[tex]y = - 9x - 4[/tex]

The slope of the perpendicular line is the interverse(opposite) of the slope of the original line.

Hope this helps!

Have any questions? Feel free to ask!

Answer:

33.33

Step-by-step explanation:

    Well, we already know the amount of time it takes to make the lemonade, and we know the rate as well.

We need to find out how many pitchers you can make in an hour. If you can make one pitcher in 10 minutes, then you can make 6 pitchers in 60 minutes, AKA an hour.

So we make 6 pitchers per hour.

If we make 6 an hour, then we multiply 6 by 200 to get the number of pitchers we can make in 200 hours.

We can make 33.33 pitchers in 200 hours

Answer:The answer is actually 19.6

Step-by-step explanation:

trust me I did the test

unlike others

the height will be 50 ft after 0.575 seconds and after 5.424 seconds.

The equation that models the height is:

h(t) = 96*t - 16*t^2

Here we want to find the time such that the height is 50ft, then we need to solve the equation:

h(t) = 50 = 96*t - 16*t^2

Now we can solve that for t, we have the quadratic equation:

50 = 96*t - 16*t^2

0 = -16*t^2 + 96*t - 50

Then the solutions are:

t = (-96 ± √(96*96 - 4*(-50)*(-16))/(2*-16)

t = (-96 ± 77.6)/(-32)

The two solutions are:

This means that the height will be 50 ft after 0.575 seconds and after 5.424 seconds.

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Answer:

-4.

Step-by-step explanation:

When you add two negatives together it stays a negative. So the answer is -4.

The profit in 2012 must be 4.5(in hundreds of dollars) to break even.

Profit for a company is defined as the difference between the revenue and cost of a product.

Let us take an example to learn more about profit.

A company sells a product at a price of 'S' dollars and the company spends 'C' dollars to manufacture the product. Here C includes the cost of raw materials, labor cost, transport etc.. Now the net profit for the company for selling one unit of this particular product is given by

Profit=S-C

In the given chart it is shown the profit of the company is given as

2.5,1.75,-3.3 and -1.4 (in thousands of dollars) over 4 years.

Net profit over four years=[tex]2.5+1.75-3.3 -1.4=-0.45[/tex]

So over the past four years the company incurred a loss of

[tex]=0.45\times 1000\\[/tex]

[tex]=450[/tex] dollars

Now the company needs to break even in 2012.

So net profit must be zero over 5 years.

Therefore Profit in 2012=0-(-450)=450 dollars

Profit=4.5 (hundreds of dollars)

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Answer:

Approximately 1.86 times smaller

Step-by-step explanation:

5×10-7 = (5×10=50) = 50-7 = 43

8.5×10-4 = (8.5×10=84) = 84- 4 = 80

80/43 = approximately 1.86

43×1.86 = approximately 80

80/1.86 = approximately 43

Let the terms be a, ar, ar², ...

The sum to infinity being 27 means that:

[tex]\frac{a}{1-r}=27 \implies a=27(1-r)[/tex]

The sum of the first two terms being 15 means:

[tex]a+ar=15 \\ \\ (27-27r)+r(27-27r)=15 \\ \\ 27-27r+27r-27r^2 =15 \\ \\ 27r^2 -12=0 \\ \\ 27r^2 =12 \\ \\ r^2=\frac{12}{27}[/tex]

Since all the terms are positive, r>0.

[tex]\boxed{r=\frac{2}{3}} \\ \\ \implies a=27\left(1-\frac{2}{3} \right)=\boxed{9}[/tex]

answer and steps to get it in the picture

For the time period of  the projectile at least 192 feet above the ground is between 2 second and 6 second

Since s=128t-16t2, we want to know the 2 times where 128t-16t2=192, as those two values will be the begin and end times of the interval in question. We can rewrite that equation in standard quadratic equation form:

16t2-128t+192=0 or, simplified, 8t2-64t+96=0

or t2-8t+12=0

since it is now in quadratic form, we can solve using the quadratic formula:

t = 2 and 6

So the time period from approximately 2 second and 6 second has the projectile above 192 feet.

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You have 120 cups. The cups of beverage will be enough.

In order to determine the cups of beverage I have, the first step is to convert gallons and oz to cups.

1 gallon = 16 cups

16 x 4 = 64 cups

1 fluid ounce = 0.125 cups

64 x 0.125 x 7 = 56 cups

Total cups you have = 56 + 64 = 120 cups

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Answer:

x = 6

Step-by-step explanation:

4x/3 = 8

Solve for x

Multiply each side by 3

4x/3 *3 = 8*3

4x = 24

Divide each side by 4

4x/4 = 24/4

x = 6

Answer:

[tex] \sf \large \: x=6[/tex]

Step-by-step explanation:

Let's solve your equation step-by-step.

4x/3 =8

Step 1: Multiply both sides by 3.

4x/3=8

(4x/3)*(3)=(8)*(3)

4x=24

Step 2: Divide both sides by 4.

4x/4 = 24/4

x=6

Answer:

it's is the way to do it

Step-by-step explanation:

the equation is X-4y-35

Answer: irrational

Step-by-step explanation:

because it still cant be turned into a fraction after adding 9 to pi

[tex]{\sf{12 \times ( - 4) = -48}}[/tex]

m= 15

b= 45

y= 15x +45

Overall equation

The equation of a line in slope-intercept form is given by y= mx +b, where m is the slope and b is the y-intercept.

Slope, m

The slope is the measure of how steep the line is. It also defines how the y-axis changes with respect to the x-axis. The formula for finding slope is as shown below.

[tex]\boxed{\text{Slope}=\frac{y_1-y_2}{x_1-x_2} }[/tex]

where [tex](x_1,y_1)[/tex] is the first coordinate and [tex](x_2,y_2)[/tex] is the second coordinate

y-intercept, b

This is the y-value in which the line cuts through the y-axis (vertical axis).

Slope, m

Let's identify two pairs of coordinates on the line.

2 square units on the y-axis represent $30. Thus, 1 unit on the y-axis represents $15.

The 2 points are: (0, 45) and (5, 120)

Substitute the 2 points into the slope formula:

Slope

= [tex]\frac{120-45}{5-0}[/tex]

= [tex]\frac{75}{5}[/tex]

= 15

Thus, m= 15.

y-intercept, b

From the graph, the line cuts through the y-axis at y= 45.

Thus, b= 45.

Overall equation

Substitute m= 15 & b= 45 into y= mx +b:

∴ The equation of the line is y= 15x +45.

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Answer:

a=A/8

Step-by-step explanation:

This MIGHT be the answer, I am 99% sure!

Answer:

120

Step-by-step explanation:

60 gallons of 30% antifreeze means there are 18 gallons of antifreeze.

Let the number of gallons of 60% antifreeze be x. Then,

[tex]\frac{0.6x+18}{x+60}=\frac{1}{2} \\ \\ 1.2x+36=x+60 \\ \\ 0.2x=24 \\ \\ x=129[/tex]

[tex]i {}^{372} = 1 \\ i {}^{527 } = - i[/tex]

Step-by-step explanation:

Greetings !

Note that:-

[tex]i {}^{2n} = 1.\: if \: n \: is \: even \\ \: \: \: \: \: \: \: = - 1. \: if \: n \: is \: odd[/tex]

And also

[tex]i {}^{2n + 1} = i. \: if \: n \: is \: even \\ \: \: \: \: \: \: \: \: \: \: = - i.\: if \: n \: is \: dd[/tex]

Thus, remembering these general formula plug in values in the place of n and solve.

Therefore,

[tex]i {}^{2(186)} = i {}^{372} = 1[/tex]

where n is even thus its 1 from the above explanation

[tex]i {}^{2(263) + 1} = i {}^{527} = - i[/tex]

following the same procedure the values exceeds to be -i.

Hope it helps !!!

The equivalent expression of 4 − 3(x + 2) + 2(3 − 2x) is -7x + 4

The expression is given as:

4 − 3(x + 2) + 2(3 − 2x)

Open the brackets

4 − 3(x + 2) + 2(3 − 2x) = 4 - 3x - 6 + 6 - 4x

Collect the like terms

4 − 3(x + 2) + 2(3 − 2x) = - 3x - 4x + 4 - 6 + 6

Evaluate the like terms

4 − 3(x + 2) + 2(3 − 2x) = -7x + 4

Hence, the equivalent expression of 4 − 3(x + 2) + 2(3 − 2x) is -7x + 4

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