Mathematics College

derivative, by first principle
[tex] \tan( \sqrt{x } ) [/tex]

Answers

Answer 1
[tex]\displaystyle\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan x}h[/tex]

Employ a standard trick used in proving the chain rule:

[tex]\dfrac{\tan\sqrt{x+h}-\tan x}{\sqrt{x+h}-\sqrt x}\cdot\dfrac{\sqrt{x+h}-\sqrt x}h[/tex]

The limit of a product is the product of limits, i.e. we can write

[tex]\displaystyle\left(\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan x}{\sqrt{x+h}-\sqrt x}\right)\cdot\left(\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt x}h\right)[/tex]

The rightmost limit is an exercise in differentiating [tex]\sqrt x[/tex] using the definition, which you probably already know is [tex]\dfrac1{2\sqrt x}[/tex].

For the leftmost limit, we make a substitution [tex]y=\sqrt x[/tex]. Now, if we make a slight change to [tex]x[/tex] by adding a small number [tex]h[/tex], this propagates a similar small change in [tex]y[/tex] that we'll call [tex]h'[/tex], so that we can set [tex]y+h'=\sqrt{x+h}[/tex]. Then as [tex]h\to0[/tex], we see that it's also the case that [tex]h'\to0[/tex] (since we fix [tex]y=\sqrt x[/tex]). So we can write the remaining limit as

[tex]\displaystyle\lim_{h\to0}\frac{\tan\sqrt{x+h}-\tan\sqrt x}{\sqrt{x+h}-\sqrt x}=\lim_{h'\to0}\frac{\tan(y+h')-\tan y}{y+h'-y}=\lim_{h'\to0}\frac{\tan(y+h')-\tan y}{h'}[/tex]

which in turn is the derivative of [tex]\tan y[/tex], another limit you probably already know how to compute. We'd end up with [tex]\sec^2y[/tex], or [tex]\sec^2\sqrt x[/tex].

So we find that

[tex]\dfrac{\mathrm d\tan\sqrt x}{\mathrm dx}=\dfrac{\sec^2\sqrt x}{2\sqrt x}[/tex]

Related Questions

hey can you please help me posted picture of question

Answers

To find if the pairs are inverse of each other, find their composition. The composition of the inverse function will always be equal to x.

For first option, the composition will be:

[tex]f(g(x))=2+ \sqrt[3]{2- x^{3} } [/tex]

For second option, the composition will be:

[tex]f(g(x))= \frac{6x-8}{6}+8= \frac{6x-8+48}{6} = \frac{6x-40}{6} [/tex]

For third option, the composition will be:

[tex]f(g(x))= \frac{49}{x+9}-9 [/tex]

For Fourth option, the composition will be:

[tex]f(g(x))=5( \frac{x+11}{5})-11=x+11-11=x [/tex]

So the pair of function given in Fourth option are inverse of each other

According to the distributive property, 6(a + b) =

Answers

With the distributive property, you multiply whatever is one the outside of the parenthesis, by each term on the inside.

6a + 6b

Also, whats with the image? Lol.
I'm not sure why you attached that particular screenshot. However, if you wanted to distribute, then you simply multiply each inner term by the outer term 6

6 times a = 6a
6 times b = 6b
we don't know what 'a' or 'b' are, so we leave the expressions as is

Therefore, 6*(a+b) = 6a+6b

please help!!! trying to finish early

Answers

To solve this problem you must apply the proccedure shown below:

 The linear functions have the form y=mx+b

 where m is the slope and b is the y- intercept

 Therefore, you have:

 1) The slope -6/5 and b=(0,8) is:

 y=(6/5)x+8

 2) You have 3x-2y=1. Then, you only have to clear y:

 y=(3/2)x-1/2

what is the last step in planning your budget

Answers

calculating the cost of things and adding totals and finding out how much you will have left over

Answer:

review and revisions

Step-by-step explanation:

Help see photo attached

Answers

The first pair of statements is the best option. h(0) based on the given equation would be the square root of -3, which is an imaginary number. Also, w(0) should be 0, since if nobody is working on making wrenches, none can be made. This is despite the graph having a value at x = 0.

The second statement in the second pair is false, since 0 is not an imaginary number.

find the lateral area of the prism 3m 6m 3m 6m 15m

Answers

To find the lateral area of a prism, we calculate the combined area of its vertical rectangular faces, excluding the top and bottom faces. This prism has dimensions of 3m x 6m x 15m. It has two pairs of congruent rectangular faces: one pair with dimensions 3m x 15m and the other with dimensions 6m x 15m. The area of each face is determined by multiplying its length by its height. For the first pair of faces, the area is 3m * 15m = 45 square meters each. Similarly, for the second pair, the area is 6m * 15m = 90 square meters each. Adding these areas together (2 * 45 + 2 * 90), we get a lateral area of 270 square meters.

Complete answer;

find the lateral area of the prism 3m 6m 3m 6m 15m

Nina wants to put color tiles on a square. 3 color tiles fit across the top of the square. How many rows and columns of square will Nina need? How many color tiles will she use in all?

Answers

The area is a square. If she uses 3 tiles to fit length-wise, she will need 3 tiles to fit width-wise.

Total tiles needed = 3 x 3 = 9 tiles.

Answer: 3 rows of 3 tiles. Total of 9 tiles needed.

Nina will need 3 rows and 3 columns of square tiles. She will use 9 color tiles in total to cover the square.

To determine how many rows and columns of square tiles Nina needs, and how many color tiles in total she will use, let's break down the problem step by step:

1. Width and Height of the Square:

Nina can fit 3 color tiles across the top of the square. This implies the width of the square is 3 color tiles.

2. Rows and Columns Calculation:

  Since the square has 3 tiles across the top, and assuming the square is symmetric (same number of tiles in width and height):

Number of columns(width) = 3 tiles Number of rows (height) = 3 tiles So, Nina will need 3 columns and 3 rows of square tiles.

3. Total Number of Color Tiles:

To find out how many color tiles Nina will use in total, calculate the area of the square:

Area of the square = [tex]\( \text{Number of rows} \times \text{Number of columns} \)[/tex]

Area = [tex]\( 3 \times 3 = 9 \)[/tex] color tiles

Therefore:

Nina will need 3 rows and 3 columns of square tiles.

She will use 9 color tiles in total to cover the square.

5 Stars and Brainliest to correct answer

Answers

At θ = 90°, the equation must evaluate to zero. The only one that does that is the 1st choice ...
  r = 2 - 2sin(θ)

hey can you please help me posted picture of question

Answers

The answer is B. Hope this was helpful.
Answer:
144

Explanation:
In the closet, we have:
6 pairs of pants
8 shirts
3 pairs of shoes

To know the number of possible combinations that could be made from the available objects, all we have to do is multiply all three numbers together.
Therefore:
possible outcomes = number of pants * number of shirts * number of shoes
possible outcomes = 6 * 8 * 3
possible outcomes = 144

This means that the tree diagram would have 144 leaves to represent all possible combinations.

Hope this helps :)

what is -1/2x+1=9 with work shown.

Answers

[tex]-1/2x+1=9 (-1/2x+1=9)2 -x+2 = 18 -x = 16 x = -16[/tex]

The scores of individual students on the act program composite college entrance examination have an approximately normal distribution with mean 18.6 and standard deviation 6.0 and northside high 30 16 years take the test as soon that the scores at this school have the same distribution as national scores what is the probability that a randomly-chosen student has a score of higher than 20 points

Answers

The probabiity of a score greater than 20 from a normal distribution with a mean of 18.6 and a standard deviation of 6 is about 0.4078.

(a^3 / 4)^2
Simplify

Answers

Well, you can change this to (a^3/4)(a^3/4), which is equal to a^6/16, because a^3 times a^3=a^6, and 4 times 4=16.
I hope this helps!
-Raiden
Multiply the exponents in (a^3/4)^2 =a^3/2 simplified 





write an inequality to represent the situation. the temperature stayed above -15 degrees

Answers

t will represent temperature so

t > -15°

t will represent temperature  

t > -15°

Use a proportion to convert 18 grams to milligrams. [Hint: 1,000 mg = 1 g.]

Answers

x/18=1000/1
18000=1x
18000mg
Our proportion would be: 

[tex] \frac{1g}{1000mg} = \frac{18g}{x} [/tex]

We would then cross-multiply and our answer would be 18,000 milligrams. 

Write out the first four terms of the maclaurin series of f(x) if f(0)=−13,f′(0)=−6,f″(0)=5,f‴(0)=6 f(x) = +⋯

Answers


[tex] -13 - 6x + \frac{5}{2} {x}^{2} + {x}^{3} [/tex]

a local club is arranging a charter flight to hawaii. the cost of the trip is $574 each for 81 passengers, with a refund of $5 per passenger for each passenger in excess of 81.
find the number of passengers that will maximize the revenue recieved from the flight
also find the maximum revenue.

Answers

Let x be the number of passengers.
The discounted cost is given by,

Cost = 574-5(x-81) = 574-5x + 405 = 979 -5x

Total revenue, R(x) = x(979 - 5x) = 979x - 5x^2

At maximum revenue, dR/dx = 0 => dR/dx = 979 - 10x = 0 =>10x = 979 => x =979/10 = 97.9 ≈ 98 passengers

Therefore,
Revenues, R(97) = 98[574 -5(98-81)] = 98[574-85] = $47,922

Find the work done by the force field f(x,y,z)=3xi+3yj+7kf(x,y,z)=3xi+3yj+7k on a particle that moves along the helix r(t)=3cos(t)i+3sin(t)j+3tk,0≤t≤2π

Answers

Call the path [tex]\mathcal C[/tex]. Then the work done by [tex]\mathbf f(x,y,z)[/tex] along [tex]\mathcal C[/tex] is given by the line integral,[tex]\displaystyle\int_{\mathcal C}\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=2\pi}\mathbf f(x(t),y(t),z(t))\cdot\dfrac{\mathrm d\mathbf r}{\mathrm dt}\,\mathrm dt[/tex]
Swapping the [tex]\mathbf{ijk}[/tex] notation out for ordered component notation, I'll write
[tex]\mathbf r(t)=(x(t),y(t),z(t))=(3\cos t,3\sin t,3t)[/tex]
so that
[tex]\dfrac{\mathrm d\mathbf r}{\mathrm dt}=(-3\sin t,3\cos t,3)[/tex]
The line integral reduces to[tex]\displaystyle\int_0^1(9\cos t,9\sin t,7)\cdot(-3\sin t,3\cos t,3)\,\mathrm dt[/tex][tex]=\displaystyle\int_0^1(-27\cos t\sin t+27\sin t\cos t+21)\,\mathrm dt[/tex][tex]=\displaystyle21\int_0^1\mathrm dt=21[/tex]

You invest $230 to start a sandwich stand and decide to charge $4.25 per sandwich. How much money will you make if you sell 75 sandwiches?

Answers

You should do 4.25*75= 318.75
So you'll make $318.75

We have been given that the cost to start the sandwich stand is $230.

Now, the cost of one sandwich is given by $4.25

Also, it has been given that total 75 sandwiches sold.

Thus, the cost of 75 sandwiches is given by

[tex]75 \times 4.25\\ \$ 318.75[/tex]

Now, the sandwich stand cost $230 and the earning in selling of 75 sandwiches is $ 318.75.

Therefore, the profit is given by

[tex]318.75-230 =\$88.75[/tex]

Hence, I can make $88.75 on selling of 75 sandwiches.

how to get arc length when central angle is 139

Answers

So, this is a really good video to help you. I hope you get the answer, if you still don’t- I’ll answer your question in the comments. https://youtu.be/lBOy62v-9Uk
Final answer:

To find the arc length when the central angle is 139 degrees, multiply the fraction of the circle that the angle represents (139/360) by the circle's circumference formula (2πr), where r is the radius.

Explanation:

Calculating Arc Length with a Known Central Angle

To calculate the arc length when the central angle is 139 degrees, you need to know the radius of the circle. The arc length, As, is the distance traveled along the circumference of a circle as the radius is rotated through a given angle. Since there are 360 degrees in a full revolution, a central angle of 139 degrees represents a fraction 139/360 of the full circle.



The circumference of the entire circle is given by the formula 2πr, where r is the radius. Therefore, the arc length can be calculated by multiplying the fraction of the circle represented by the angle (in this case, 139/360) by the total circumference. The formula to determine the arc length As is:



As = (139/360) × 2πr



By knowing the radius and the central angle, you can substitute these values into the formula to find the arc length for any circular path.

Ginger is filing her federal income tax return with the 1040EZ form, and she received two W-2 forms, with the federal income tax withheld on one being $4530, and the federal income tax withheld on the other being $540. If Ginger received a Making Work Pay credit of $400, and if lines 9a and 9b in the Payments, Credits, and Tax section below are $0, what should she enter on line 10?
A. $4670
B. $4930
C. $5470
D. $5070

Answers

The answer is C. 5470

Answer - C

Add 4530 + 540 + 400 = 5470

Which of the choices below represents the definition of information secrecy?

Answers

Information secrecy is the practice of hiding information from certain individuals or groups who do not have the "need to know", perhaps while sharing it with other individuals. That which is kept hidden is known as the secret. Choose the answer that meet this definition

Secrecy (also called clandestinity or furtiveness) is the practice of hiding information from certain individuals or groups who do not have the "need to know", perhaps while sharing it with other individuals. That which is kept hidden is known as the secret.

one of the largest animal rehab places facilities in Montana. If there success rate, depending on the year is between 35 and 40 percent. According to the website is 2015, 234 raptors were taken in. Based on this data about how many rapotors were successful that year?

Answers

Bar models help try them out :)

Final answer:

To estimate the number of successful raptors in 2015, we can use the success rate provided by the website. If we assume a success rate of 35 percent, approximately 82 raptors were successful that year.

Explanation:

To estimate the number of successful raptors in 2015, we can use the success rate provided by the website. The success rate is between 35 and 40 percent. So, if we assume a success rate of 35 percent, we can calculate the number of successful raptors as follows:

Number of successful raptors = 35% of 234 raptors

Number of successful raptors = 0.35 * 234

Number of successful raptors = 81.9

Rounding to the nearest whole number, approximately 82 raptors were successful in 2015.

x-5 is less than or equal to 1/3 (9x-6)

Answers

x-5 is less than 1/3 (9x-6) so that problem is correct

Clem Colfax had $10 to buy groceries. He needed milk costing 70 cents a carton, bread costing 60 cents a loaf, breakfast cereal costing 50 cents a box, and meat costing $1.50 a pound. He bought twice as many cartons of milk as he did loaves of bread, the number of boxes of cereal was one more than the number of loaves of bread, and the number of pounds of meat was the same as the number of boxes of cereal. How many of each did he buy if the total he spent was exactly $10?

Answers

The first thing we must do for this case is to define variables:
 w: pounds of meat
 x: carton of milk
 y: loaf of bread
 z: cereal breakfast box
 We write the system of equations:
 0.70x + 0.60y + 0.50z + 1.50w = 10
 x = 2y
 z = y + 1
 w = z
 Solving the system we have:
 w = 3
 x = 4
 y = 2
 z = 3
 Answer:
 3 pounds of meat
 4 cartons of milk
 2 loaves of bread
 3 boxes of breakfast cereal

The measure of forecast error when the average absolute error of each forecast is shown as a percentage of demand is

Answers

The answer is:  "mean absolute percentage error {"MAPE"} .
_________________________________________________________

15 points!!! GEOMETRY help please!!
Questions attached!

Answers

9.
[tex]\text{If QR is tangent to circle then m}\angle PQR=90^o[/tex]

[tex]\text{We can use the converse of Pythagorean theorem}[/tex]

[tex]8\ and\ 4\sqrt3\\\\check:\ 4^2+(4\sqrt3)^2=8^2\\16+16\cdot3=64\\16+48=64\\64=64\ -O.K.\\\\2\ and\ 4\sqrt2\\\\check:\ 4^2+2^2=(4\sqrt2)^2\\16+4=16\cdot2\\20=32\ -FALSE\\\\3\ and\ 5\\\\check:\ 3^2+4^2=5^2\\9+16=25\\25=25\ -O.K.\\\\2\ and\ 2\sqrt5\\\\check:\ 4^2+2^2=(2\sqrt5)^2\\16+4=4\cdot5\\20=20\ -O.K.\\\\6\ and\ 10\\check:\ 4^2+6^2=10^2\\16+36=100\\52=100\ -FALSE[/tex]

[tex]\text{Answer:}\boxed{8\ and\ 4\sqrt3;\ 3\ and\ 5;\ 2\ and\ 2\sqrt5}[/tex]


10.
[tex]\text{Look at the pictures}[/tex]

[tex]\text{Therefore}\\\\C=2\cdot8+2\cdot16+2\cdot9+2\cdot6=16+32+18+12=78cm[/tex]

[tex]Other\ method:\\C=2\cdot(8+16+6+9)=2\cdot39=78cm[/tex]

Answer:

\text{If QR is tangent to circle then m}\angle PQR=90^o

\text{We can use the converse of Pythagorean theorem}

8\ and\ 4\sqrt3\\\\check:\ 4^2+(4\sqrt3)^2=8^2\\16+16\cdot3=64\\16+48=64\\64=64\ -O.K.\\\\2\ and\ 4\sqrt2\\\\check:\ 4^2+2^2=(4\sqrt2)^2\\16+4=16\cdot2\\20=32\ -FALSE\\\\3\ and\ 5\\\\check:\ 3^2+4^2=5^2\\9+16=25\\25=25\ -O.K.\\\\2\ and\ 2\sqrt5\\\\check:\ 4^2+2^2=(2\sqrt5)^2\\16+4=4\cdot5\\20=20\ -O.K.\\\\6\ and\ 10\\check:\ 4^2+6^2=10^2\\16+36=100\\52=100\ -FALSE

\text{Answer:}\boxed{8\ and\ 4\sqrt3;\ 3\ and\ 5;\ 2\ and\ 2\sqrt5}

10.

\text{Look at the pictures}

\text{Therefore}\\\\C=2\cdot8+2\cdot16+2\cdot9+2\cdot6=16+32+18+12=78cm

Other\ method:\\C=2\cdot(8+16+6+9)=2\cdot39=78cm

Step-by-step explanation:


Scientists discovered a large star 400,000 light years from Earth.A light-year is about 5,880,000,000,000 miles.What is the distance of the large star from Earth in scientific notation?

Answers

So if you want to find the distance, you would take 400,000x5,880,000,000,000=
2.35200000E+18. So your answer would be 2.35200000E+18 in scientific notation it would be 2.35200000x10^18. I hope this helps!

You spend $77 shopping. You buy some shirts at $12 each, and spend $5 on food. How many shirts did you buy?
A.
5x + 12 = 77
B.
x(12 + 5) = 77
C.
12x + 5 = 77
D.
12*5+x=77

Answers

Fam I'm ganna say the answer is C✌
In order to find this out,, we need to know that $12 is the changing number. This number depends on how many shirts were bought,, so we must put this with x. We also know that $5 is our constant (does not change) and we spent a total of $77 so our equation should equal to 77. This means that our equation should look like the following:
12x + 5 = 77
This means that the correct answer to your question is going to be option C.
Let me know if you have any further questions.
:)

State whether the given measurements determine zero, one, or two triangles. B = 86°, b = 21, c = 18


In need of help asap

Answers

we know that
Applying the law of sines
b/sin B=c/sin C-----> b*sin C=c*sin B-----> sin C=c*sin B/b
c=18
b=21
B=86°
sin C=18*sin 86/21-----> sin C=0.8550------> C=arc sin(0.8550)
C=58.77°------> C=58.8°
calculate angle A
A+B+C=180--------> A=180-B-C----> A=180-86-58.8----> A=35.2°
b/sin B=a/sin A-------> a=b*sin A/sin B----> a=21*sin 35.2/sin 86
a=12.13-------> a=12

the measure of the first triangle are
a=12
b=21
c=18
A=35.2°
B=86°
C=58.8°

calculate the measures of the second triangle
b=21
c=18
B=86°
C=180-58.8----> C=121.2°
A+B+C=180-----> A=180-86-121.2----> A=-27.2----> It is not possible to build a second triangle

the answer is
it is possible to assemble only a single triangle
a=12
b=21
c=18
A=35.2°
B=86°
C=58.8°

Answer:

one triangle

Step-by-step explanation:

The digits of a certain three-digit number are consecutive odd numbers. If the sum of the digits is 15, find the number.

Answers

The answer is 5 because it is consecutive 3 times and it is an odd number

Let the digits of the 3 digit number be [tex] x-2,x,x+2 [/tex]. where [tex] x [/tex] is an integer greater than 1. It is given that the sum of the 3 digits is 15. Hence,

[tex] x-2+x+x+2=15\\
3x=15\\
x=5 [/tex]

Thus the digits are [tex] 3,5,7 [/tex].

Therefore, the number is [tex] 357 [/tex].

We assume that the digits of the number is increasing from the 100th place.

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Marilyn has had a lump removed from her breast and sent off to a pathologist for examination. what description of the tumor would marilyn consider good news? the tumor is benign the tumor is malignant neither of the above is good news cancer is cancer The function rule for the following graph is y = 2x + 2. What is the value of y in the ordered pair (2, y)? 24 4 10 6 In the compound cah2 calcium has an oxidation number of 2+ and hydrogen has an oxidation number of Scientists discovered a large star 400,000 light years from Earth.A light-year is about 5,880,000,000,000 miles.What is the distance of the large star from Earth in scientific notation? An element's atomic number is the _____? What is the function of reverse transcriptase in retroviruses? it converts host cell rna into viral dna. it uses viral rna as a template for dna synthesis. it hydrolyzes the host cell's dna. it translates viral rna into proteins. it uses viral rna as a template for making complementary rna strands? Many workplace documents are procedural to A. help employees accomplish a goal with sequential steps. B. state the writer's main idea clearly. C. tell a story with a structured plot. D. support an argument with valid evidence. You can infer that a person who has lost his or her sense of smell is also likely to have a poor a. sense of balance b. sense of taste c. sense of touch d. sense of hearing 5 Stars and Brainliest to correct answer Which element of government does the Constitution give full sovereignty to? the state legislatures the president of the United States the Supreme Court the national government Why does decreasing the temperature of orange juice increase its pH level? Is it something to do with the chemical equilibrium? Whose job is it to review an athletes or entertainers contract for legitimacy? As an action potential occurs, the neuron's electrical charge changes from _____ to _____.a. negative; more negativeb. positive; negativec. negative; positived. positive; more positive One reason that the british did not try to break the union blockade of the south during the civil war was that help please1.Which of the following is a reason to use a direct quotation?a.when its the only support you have for a pointb.when the quote is a good length and fits well into the paragraphc.when the words of an authority are needed to emphasize the pointd.when you dont understand an idea or know how to state your point2.What is the best use of a style manual?a.to see if the papers supporting details are correctb.to make sure the spelling is correct throughoutc.to establish the specific formatting of the paperd.to proofread the paper for typos and errors please help!!! trying to finish early What is the whisker length of a sea lion with the two-locus genotype aabb? Please help with geometry Personal computer hard disk platters typically have storage capacities ranging from 40 gb to ____. Which of these island nations is not within the zone of persisting u.s. influence? which of these island nations is not within the zone of persisting u.s. influence? america samoa palau federated states of micronesia marshall islands fiji?