Derivative of six inverse trigonometric functions are as follow:
d/dx(sin⁻¹x) = 1 / √1 - x²
d/dx(cos⁻¹x) = -1 / √1 - x²
d/dx(tan⁻¹x) = 1 / √1 + x²
d/dx(cosec⁻¹x) = -1 / |x|√ x² - 1
d/dx(sec⁻¹x) = 1 / |x|√ x² -1
d/dx(cot⁻¹x) = -1 / √1 + x²
As given in the question,
Explanation of the derivative of six inverse trigonometric functions are as follow:
1. y = sin⁻¹x
⇒ x = siny
⇒d/dx(x) = d/dx(siny)
⇒ 1 = cosy dy/dx
⇒dy/dx = 1/cosy
⇒dy/dx = 1/√1-sin²y
⇒dy/dx = 1/√1-x²
⇒d/dx( sin⁻¹x) = 1/√1-x²
2. y = cos⁻¹x
⇒ x = cosy
⇒d/dx(x) = d/dx(cosy)
⇒ 1 =-siny dy/dx
⇒dy/dx = -1/siny
⇒dy/dx = -1/√1-cos²y
⇒dy/dx = -1/√1-x²
⇒d/dx( cos⁻¹x) = -1/√1-x²
3. y = tan⁻¹x
⇒ x = tany
⇒d/dx(x) = d/dx(tany)
⇒ 1 =sec²y dy/dx
⇒dy/dx = 1/sec²y
⇒dy/dx = 1/√1+ tan²y
⇒dy/dx = 1/√1 + x²
⇒d/dx( tan⁻¹x) = 1/√1+x²
4. y = cosec⁻¹x
⇒ x = cosecy
⇒d/dx(x) = d/dx(cosecy)
⇒ 1 =-cosecy coty dy/dx
⇒dy/dx = -1/cosecy coty
⇒dy/dx = -1/cosecy√cosec²y-1
⇒dy/dx = -1/|x|√x²-1
⇒d/dx( cosec⁻¹x) = -1/|x|√x²-1
5. y = sec⁻¹x
⇒ x = secy
⇒d/dx(x) = d/dx(secy)
⇒ 1 =secy tany dy/dx
⇒dy/dx = 1/secy tany
⇒dy/dx = 1/secy√sec²y-1
⇒dy/dx = 1/|x|√x²-1
⇒d/dx( sec⁻¹x) = 1/|x|√x²-1
6. y = cot⁻¹x
⇒ x =coty
⇒d/dx(x) = d/dx(coty)
⇒ 1 =-cosec²y dy/dx
⇒dy/dx = -1/cosec²y
⇒dy/dx = -1/√1+ cot²y
⇒dy/dx = -1/√1 + x²
⇒d/dx( cot⁻¹x) = -1/√1+x²
Therefore, the derivative of the six inverse trigonometric functions are given below:
d/dx(sin⁻¹x) = 1 / √1 - x²
d/dx(cos⁻¹x) = -1 / √1 - x²
d/dx(tan⁻¹x) = 1 / √1 + x²
d/dx(cosec⁻¹x) = -1 / |x|√ x² - 1
d/dx(sec⁻¹x) = 1 / |x|√ x² -1
d/dx(cot⁻¹x) = -1 / √1 + x²
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suppose a license plate that has 2 length of 7 characters can only contain digits and upper-case vowels and must contain at least one digit and at least one vowel. how many different legitimate license plates are available?
The total number of possible legitimate license plate numbers possible are (15)⁵.
The license plate has a length of 7 character for the legitimate license plates.
So, there are a total 7 position, in these positions, on has to be a upper case vowel and one has to be an integer digit.
Now, we have only 5 position are left.
There are only 5 vowels and 10 integers digits.
So, we are assuming that the repetition is allowed because nothing else is mentioned.
So, total possible combinations are 1 x 1 x 15 x 15 x 15 x 15 x 15.
Because repetition is allowed, we don't have to consider the factor of internal arrangement.
So, the total number of legitimate license plates possible are (15)⁵.
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A casino only has red playing chips for $10 each and blue playing chips for $5 each. Jack paid $400 to purchase 50 playing chips at the cashier's window. How many red playing chips were given to him?
Answer: 30
Step-by-step explanation:
1 red chip = $10, so 30 red chips = $ 300
1 blue chip = $5, so 20 blue chips = $100
Add the cost of both we get $400, matching our question requirements.
Help me out please
answers:
-1/3
3
1/3
-3
please do x to
Answer:
slope = 3
y-intercept = 0
Step-by-step explanation:
Which is the polynomial function of lowest degree with rational real coefficients, a leading coefficient of 3 and roots and 2?.
On solving the given question, we got that - the polynomial function of lowest degree with rational real coefficients is [tex]= > P(x) = 3x^{2} - 21x + 30[/tex]
What is Polynomial Equation?Polynomial equations are those created using variables, exponents, and coefficients. The higher exponent, known as the equation's degree, can have a variety of exponents. By factoring polynomials based on the degree and variables in the equation, we can solve them.
We are aware that the lowest degree polynomial function, denoted by p(x), has the following properties:
[tex]= > P(x) = c(x - a)(x - b)\\[/tex]
The leading coefficient is 3, the roots are 5, and the roots are 2, respectively.
Therefore, the lowest degree polynomial with the aforementioned characteristics is:
[tex]= > P(x) = 3(x - 5)(x - 2))\\= > P(x) = 3(x^{2} - 7x + 10)\\= > P(x) = 3x^{2} - 21x + 30\\[/tex]
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Please help! I need credit to graduate
The annual percentage rate (APR) for this loan is equal to 412.7%.
What is APR?
APR is an abbreviation for annual percentage rate and it can be defined as a measure of an amount of money based on a yearly rate of interest, which a borrower must pay on a loan or credit card.
Mathematically, the annual percentage rate (APR) of a loan can be calculated by using this formula:
APR = [((simple interest/principal)/n) × 365] × 100
Where:
n represents the number of days.
Note: n is equal to 14 days (2 weeks).
Substituting the given parameters into the formula, we have;
APR = [((95/600)/n) × 365] × 100
APR = 0.01130 × 36500
APR = 412.7%.
the answer should be $568
Step-by-step explanation:
How would we write this as an expression? Be sure to define any variables used
Paul worked five more than twice as many hours as Austin
A: What is the solution to b² = 196?
Answer:
b=14
Step-by-step explanation:
b^2=196
Square root both sides
b=14
Hopes this helps please mark brainliest
Answer:
14
Step-by-step explanation:
You're just looking for the square root of 196, and [tex]\sqrt{196} = 14[/tex]
Solve for y given that the height of the triangle is 4 miles and each of the legs are 7 miles.
Answer:
Step-by-step explanation:
x^2 + y^2 = z^2 is the pythagorean theorem.
If you divide the large triangle into two separate ones, with sides 7, 4 and 0.5y, you can use the pythagorean theorem to figure out 0.5y. So:
7^2 = 4^2 +(0.5y)^2
49-16 = 33
so y=2sqrt(33)
the u.s. energy information administration claimed that u.s. residential customers used an average of 10,400 kilowatt hours (kwh) of electricity this year. a local power company believes that residents in their area use more electricity on average than eia's reported average. to test their claim, the company chooses a random sample of 163 of their customers and calculates that these customers used an average of 10,782kwh of electricity last year. assuming that the population standard deviation is 2768kwh, is there sufficient evidence to support the power company's claim at the 0.01 level of significance?
There is not sufficient evidence to support the power company's claim.
According to the information given in the question, we can frame the hypothesis as follows-
Null hypothesis ⇒ : H = 10400
Alternate hypothesis ⇒ Ha: > 10400
This is the right-tailed test.
Test statistic = z
[(x - μ)√n]/ σ
where x ⇒ sample mean = 10,782
σ ⇒ standard deviation = 2768
= (10782 - 10400)√163/ (2768)
= 382* 12.76/(2768)
= 1.76
⇒ Test statistic z = 1.76
Now, P( z > 1.76) = 1 - P( z < 1.76)
= 1 - 0.9608
= 0.0392
using z tables-
p-value = 0.0392
a = 0.01
we see that p-value > a
Thus, we fail to reject our null hypothesis.
Therefore, there is not sufficient evidence to support the power company's claim.
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One leg of a right triangle is 6 units long, and its hypotenuse is 12 units long. What is the length of the other leg? Round to the nearest whole number.
10 units
13 units
18 units
4 units
The length of the other leg of the right triangle, rounded to the nearest whole number, is 10 units.
To find the length of the other leg of the right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's denote the length of the other leg as x. Given that one leg is 6 units long and the hypotenuse is 12 units long, we can set up the equation:
6^2 + x^2 = 12^2
Simplifying this equation, we have:
36 + x^2 = 144
Subtracting 36 from both sides, we get:
x^2 = 108
Taking the square root of both sides, we find:
x = √108
Rounding this value to the nearest whole number, we have:
x ≈ 10.392
The length of the other leg of the right triangle, rounded to the nearest whole number, is 10 units.
The correct option is: 10 units.
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Pressley is working two part-time jobs. He makes $18 per hour lifeguarding and $12 per
hour as a cashier at the local grocery store. During spring break from school, he worked
twice as many hours lifeguarding as he did at the local grocery store and earned $144.
How many hours did he work at each job?
Answer: He spent 3 hours as a cashier and 6 hours as a lifeguard
Step-by-step explanation:
First, you figure out his time as a cashier; 12($ per hour) x 3(hours working) = $36. Then, since it states he spent twice as many hours being a lifeguard, you double his hours when figuring out his time as a lifeguard; 18($ per hour) x 6(hours working) = $108. Add up both of your totals, $108 + $36 = $144.
What is an equation of the line that passes through the point (-5,7) and is parallel
to the line x + 5y = 10?
Answer y=-1/5x+6
Step-by-step explanation:
Rewrite in slop intercept form y=mx+b
5y=10-x
y=2-(x/5)
y=-1/5x+2
Slope = -1/5
Substitute the 2 numbers in the point and solve
y-7=-1/5(x+5)
y=-1/5x+6
Answer:
This exact question has already been answered on the site, but here it is again.
2x2−4x=t In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?A. −3
B. −1
C. 1
D. 3
The correct option A. -3, is the value of constant 't', for which the equation has no real solutions.
Define the term real solutions?In algebra, a real solution is just an answer to an equation which is a real number. The number of solutions to the provided quadratic equation depends on the discriminant, but can be positive, 0, or negative. A quadratic equation with a positive discriminant has two unique real number solutions.A repeating real number solution to the quadratic equation is indicated by a discriminant of zero.For the stated question-
The quadratic equation is given as-
2x² - 4x = t
2x² - 4x - t = 0
The standard from of quadratic equation is-
ax² + bx + c = 0
For no solution;
Discriminant is less than zero.
D < 0
b² - 4ac < 0
Comparing both equations-
b = -4, a = 2 and c = -t
Put value in D.
(-4)² - 4(2)t < 0
Solve the inequality
16 - 8t < 0
t < -2
Thus, the value of constant 't', for which the equation has no real solutions.
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The correct question is-
2x² - 4x = t,In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?
A. -3
B. -1
C. 1
Percentage,base,rate
Answer:125
Step-by-step explanation:
when the null hypothesis is not rejected, it is possible a type i error has occurred. possible a type ii error has occurred. possible either a type i or a type ii error has occurred. not possible a type ii error has occurred.
When a null hypothesis is not rejected, a type ii error has occurred.
What is a null hypothesis?A null hypothesis being a statistical hypothesis, states that no statistical significance can be present in a set of observations. Usually, hypothesis testing is a method used to judge the veracity of a given sample.
Sometimes, it is called just null, or even it has a symbol, viz, [tex]H_0[/tex].
We consider generally 4 cases:
[tex]H_0[/tex] is accepted when [tex]H_0[/tex] is true ⇒ A correct decision[tex]H_0[/tex] is rejected but is true⇒ Type i error has occurred and is rejected by us, null hypothesis is present[tex]H_0[/tex] is false but is accepted by us⇒ Type ii error has occurred, null hypothesis is not present. [tex]H_0[/tex] is false and is rejected⇒ A correct decision.According to the given question, we have not rejected the null hypothesis.
Thus, we have two options present, either the type ii error has occurred and null hypothesis is not present, or the null hypothesis is present.
Concluding, when a null hypothesis is accepted, it is possible that a type ii error has occurred.
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Nia's normal running
route is 3 1/3 miles.
If she ran this route 3 3/4 times
this week, find the total
number of miles she ran.
If malik purchases an iced coffee drink for $5. 69 and leaves a $1 tip, what percentage did he tip? show all your work based upon your knowledge of proportions.
The percentage of his tip compared to the cost of the iced coffee drink is 17.57%.
Percentage is defined as part of a whole expressed in hundredths or out of 100. It is usually written with the symbol "%".
To determine the percentage, we must first determine the proportion of the amount of the tip compared to the cost of the drink he purchased by dividing the two values. After that, multiply the answer by 100.
percentage = (amount of the tip/cost of the iced coffee drink) x 100
percentage = ($1/$5.69) x 100
percentage = 0.175747 x 100
percentage = 17.57%
Hence, he tip 17.57% of what he purchased.
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i need help plssssssssssss
Answer:
I can't well see question but, their ratios aren't the same
what is the area of semicircle if the arc is 35 pi
Answer:
612½ pi units² or 1920 units²
3 times a number is 28 less than the square of that number. Find the negative solution.
The solution that is "3 times a number is 28 less than the square of that number" in the negative is -4.
What is polynomial?A polynomial is a mathematical statement made up of coefficients and indeterminates that uses only the operations addition, subtraction, multiplication, and powers of positive integers of the variables. x2 4x + 7 is an illustration of a polynomial with a single indeterminate x. Sums of terms of the form kxn, where k is any number and n is a positive integer, make up polynomials. For instance, the polynomial 3x+2x-5. a description of polynomials. The concepts degree, standard form, monomial, binomial, and trinomial are all covered in this video.
Here,
Let the number be n.
3n=n²-28
n²-3n-28=0
n²-7n+4n-28=0
n(n-7)+4(n-7)=0
(n+4)(n-7)=0
n=-4,7
The negative solution to the number that is "3 times a number is 28 less than the square of that number" is -4.
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Answer:
-4 is correct
Step-by-step explanation:
Got it correct on delta 2023
Help me......Solve for each variable. Show your work and explain in paragraph form how you solved each problem.
Base Angles Theorem
Converse to Base Angles Theorem
Corollary
Converse to the corollary
Sum of Interior Angles
Exterior angles
1) The value of x is 30 degree.
2) The value of x is 32.5 degree.
What is Base Angle Theorem?A triangle's opposite angles are also congruent if its two sides are.
Given:
Using Base Angle Theorem
Two Angles are 35 and 35.
Using Angle Sum Property
3x+ 20 +35+ 35 = 180
3x+ 90 = 180
3x = 90
x= 30
Again Using Base Angle Theorem
let the Angles in left triangle are 50, y and y
Using Angle Sum Property
50+ 2y= 180
2y= 130
y= 65
Again Using Angle Sum Property in Big Triangle
50+x + 65 + x = 180
2x = 65
x= 32.5 degree
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A very large study showed that aspirin reduced the rate of first heart attacks by %. A pharmaceutical company thinks they have a drug that will be more effective than aspirin, and plans to do a randomized clinical trial to test the new drug. What is the null hypothesis the company will use?.
the null hypothesis the company will use p =0.44.
What is z test?
If the data is distributed normally, a z test can be used to determine whether the means of two populations differ from one another. The z test statistic's value must be determined, together with the null hypothesis and alternative hypothesis setups, for this reason. On the z critical value, the decision criterion is based. The distribution graph is divided into acceptance and rejection zones during hypothesis testing by the z critical value. The null hypothesis can be rejected if the test statistic lies inside the rejection region; otherwise, it cannot.
Here the data is normally distributed and using the sample mean.
This is the right-tailed test.
The null hypothesis is
H0 : p =0.44
Hence, the null hypothesis the company will use p =0.44.
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Alex rides his bike to visit two friends and returned home. Interpret the graph to answer questions about his activities.
1. How long did it take Alec to get to his friends house?
2. How long did he stay at his first friends house?
3. How long did it take alec ti ride his bike from his first friends house to his second friends house?
4. How much longer did alec stay at his first friends house than at his second friends house?
5. In which parts of the graph is alec riding his bike toward his home?
6. In which parts of the graph is alec traveling the fastest?
7. How long was alec away from his house?
The required question of the given problem are solved below.
Explain is distance vs time graph?A distance-time graph shows how far an item has gone in a given time. It is a straightforward line diagram that signifies distance versus time discoveries on the chart. Distance is plotted on the Y-pivot. Time is plotted on the X-pivot.
According to question:1) he take 20 minutes to get to his friends house.
2) he stayed 60 - 20 = 40 minutes at his first friends house.
3) he take 70 - 60 = 10 minutes to ride his bike from his first friends house to his second friends house.
4) 30 minutes, he stayed his second friend
Then
Difference = 40 - 30 = 10 minutes
So, he stay 10 minutes longer than first friend.
5) In part E, he is riding towards his home.
6) In part A of the graph he is travelling faster.
7) he went 10 kilometer away from his house.
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If a categorical independent variable contains 4 categories, then ________ dummy variables will be needed to uniquely represent these categories. a. 1 b. 2 c. 3 d. 4
If a categorical independent variable contains 4 categories, then four dummy variables will be needed to uniquely represent these categories. a. 1 b. 2 c. 3 d. 4
How many dummy variables are needed to represent the categorical variable?
We need 1 less dummy variable than outcomes for the categorical variable.
categorical variable has 2 outcomes
So, we need 2-1 = 1 dummy variable.
Note: If our categorical variable had 5 outcomes, we would need 5-1 = 4 dummy variables.
b. Write the multiple regression equation relating the dependent variable to the independent variables.
Note: Depending your your professor, this can be written one of the following ways...
yhat = a + b1 x1 + bx2
yhat = b0 + b1 x1 + b2 x2
yhat = beta0hat + beta1hat x1 + betahat2 x2
where x1 is the ratio variable
and x2 is the dummy variable
c. Interpret the meaning of the coefficients in the regression equation.
b0 = the predicted value of y when x1 and x2 both equal 0.
b1 = the change in y when x1 increases by 1 unit holding x2 constant.
b2 = the predicted difference in y between the two possible outcomes of the categorical variable holding x1 constant
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Write an equation in point-slope form for the line that passes through the point with the given slope. (–2, 11).m=4/3
Answer:
y
+
2
=
3
4
(
x
−
11
)
Explanation:
Point slope form is given by the following formula:
y
−
y
1
=
m
(
x
−
x
1
)
In order to use the point slope formula we need two things:
1. The slope or
m
which in this case is
4
3
2. A point on the graph which we are given
(
−
2
,
11
)
Along with the slope, if we let
(
−
2
,
11
)
→
(
x
1
,
y
1
)
we can substitute all this information into the point-slope formula:
y
−
−
2
=
3
4
(
x
−
11
)
Simplify:
y
+
2
=
3
4
(
x
−
11
)
hope I helped you:)
Solve a.
a+4/5= 1 1/2
Answer:
Step-by-step explanation:
a+4/5=11/2
-4/5. -4/5
_____. _____
a= 11/2 *5/5. = 55/10
-4/5 * 2/2. -8/10
______________
a= 47/10
Write the equation of the line passing through the points (-1,9) and (-9,1).
The equation of the line is…
Answer:
y=x+10Step-by-step explanation:
m=y2- y1/x2 -x1y2=1,y1=9,x2=-9,x1= -1m=1 -9/-9-(-1)m=1m= y-y1/x-x11 =y- 9/x -(-1)y-9 =x+1y =X +10what is the quotient of 1/4 divided by 3/5
Answer:
[tex]\frac{5}{12}[/tex]
Step-by-step explanation:
1/4 / 3/5
To divide fractions we flip the second one and then multiply them both and this method works for any number
1/4 x5/3
5/12
Hopes this helps please mark brainliest
Determine the amount of potential energy of a 5. 0-n book that is moved to three different shelves on a bookcase. The height of each shelf is 1. 0 m, 1. 5 m, and 2. 0 m.
The potential energy of book at heights 1m, 1.5m and 2m are 5J, 7.5J and 10J respectively.
Given,
weight of book =5
Heights of shelves are 1m, 1.5m and 2m
Potential energy can be determined by using formula, E=m*g*h
where, m*g is weight and h is height
amount of potential energy,
at height=1m
E=5*1=5 J
at height=1.5m
E=5*1.5=7.5J
at height=2m
E=5*2=10J
Thus, the potential energy of book at heights 1m, 1.5m and 2m are 5J, 7.5J and 10J respectively.
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a confidence interval for a population mean has a margin of error of 6.4. determine the length of the confidence interval.
the length of the confidence interval is 12.8
What 95% confidence interval means?You have a 5% chance of being incorrect with a 95% confidence interval. You have a 10% chance of being incorrect with a 90% confidence interval. A 95% confidence interval is narrower than a 99% confidence interval.
How do you interpret a confidence interval?The likely location of the population parameter is shown by a confidence interval. For instance, the population mean is between 9 and 11, according to the 95% confidence interval of the mean [9 11].
According to the question,
Length of confidence interval = 2*(6.4)
Length of confidence interval = 12.8
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