[tex]$W(\mathbf{x})$[/tex]is a linear transformation. it is not onto .The given statement is false.
The objective is to determine whether the following statement is true or false.
"If [tex]$$T_1(\mathbf{x})$ and $T_2(\mathbf{x})$[/tex] are onto linear transformations from [tex]$\mathbf{R}^n$[/tex] to [tex]$\mathbf{R}^m$[/tex] then [tex]$W(\mathbf{x})=T_1(\mathbf{x})+T_2(\mathbf{x})$[/tex] is an onto linear transformation."
To check whether [tex]$W(\mathbf{x})$[/tex] is a linear transformation or not:
For a, b ∈ R and for x, y ∈ [tex]$\mathbf{R}^n$[/tex],
[tex]$$\begin{aligned}W(a \mathbf{x}+b \mathbf{y})= & T_1(a \mathbf{x}+b \mathbf{y})+T_2(a \mathbf{x}+b \mathbf{y}) \quad\left(\text { Since } W(\mathbf{x})=T_1(\mathbf{x})+T_2(\mathbf{x})\right) \\= & {\left[a T_1(\mathbf{x})+b T_1(\mathbf{y})\right]+\left[a T_2(\mathbf{x})+b T_2(\mathbf{y})\right] } \\& \quad\left(\text { Since } T_1(\mathbf{x}) \text { and } T_2(\mathbf{x}) \text { are onto linear transformations }\right) \\\end[/tex]
[tex]= & a\left[T_1(\mathbf{x})+T_2(\mathbf{x})\right]+b\left[T_1(\mathbf{y})+T_2(\mathbf{y})\right] \\\\= & a W(\mathbf{x})+b W(\mathbf{y})\end{aligned}$$[/tex]
Therefore, [tex]$W(\mathbf{x})$[/tex] is a linear transformation.
To check whether [tex]$W(\mathbf{x})$[/tex] is onto:
Define the linear transformations:
[tex]$$T_1: \mathbf{R} \rightarrow \mathbf{R}$ such that $T_1(\mathbf{x})=\mathbf{x}$[/tex]
[tex]$$T_2: \mathbf{R} \rightarrow \mathbf{R}$ such that $T_2(\mathbf{x})=-\mathbf{x}$[/tex]
Then, [tex]$T_2(\mathbf{x})=-\mathbf{x}=-T_1(\mathbf{x}) \Rightarrow T_2(\mathbf{x})=-T_1(\mathbf{x})$[/tex]
The range of both functions [tex]$T_1, T_2$ is $\mathbf{R}$[/tex] This is same as the co-domain of [tex]$T_1, T_2$[/tex]. So, they are onto functions.
If A and B are the two sets, if for every element of B, there is at least one or more element matching with set A, it is called the onto function.
Therefore, [tex]$T_1, T_2: \mathbf{R} \rightarrow \mathbf{R}$[/tex]are onto linear transformations.
And, [tex]$W=T_1+T_2: \mathbf{R} \rightarrow \mathbf{R}$[/tex] is a mapping.
Now, [tex]$W(\mathbf{x})=T_1(\mathbf{x})+T_2(\mathbf{x})$[/tex]
[tex]$$\begin{aligned}& =T_1(\mathbf{x})-T_1(\mathbf{x}) \\& =0\end{aligned}$$[/tex]
So, the range of [tex]$$W(\mathbf{x})$ is $\{0\} \neq \mathbf{R}($[/tex] co-domain of [tex]$W(\mathbf{x}))$[/tex]
Then, is [tex]$W(\mathbf{x})$[/tex] not onto.
So, the given statement is false.
Hence, the correct answer is 5th option
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Tommy i buying ued book the price of the book are 180. He ue the 30% off tudent dicount and there i a 4 % ale tax. What i the total price
Using percentages we know that the total price of the book that Tommy will pay is $131.04.
What is the percentage?
In mathematics, a percentage is a number or ratio that is expressed as a fraction of 100.
Although "pct," "pct," and occasionally "pc" are also used as abbreviations, the percent symbol "%" is most usually used to denote it.
A% is a number that has neither dimensions nor a defined unit of measurement.
For instance, if you properly answered 75 out of 100 questions on a test, you would have received a 75% grade (75/100).
So, we know that the price of the book is 180.
Sales tax is 4%. So, the total price:
180/100 * 4 = 7.2$ ⇒ $187.2
Now, there is a 30% discount:
187.2/100 * 30 = $56.16
⇒ 187.2 - 56.16 = $131.04
Therefore, using percentages we know that the total price of the book that Tommy will pay is $131.04.
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Find the value of the variable
A. 180 degree
B. 25 degree
C. 55 degree
D. 125 degree
Help!!!!!
Answer:
120°
Step-by-step explanation:
125° + ( y + 5 )° + y = 360°
or, 125° + y + 5° + y = 360°
or, 2y + 130° = 360°
or, 2y = 360° - 130°
or, 2y = 230
or, y = 230/ 2
. y = 115°
. .
. ( y + 5 )°
. . ( 115 + 5 )°
.
. . 120°
Option are wrong.
discuss the continuity of the function. f(x, y) = sin(xy) xy , xy ≠ 0 1, xy = 0
The function f(x,y) is at origin [tex]\left|\frac{\sin x y}{x y}-1\right| < \varepsilon[/tex].
We can treat this function as h=xy and then it will looks like sin h/h because if we choose any path passing through original it will always continuous so above f(x,y) is continuous
[tex]$$f(x, y)= \begin{cases}\frac{\sin x y}{x y,} & \text { if } x y \neq 0 \\ 1, & \text { if } x y=0\end{cases}$$[/tex]
Choose y=mx path y→0, x→0
[tex]$$\begin{aligned}\lim _{\substack{x \rightarrow 0 \\y=\infty}} f(x, x) & =\lim _{x \rightarrow 0} \frac{\sin m x^2}{m x^2} \\& =\lim _{x \rightarrow 0} \frac{\cos m x^2 \cdot 2 m x}{2 m x}=1\end{aligned}$$[/tex]
Now consider,
[tex]$|f(n, y)-L 1=| \frac{\sin x-1}{n y}-1 \mid < \varepsilon$[/tex]
[tex]$$$\forall \varepsilon > 0$, and $|n| < \delta,|y| < d$$$=\left|\frac{\sin x y}{x y}-1\right| < \varepsilon$$[/tex]
Hence f(x,y) is continues at origin.
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Discuss the continuity of the function:
[tex]$$f(x, y)= \begin{cases}\frac{\sin x y}{x y,} & \text { if } x y \neq 0 \\ 1, & \text { if } x y=0\end{cases}$$[/tex]
Simplify the expression: (6 − 2i)(4 − 5i). a. 34 − 17i b. 14 − 17i c. 34 − 38i d. 14 − 38i
question and ente your response in the box provided. Use the
Factor the following expression: x² + x - 6
Answer:
-3
Step-by-step explanation:
x² + x = 3
3 - 6 = -3
two arithmetic sequences $a$ and $b$ both begin with $30$ and have common differences of absolute value $10$, with sequence $a$ increasing and sequence $b$ decreasing. what is the absolute value of the difference between the $51$st term of sequence $a$ and the $51$st term of sequence $b$?
The difference between 51th term of a and 51th term of b is 1000.
nth term of an arithmetic series is given by [tex]a_n = a_0 +( n-1)* d[/tex]
where [tex]a_0[/tex] is first term of arithmetic series , [tex]a_n[/tex] is nth term of the series and d is the common difference
given [tex]a_0 = b_0 = 30[/tex]
and given that both series have common difference have absolute value of 10.
common difference of series a is 10 as this series is increasing
similary common difference of series b is -10 as it is decreasing
putting the values in the above equation we get:
[tex]a_5_1[/tex] = [tex]a_0[/tex] +(n-1)*d
=> 30 + (51-1)*10
=> 30 + 500
so [tex]a_5_1[/tex] = 530
[tex]b_5_1[/tex] = [tex]b_0[/tex] +(n-1)*d
=> 30 + (51-1)*-10
=> 30 -500
so [tex]b_5_1[/tex] = -470
so we have to find value of [tex]a_5_1 - b_5_1[/tex] which is 530 - (-470) =530 +470 =1000
so the differnce of 51th term of a and b is 1000
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a
b
c
whuch is the answer look at the picture
d
Answer:
Step-by-step explanation:
the equation is given: a^2x-a=ax. for some values of a this equation has a unique solution. Solve the equation for these values of a
The solution of the equation for values of a is 1/(x - 1)
How to solve the equation for these values of aFrom the question, we have the following parameters that can be used in our computation:
a^2x - a = ax
Rewrite properly as
a²x - a = ax
Divide through by a
So, we have
ax - 1 = x
Collect the like terms
ax - x = 1
Factor out x
This gives
a(x - 1) = 1
Divide both sides by x - 1
a = 1/(x - 1)
Hence, the solution is 1/(x - 1)
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The perimeter of a regular pentagon is 100 centimeters. how long is each side?
a. 100 cm
b. 40 cm
c. 20 cm
d. 80 cm
The correct option c. 20 cm, is the length of each sides of the regular pentagon.
Explain the term regular pentagon?Every side of a regular pentagon has the same length, and its five angles are all the same size. A pentagon is said to as irregular if its side length as well as angle measurement are not equal.A quadrilateral with five equal sides is known as a regular Pentagon.
Regular pentagon perimeter equals 5*side
Since we already know that a normal pentagon's perimeter is 100 cm.
Hence, 100 = 5*side
Side = 20 cm
Each side is 20 cm long.
As a result, the side of a regular pentagon is 20 cm long.
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Resolver:
|3x − 4| = 9 - 5x
—
Answer:
Step-by-step explanation:
Which of the following is an example of an arithmetic sequence?
a.1/2, 1/4, 1/6, 1/8, ...
b. 3, 5, 7, 9, 11, ...
c.2, 6, 18, 54, ...
d.64,32,16, 8, ...
An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed number, called the common difference, to the previous term. For example, the sequence 1, 2, 3, 4, … is an arithmetic sequence because each term is obtained by adding 1 to the previous term, which is the common difference.
Out of the four given options, only the sequence 1/2, 1/4, 1/6, 1/8, … is an arithmetic sequence. The common difference of this sequence is 1/4-1/2 = -1/4, so each term is obtained by subtracting 1/4 from the previous term. The other three sequences, 3, 5, 7, 9, 11, …, 2, 6, 18, 54, …, and 64, 32, 16, 8, … are not arithmetic sequences because the difference between each consecutive pair of terms is not constant. Therefore, the correct answer is a.
Given: AB = CD
Prove: AC = BD
What reason can be used to justify statement 4 in the proof above?
the reflexive property the symmetric property the transitive property the segment addition property
Step-by-step explanation:
To prove that AC = BD, we can use the symmetric property. The symmetric property states that if two quantities are equal, then their opposites are also equal.
In this case, we are given that AB = CD. Since AB and CD are opposite quantities (they are the lengths of the diagonals of a rectangle), we can use the symmetric property to conclude that AC = BD.
Therefore, the reason that can be used to justify statement 4 in the proof is the symmetric property.
3-quart carton of milk costs $4.92. What is the price per cup?
Answer:
The price per cup would be 0.41 cents.
Step-by step explaination:
Answer:
$0.41
Step-by-step explanation:
To find the price per cup of milk, we first need to know how many cups are in a 3-quart carton.
1 quart is equivalent to 4 cups, so a 3-quart carton contains:
[tex]\sf 3\;quarts=3 \times 4 \;cups= 12\;cups[/tex]
Now, we can calculate the price per cup by dividing the total price by the number of cups:
[tex]\begin{aligned}\textsf{Price per cup} &= \dfrac{\textsf{Total price}}{\textsf{Number of cups}} \\\\&=\sf \dfrac{\$4.92}{12}\\\\&=\sf \$0.41\end{aligned}[/tex]
Therefore, the price per cup of milk is $0.41.
find the exact length of the curve. x = et − t, y = 4et/2, 0 ≤ t ≤ 4
The exact length of the curve is [tex]& \mathbf{L}=\mathbf{e}^5+\mathbf{4}[/tex].
Let the given equation is [tex]x=e^t-t, y=4 e^{\frac{t}{2}}, 0 \leq t \leq 5[/tex].
Length of Parametric Curve: A parametric curve is a function expressed in components form, such that x=f(t),y=g(t). The length of a parametric curve on the interval a ≤ t ≤ b is given by the definite integral [tex]$L=\int_a^b \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t$[/tex]
Let’s begin by getting the first derivative of the components of the function with respect to the variable t.
[tex]$$\begin{aligned}x & =e^t-t, y=4 e^{\frac{t}{2}} \\\frac{d x}{d t} & =\frac{d}{d t}\left(e^t-t\right)=\frac{d}{d t}\left(e^t\right)-\frac{d}{d t}(t)=e^t-1 \\\frac{d y}{d t} & =\frac{d}{d t}\left(4 e^{\frac{t}{2}}\right)=\left(4 e^{\frac{t}{2}}\right) \frac{d}{d t}\left(\frac{t}{2}\right)=\left(4 e^{\frac{t}{2}}\right)\left(\frac{1}{2}\right)=2 e^{\frac{t}{2}}\end{aligned}$$[/tex]
Substitute the derivatives into the following definite integral which computes the length of the parametric curve on the interval [a,b]=[0,5].
[tex]$$\begin{aligned}L & =\int_a^b \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t \\& =\int_0^5 \sqrt{\left(e^t-1\right)^2+\left(2 e^{\frac{t}{2}}\right)^2} d t \\& =\int_0^5 \sqrt{\left(e^{2 t}-2 e^t+1\right)+\left(4 e^t\right)} d t \\& =\int_0^5 \sqrt{\left(e^{2 t}+2 e^t+1\right)} d t \\& =\int_0^5 \sqrt{\left(e^t+1\right)^2} d t \\& =\int_0^5\left(e^t+1\right) d t\end{aligned}$$[/tex]
We need to find the value of the definite integral to get the exact length of the curve.
Take out the limits of integration and evaluate the resulting indefinite integral to solve.
[tex]$$\begin{aligned}L & =\left.\left[\int\left(e^t+1\right) d t\right]\right|_0 ^5 \\& =\left.\left[\int e^t d t+\int 1 d t\right]\right|_0 ^5 \\& =\left.\left[e^t+t\right]\right|_0 ^5\end{aligned}$$[/tex]
Evaluate the solution at the limits of integration to get the length.
[tex]$$\begin{aligned}& L=\left[e^{(5)}+(5)\right]-\left[e^{(0)}+(0)\right] \\& L=\left(e^5+5\right)-\left(e^0+0\right) \\& L=e^5+5-(1+0) \\& L=e^5+5-1 \\& \mathbf{L}=\mathbf{e}^5+\mathbf{4}\end{aligned}$$[/tex]
Therefore, the exact length of the curve is [tex]& \mathbf{L}=\mathbf{e}^5+\mathbf{4}[/tex].
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what is the simple linear regression model for the response variable of annual salary (as) and explanatory variable of years of education (ed)? multiple choice question.
The variable that researchers are trying to explain or predict is called the response variable. It is also sometimes called the dependent variable because it depends on another variable.
Variables used to explain or predict a response variable are called explanatory variables.
The students want to predict age using their height, so the explanatory variable is height and the response variable is age.
The values of an ordinal variable have a meaningful order. For example, an education level (with possible values of high school, bachelor's degree, and graduate degree) can be an ordinal variable.
"Hand" belongs to "Clock". "Experience" also belongs to "Year". So if he has 10 years of experience in the industry, no apostrophes are needed. An apostrophe is required if you have more than 10 years of experience.
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find the exact length of the curve. y = 4 + 6x3/2, 0 ≤ x ≤ 1
[tex]\frac{2}{243} (82^\frac{3}{2} -1)[/tex] is the required length of the curve y = 4 + 6[tex]x^\frac{3}{2}[/tex]
Given equation of curve:
y = 4 + 6[tex]x^\frac{3}{2}[/tex]
length of the curve is given by L= [tex]\int\limits^a_b {\sqrt{1+(y^')^2} } \, dx[/tex]
where [tex]y^'[/tex] is derivative of y with respect to x
so [tex]y^'[/tex] = 6*(3/2) * [tex]\sqrt{x}[/tex]
=> 9 [tex]\sqrt{x}[/tex]
given a=1 and b=0
so L= [tex]\int\limits^1_0 {\sqrt{1+(9\sqrt{x})^2 } } \, dx[/tex]
L= [tex]\int\limits^1_0 {\sqrt{1+81x } } \, dx[/tex]
let 1+81x be t
81dx = dt
as the value of x changes after the substitution of the value.
so now the L changes as:
L= 1/81 [tex]\int\limits{\sqrt{t } } \, dt[/tex]
L=1/81* [tex]\left \{ {{t=82} \atop {t=1}} \right.[/tex] [tex]\frac{t^\frac{3}{2} }{\frac{3}{2} }[/tex]
the min value of t=1 and max value of t=82
L=[tex]\frac{2}{243} (82^\frac{3}{2} -1)[/tex]
so the exact length of the curve is given by L=[tex]\frac{2}{243} (82^\frac{3}{2} -1)[/tex]
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14. 8:14
16. There are 16 pints in 2 gallons. Complete the
table to write three ratios of pints to gallons.
Number of
Pints
Number of
Gallons
16
2
I don’t understand what to do
Answer:
Step-by-step explanation:
Number of pints | 16 | 32| 48|
Number of galll. | 2 | 4 | 6 |
Those are 3 ratios
16:2
32:4
and 48:6
You multiply by 2 to get 3 ratios
Solve the equation: logsx + 3logsx = 1.
{-√5, √5)
{√5)
(-√5)
Ø
Enter the value of x below.
x+6=9
Answer:
x = 3
Step-by-step explanation:
x + 6 = 9
Subtract 6 from both sides :
x = 9 - 6
x = 3
Hope this helped and have a good day
Answer:
x = 3
Step-by-step explanation:
x + 6 = 9 Subtract from both sides of the equation
x + 6 - 6 += 9 - 6
x = 3
Check:
x + 6 = 9
3 + 6 = 9
9 = 9 Checks
Please help with questions 1 2 3 5
An integer to represent the situation, the temperature was 3° below zero is -3.
What is an Integer?Integers include all whole numbers and negative numbers. This means if we include negative numbers along with whole numbers, we form a set of integers.
1) The stock market increased 75 points
Integer to represent the given situation is +75 or 75.
2) A loss of 15 yards
Integer to represent the given situation is -15
3) Nancy owes her friend $10
Integer to represent the given situation is -10
4) Frederick is located 290 feet above sea level
Integer to represent the given situation is 290 or +290
5) The temperature was 3° below zero
Integer to represent the given situation is -3.
Therefore, an integer to represent the situation, the temperature was 3° below zero is -3.
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There are x melons and y oranges. The product of integers x and y is
539. The number of melons is 11 times of that of oranges, How many
melon(s) is / are there?
Answer:
77 melons
Step-by-step explanation:
[tex]11x \times x = 539 \\ 11 {x}^{2} = 539 \\ \frac{11x}{11} = \frac{539}{49} \\ {x}^{2} = 49 \\ \sqrt{ {x}^{2} } = \sqrt{49} \\ x = 7[/tex]
The oranges are 7
And the melons are 11 times the number of the oranges hence
11 x 7= 77
Answer:
There are 77 melons
Step-by-step explanation:
See picture below :)
The tens digit of a certain two-digit number is 1/3 of the units digit. When the digits
are reversed, the new number exceeds twice the original number by 2 more than the
sum of the digits. Find the original number
Answer:
Step-by-step explanation
The tens digit of a certain two-digit number is 1/3 of the units digit. When the digits are reversed, the new number exceeds twice the original number by 2 more than the sum of the digits. Find the original number.
Annual salary at r rupees per month along with a Christmas bonus of Rs2000. Find the total annual salary with a festive bonus.
The total annual salary with a festive bonus is Rs (12r+2000)
What is multiplication?In maths, multiply means the repeated addition of groups of equal sizes.
Given that, there is an annual salary at r rupees per month along with a Christmas bonus of, Rs 2000.
Here, a person is getting monthly salary = Rs r
Therefore, his annual salary = Rs r × 12
= Rs 12r
He is also getting a bonus for Christmas = Rs 2000
So, in total, he will get = Rs (12r+2000)
Hence, The total annual salary with a festive bonus is Rs (12r+2000)
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Complete the square-
x^2-1/2x=8/16
thank you :)))
Answer: (x- 1/4)^2=9/16
Step-by-step explanation:
x^2 -1/2x =8/16
(x^2-1/2x +1/16)^2= 8/16+1/16)
(x- 1/4)^2=9/16
proof: fill in the reasons for the proof
∠1≅∠2 if RECT is the rectangle and M is the midpoint of EC.
What is Triangle?A triangle is a three-sided polygon that consists of three edges and three vertices.
RECT is the rectangle and M is the midpoint of EC.
we need to prove angle one is congruent to angle two.
By given,RECT is the rectangle and M is the midpoint of EC.
∠E≅∠C
In the rectangle all the angles are 90 degrees.
∠EM≅∠MC.
As M is midpoint, it has equidistant from E and C.
ER≅CT both sides are congruent.
ΔERM≅ΔCTM both are similar triangles.
RM≅TM both sides are congruent.
∠1≅∠2 by SAS postulate.
Hence, RECT is the rectangle and M is the midpoint of EC then ∠1≅∠2
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In the entence, the author want to ue alliteration. Which choice provide the bet example of thi?
Ted abolutely aced hi teaching exam
The choice that provides the best example of alliteration for the given sentence is tested terrifically on teaching. (Option C)
Alliteration is a literary device in which there is a noticeable repetition of initial consonant sounds in consecutive or nearby words in a phrase. It refers to the occurrence of the same sound or letter at the beginning of adjacent or closely connected words. Alliteration is created by repeated sound at the start of the words and not the repeated letter. For example, the phrase “kids’ coats” is alliterative, and the phrase “phony people” is not alliterative. In the given sentence, author can create alliteration by using tested terrifically in place was absolutely aced.
Note: The question is incomplete. The complete question probably is: In the sentence, the author wants to use alliteration. Which choice provides the best example of this? Ted absolutely aced his teaching exam. A) NO CHANGE B) was astounded at how well he did on C) tested terrifically on teaching.
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[tex]evaluate 2^{-1}(\alpha +\beta)^{2} if \alpha and \beta are the roots of the quadratic equaton x^{2}-4x+1=0[/tex]
The value of α + β is 4. Then the value of the expression 2⁻¹ x (α + β)² will be 8.
What are the sum and product of the roots?Let the equation be ax² + bx + c = 0 and the roots are α and β.
Then the sum of the roots will be
α + β = - b / a
And the product of the roots will be
α · β = c / a
The quadratic equation is given below.
x² - 4x + 1 = 0
The sum of the roots of the equation is given as,
α + β = - (-4) / 1
α + β = 4
Then the value of the expression will be given as,
⇒ 2⁻¹ x (α + β)²
⇒ 2⁻¹ x (4)²
⇒ 16 / 2
⇒ 8
The value of the expression 2⁻¹ x (α + β)² will be 8.
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Wanda sews small and large gloves. It takes her 45 minutes to sew a small pair of gloves and 120 minutes to sew a large pair of gloves. The costs of producing the gloves are $2 for a small pair and $4 for a large pair. Wanda has 16 hours available to sew gloves. The materials to make the gloves must cost at most $40. The system of linear inequalities represents this situation. {45x+120y≤9602x+4y≤40 What does the solution (16, 2) represent?
Suppose n represents a power of 10.
What is the value of n when 3562 is rounded to the nearest power of 10?
Answer:
We get the approximate value of [n] as 5.
What is logarithm? What is a mathematical equation and expression?
A quantity representing the power to which a fixed number (the base) must be raised to produce a given number. We can write -
$$$\log_{b}({b^x})=x$$
A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions.
A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have 31,100 is rounded to the nearest power of 10 and [N] represents a power of 10.
We can write -
We can write -10ⁿ = 31100
We can write -10ⁿ = 3110010ⁿ = 311 x 10²
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311log(10ⁿ ⁻ ²) = log(311)
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311log(10ⁿ ⁻ ²) = log(311)(n - 2) log 10 = log (311)
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311log(10ⁿ ⁻ ²) = log(311)(n - 2) log 10 = log (311)n - 2 = log(311)/log(10)
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311log(10ⁿ ⁻ ²) = log(311)(n - 2) log 10 = log (311)n - 2 = log(311)/log(10)n - 2 = 2.5
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311log(10ⁿ ⁻ ²) = log(311)(n - 2) log 10 = log (311)n - 2 = log(311)/log(10)n - 2 = 2.5n = 4.5
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311log(10ⁿ ⁻ ²) = log(311)(n - 2) log 10 = log (311)n - 2 = log(311)/log(10)n - 2 = 2.5n = 4.5n = 5 (approx.)
We can write -10ⁿ = 3110010ⁿ = 311 x 10²10ⁿ ⁻ ² = 311log(10ⁿ ⁻ ²) = log(311)(n - 2) log 10 = log (311)n - 2 = log(311)/log(10)n - 2 = 2.5n = 4.5n = 5 (approx.)Therefore, we get the approximate value of [n] as 5.
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find the area of the region bounded by the given curves. y = sin2(x), y = sin3(x), 0 ≤ x ≤ π
The area of the region bounded by the given curves is 0.7123 square units
In this question we have been given two curves y = sin^2(x), y = sin^3(x), 0 ≤ x ≤ π
We need to find the area of the region bounded by the given curves.
We know that the formula for the area between two curves f and g :
A = |∫_[a to b] [f(x) - g(x)] dx|
here, a = 0, b = π, f(x) = sin^2(x) and g(x) = sin^3(x)
So, A = |∫_[0 to π] [sin^2(x) - sin^3(x)] dx|
A = |∫_[0 to π] sin^2(x) dx - ∫_[0 to π] sin^3(x) dx|
consider ∫_[0 to π] sin^2(x) dx
= ∫_[0 to π] (1 - cos(2x) / 2) dx
= π/2
Now consider ∫_[0 to π] sin^3(x) dx
= ∫_[0 to π] (1 - cos^2(x)) sin(x) dx
= 4/3
So A = |π/2 - 4/3|
= 3π - 8/2
= 0.7123
Therefore, the required area: 0.7123 square units
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