R = 1 / L = ∞. Thus, the radius of convergence of the given power series is ∞. The interval of convergence is (-1,1) is the answer.
a)The indefinite integral of x7ln(1 + x) can be obtained by using the formula for integration by parts. For the same, we need to select the parts as u and dv, such that on differentiating u and integrating dv, the obtained integrals get easier to solve.
Let us select x7 as u and ln(1 + x)dx as dv.u = x7 => du/dx = 7x6 => du = 7x6dx, and v = ∫ ln(1 + x)dx.
Using u and v, we can express the integral as,x7ln(1 + x)dx= ∫ u dv= uv - ∫ v du= x7 ln(1 + x) - ∫ 7x6/ (1 + x) dx = x7ln(1 + x) - 7 ∫ x6/ (1 + x) dxThe indefinite integral of the term ∫ x6/ (1 + x) dx can be obtained by the substitution method, let t = 1 + x, then x = t - 1, and dx = dt.∫ x6/ (1 + x) dx= ∫ (t - 1)6/t dt= ∫ (t6 - 6t5 + 15t4 - 20t3 + 15t2 - 6t + 1)/ t dt= ∫ t6/t dt - 6 ∫ t5/t dt + 15 ∫ t4/t dt - 20 ∫ t3/t dt + 15 ∫ t2/t dt - 6 ∫ t/t dt + ∫ 1/t dt= ∫ t5 dt - 6 ∫ t4 dt + 15 ∫ t3 dt - 20 ∫ t2 dt + 15 ∫ t dt - 6 ln|t| + ln|t| + C= t6/6 - 6t5/5 + 15t4/4 - 20t3/3 + 15t2/2 - 6 ln|t| + C.
Substituting the value of t, we get the indefinite integral of the original expression as,x7 ln(1 + x)dx= x7ln(1 + x) - 7 [x6/6 - 6x5/5 + 15x4/4 - 20x3/3 + 15x2/2 - 6 ln|1 + x|] + C= x7ln(1 + x) - x6 - 42x5/5 - 280x4/4 - 1125x3/3 - 1875x2/2 - 3150x - 735 ln|1 + x| + C.
Now, we need to obtain the power series for f(x) = x7ln(1 + x).
The formula to obtain the power series for f(x) = (1 / 1 - x)2 is as follows,f(x) = Σn=0 ∞ (n + 1)xn.
The integral x7ln(1 + x) can be written as Σn=1 ∞ (-1)n-1 xn / n.
Therefore, the power series for x7ln(1 + x) can be written as,f(x) = ∑n=1 ∞ (-1)n-1 xn / n= -x + x2/2 - x3/3 + x4/4 - x5/5 + x6/6 - x7/7 + ...= C + ∑n=1 ∞ (-1)n-1 xn / n, Where C is a constant, we can evaluate the value of C by substituting x = 0 in the power series. f(0) = 0, therefore, the constant C = 0.
Now, we need to obtain the radius of convergence of the obtained power series using the formula for the radius of convergence, R = 1 / lim supn→∞ |an|where an is the nth term in the power series.
In this case, |an| = |(-1)n-1 / n| = 1 / n.Let L = lim supn→∞ |an| = limn→∞ |an| = 0Therefore, R = 1 / L = ∞Therefore, the radius of convergence is ∞.
b)To obtain the power series for the given function f(x) = 13 / (x2 - 5x - 36), we need to first perform the partial fraction decomposition of the given function. The partial fraction decomposition of the given function is given as follows,f(x) = 13 / (x2 - 5x - 36)= 13 / [(x - 9)(x + 4)] = A / (x - 9) + B / (x + 4) where A and B are constants.
To obtain the values of A and B, we can equate the numerators on both sides and solve for A and B.13 = A(x + 4) + B(x - 9)At x = 9, we get 13 = 13B, B = 1.At x = -4, we get 13 = -4A, A = -13/4.
Therefore, the partial fraction decomposition of the given function is,f(x) = 13 / (x2 - 5x - 36)= -13/4 * 1 / (x + 4) + 1 / (x - 9)
Now, we can write the power series for the above partial fractions. The power series for 1 / (1 - x) is given by,f(x) = Σn=0 ∞ xn, |x| < 1
The power series for 1 / (x + 4) is given by,f(x) = -1/4 * Σn=0 ∞ (-x / 4)n, |x / 4| < 1
The power series for 1 / (x - 9) is given by,f(x) = Σn=0 ∞ (x / 9)n, |x / 9| < 1
Substituting the above power series in the original function, we get the power series for the given function as,f(x) = -1/4 * Σn=0 ∞ (-x / 4)n + Σn=0 ∞ (13 / 4) (x / 9)n= Σn=0 ∞ [-1/9 * (x / 4)n - 1/4 * (-x) n](x / 9)
Therefore, the power series for the given function is,f(x) = Σn=0 ∞ [-1/9 * (x / 4)n - 1/4 * (-x) n](x / 9)
The given function f(x) has the power series representation, f(x)= ∑n=0 ∞ [-1/9 * (x/4)n - 1/4 * (-x)n] * (x/9)n where c= 0.
Now, the radius of convergence of a power series is given by the formula,R = 1 / lim supn→∞ |an| where an is the nth term in the power series.
In this case, the nth term in the power series is given by |an| = |(-1)n-1 / 9 * 4n-1 | + |(-1)n / 4 * n|Let L = lim supn→∞ |an|. L = limn→∞ |(-1)n-1 / 9 * 4n-1 | + |(-1)n / 4 * n|L = 0 + 0 = 0.
Therefore, R = 1 / L = ∞Thus, the radius of convergence of the given power series is ∞.The interval of convergence is (-1,1)
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A carpet cleaner charges $50.97 to travel to the customers house and $15.11 per room cleaned. In the carpet cleaner charged $96.97 how many rooms did he clean?
True/False: if the probably of committing a type ii error is 0.20, then the computed value for power is 0.80.
False. Type II error and power are complementary probabilities. The power of a statistical test is the probability of rejecting the null hypothesis when it is false (i.e., correctly detecting a significant effect or difference).
On the other hand, the type II error is the probability of failing to reject the null hypothesis when it is false (i.e., failing to detect a significant effect or difference).
Therefore, if the probability of committing a type II error is 0.20, then the power of the test is 1 - 0.20 = 0.80. This means that there is an 80% chance of correctly detecting a significant effect or difference.
However, we cannot determine the power of a statistical test from the probability of committing a type II error alone. We would need additional information such as the sample size, effect size, alpha level, and variability to calculate the power of the test.
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simplify -2+3(1-4)-2
Answer:
-13
Step-by-step explanation:
−2+3(1−4)−2
=−2+(3)(−3)−2
=−2+−9−2
=−11−2
=−13
hope it helped
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Find the value of the sine for
Answer:
3/5
Step-by-step explanation:
Use Pythagorean to figure out that the is 100. Then do 60/100 and simplify
theoremhypotenuse
The length of a rectangle is three times its width. If the perimeter is 120 inches, what is the length? The width?
Answer:
Length=45 inches, width=15 inches
Step-by-step explanation:
Length=3w
Width=w
Can be written as 3w+w+3w+w
Combine like terms to get 8w
120/8=15
w=15
Width=15 inches
Length=3(15)=45
Length=45 inches
In Canada, the rate of inflation (i.E. How much things cost) is about 2.0%/year. In other words, the real value of your money is worth 2.0% less each year. Imagine you have $2250 in savings under your mattress
Answer:
Step-by-step explanation:
If you have $2250 as against an inflation rate of 2% every year, this then means that after the first year, the value of your money becomes
2250 * 2% =
2250 * 0.02 = 45
2250 - 45 = $2205
At the end second year
$2205 * 2% =
$2205 * 0.02 = 44.1
$2205 - 44.1 = 2160.9
At end of the third year
$2160.9 - 2% =
$2160.9 * 0.02 = 43.218
$2117.682
And so on, and so forth
200% of what number is 350?
Answer:
The answer is 175
1 The legs of a triangle are 6 cm and 8 cm. The hypotenuse is 14 cm. Is it a right triangle?
Answer:
this is not a right triangle
Step-by-step explanation:
^ = squared
The formula forthe Pythagorean theorem is a^+ b^ = c^
If this is a right triangle this must be true
6^ + 8^ = 14^
36 + 64 = 196
100 is not equal to 196
So this is not a right triangle
Determine whether the given differential equation is exact. If it is exact, solve it. (If it is not exact, enter NOT.) dy/ dx = 9xe^x – y + 6x2
The given differential equation is not exact. To determine if a differential equation is exact, we need to check if its partial derivatives satisfy a specific condition.
In this case, the given equation is dy/dx = 9xe^x – y + 6x^2. Let's find the partial derivatives of the expression on the right-hand side with respect to y and x.∂/∂y (9xe^x – y + 6x^2) = -1∂/∂x (9xe^x – y + 6x^2) = 9e^x + 12x
The condition for exactness is that these partial derivatives are equal: ∂/∂y (∂/∂x) = ∂/∂x (∂/∂y). However, in this case, we have -1 ≠ 9e^x + 12x, which means the equation is not exact.As the equation is not exact, we cannot directly solve it by finding a potential function. Other methods, such as using integrating factors or applying specific techniques for solving non-exact equations, may be required.
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if a club has 20 members and 4 officers how many choices are there for a secretary
Answer:
20 x 4 = 60
Step-by-step explanation:
because it's says how many
Answer:
18 maybe
Step-by-step explanation:
it's either 20,19,or 18
but the secretary is the third to be elected so the answer is 18 maybe and if that doesn't work try one of the others trust me it's one of them.
hope this helps
guys i need help ASAP!!!
Answer:
Q7 0.2
Q8 0.17
Step-by-step explanation:
Probability is the ratio of the number of possible outcome to the number of total outcome. The probability that an event will happen when added to the probability that it will not happen is 1
Given that the probability that the bulb will grow is 0.8, the probability it would not grow
= 1 - 0.8
= 0.2
A match may be won lost or drawn
Given that the probability of a win is 0.28 and that of a draw is 0.55, the probability of a loss
= 1 - (0.28 + 0.55)
= 0.17
Find the surface area (help me please ╥﹏╥ )
Answer:
100
Step-by-step explanation:
Add all of the inches together including the parts you don't see.
one side is 9 inches. there are 8 sides. so 9x8. One side in the box is 7 inches. there are 4 sides. so 7x4. 9x8 + 7x4 is equal to 72 + 28.
72 + 28=100
A spherical globe has a diameter of 10 inches. What is the approximate volume of the globe?
Answer:
523.33 in^3
Step-by-step explanation:
A sphere is a 3-dimensional version of a circle. An example of a sphere is a ball
Volume of a sphere = [tex]\frac{4}{3}[/tex]π[tex]r^{3}[/tex]
Where
π = 3.14
r = radius
the diameter is the straight line that passes through the centre of a circle and touches the two edges of the circle.
A radius is half of the diameter
diameter = 2r
Radius = 10 in / 2 = 5 inches
[tex]\frac{4}{3}[/tex] × 3.14 ×[tex]5^{3}[/tex] = 523.33 [tex]in^{3}[/tex]
if f(5) = 11, f ′ is continuous, and 6 5 f ′(x) dx = 19, what is the value of f(6)?
If f(5) = 11, f ′ is cοntinuοus, and 6 5 f ′(x) dx = 19, then f(6) = 30.
What is Fundamental Theοrem οf Calculus?Tο find the value οf f(6), we can use the Fundamental Theοrem οf Calculus. Accοrding tο the theοrem, if f(x) is cοntinuοus οn an interval [a, b] and F(x) is an antiderivative οf f(x) οn that interval, then the definite integral οf f(x) frοm a tο b is equal tο F(b) - F(a).
Given that f'(x) is cοntinuοus, we can apply the theοrem tο the integral:
∫₅₆ f'(x) dx = f(6) - f(5)
We are given that ∫₅₆ f'(x) dx = 19, and f(5) = 11. Plugging in these values, we have:
19 = f(6) - 11
Tο sοlve fοr f(6), we add 11 tο bοth sides:
f(6) = 19 + 11
f(6) = 30
Therefοre, the value οf f(6) is 30.
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Match the sides based on their relationship to A
PLEASE ANSWER ASAP
Answer:
13 hypotenuse
12 adjacent
5 opposite
will give yall brainliest if yall answer this question
The probability of a vanilla taffy is given as follows:
P(vanilla) = 0.31.
Hence the probability of a vanilla or banana taffy is given as follows:
P(vanilla or banana) = 0.52.
How to calculate a probability?The two parameters that are needed to calculate a probability are listed as follows:
Number of desired outcomes in the context of a problem or experiment.Number of total outcomes in the context of a problem or experiment.Then the probability is then calculated as the division of the number of desired outcomes by the number of total outcomes.
When we are given the relative frequencies, we must simply identify the desired outcomes, as follows:
P(banana) = 0.21.P(vanilla) = 0.31.Hence the probability is given as follows:
P(vanilla or banana) = P(vanilla) + P(banana) = 0.52.
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What is the area of the triangle in square yards?
6 yd.
15 3/4 yd.
hurry plss
Answer:
im guessing 6 is your base and your height is 15 3/4.
Formula of a triangle we must remember:
BH x 1/2 (basically dividing by 2)
94.5 (fraction form will be: 94 1/2)
94.5 divided by 2 = 47.25 <---------- decimal form
94 1/2 divided by 2 = 47 1/4 <------- fraction form
Andre has a summer job selling magazine subscriptions he earned $25 per week plus $3 for every subscription he sells.Andre hopes to make at least enough money to buy a new pair of soccer cleats
Answer:
Step-by-step explanation:
That Would be 43/3
Write the word sentence as an equation.
12 less than a number m equals 20.
An equation that represents this word sentence is
Answer:
x - 12 = 20
Step-by-step explanation:
If I'm wrong do x = 12 - 20
What is the volume of this cylinder?
6 m
7 m
Round your answer to the nearest tenth.
Answer:
(Refer below:)
Step-by-step explanation:
Because there is no image, here's what I'm going to assume: there are multiple scenarios. Refer to each one below:
1) Where r = 6 and h = 7:
V = Bh
V = (π r²) (h)
V = (6² π) (7)
V = 36π (7)
V = 252π
2) Where r = 7 and h = 6:
V = Bh
V = (π r²) (h)
V = (7² π) (6)
V = 49π (6)
V = 294π
3) Where d = 6 and h = 7:
d = 2r = 6
r = 3
V = Bh
V = (π r²) (h)
V = (3² π) (7)
V = 9π (7)
V = 63π
4) Where d = 7 and h = 6:
d = 2r = 7
r = 3.5
V = Bh
V = (π r²) (h)
V = (3.5² π) (7)
V = 12.25π (7)
V = 87.75π ≈ 269.39
a car can travel 174 miles in 4 hours how many miles can the car travel per hour
Answer:
43.5
divide 174 ÷ 4 and you get 43.5
HELPP ME PLSSS ASAP NO BOTS OR I WILL REPORT The graph below is the graph of a function.
Answer:
True
Step-by-step explanation:
That graph below is a parabola that can be a function for example f(x)=x^2+3x-12 is a graph of a parabola
Hope that helps :)
A 13 foot ladder is leaning against the wall of a house. The ladder is 5 feet away from th
base of the house. How high up the house does the ladder reach?
Answer:
12 feet
Step-by-step explanation:
4. verify the following identities 0 (a) 1 = 10 (x) + 2 (-1)"l2n(x) n=1 00 (b) e* = 10(x) +2 1. (x) = n=1 0 c) (c) e-* = 10(x) +2X(-1)"In(x) -X = n=1 0 (d) cosh x = 10(x) +2 Izn (x) X d n=1 00 (e) sinh x = 222n-1(x) - n=1
The given identities can be verified using basic rules of exponentials and algebra. Here are the steps to verify the given identities:
(a) 1 = ∑_(n=1) ^∞▒〖10^(x)+2(-1) ^nln(x)〗0
Rewrite the sum to obtain two series, one for even values of n, and one for odd values of n. ∑_(n=1)^∞▒10^(x) = 10^x+10^x+...= (2/3) (10^x) (odd terms only)∑_(n=1)^∞▒〖(-1)^nln(x)〗= ln(x)-ln(x)+ln(x)-ln(x)+...= (0) (even terms only)
Thus, we have 1= (2/3) (10^x) + (0) = (2/3) (10^x) (b) e^x = ∑_(n=1) ^∞▒〖10^x+2n(x)〗
Rewrite the sum to obtain two series, one for even values of n, and one for odd values of n.
∑_(n=1) ^∞▒10^(x) = 10^x+10^x+...= (1/2) (10^x) (even terms only) ∑_(n=1) ^∞▒〖2n(x)〗= 2(x)+2(2x) +2(3x) +...= 2x (1+2+3+...) = -x/(-1) ^2= -x
Thus, we have e^x= (1/2) (10^x) - x(c) e^(-x) = ∑_(n=1) ^∞▒〖10^x+2(-1) ^nln(x)〗0
We can use the same series from part (a) with x replaced by -x.
Thus, we have e^(-x) = (2/3) (10^(-x)) + (0) = (2/3) (1/10^x)
Similarly, e^x= (2/3) (10^x) Subtracting these two equations, we get: e^x - e^(-x) = (2/3) (10^x + 1/10^x) (answer) (d)
Cosh x = ∑_(n=0) ^∞▒〖10^x+2n(x)〗
Similar to part (b), we have two series, one for even values of n, and one for odd values of n.∑_(n=0)^∞▒10^(x) = 1+10^x+10^x+...= (1/2) (10^x) (even terms only)∑_(n=0)^∞▒〖2n(x)〗= 0+2x+2(2x)+2(3x)+...= 2x (1+2+3+...)= 2x(1/(-1)^2)= 2xThus, we have Cosh(x) = 1 + (1/2) (10^x) + 2x (e^x)(e^x - e^(-x)) / 2= 1 + (1/2) (10^x) + 2x (1 + (2/3) (10^x + 1/10^x)) (answer)(e) Sinh x = ∑_(n=1)^∞▒〖2^(2n-1)(x)〗
Rewrite the sum to obtain two series, one for even values of n, and one for odd values of n.∑_(n=1)^∞▒〖2^(2n-1)(x)〗= 2x+2^3x+2^5x+...= 2x(1+2^2+2^4+...)= 2x(2^0+2^2+2^4+...)= 2x(1/ (1-2^2))= -2x/3Thus, we have Sinh(x) = ∑_(n=1)^∞▒〖2^(2n-1)(x)〗= 2x/3 (answer)
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The length of a rectangle is 3ft less than double the width, and the area of the rectangle is 14ft^2 find the dimensions of rectangle.
Answer: [tex]4\times 3.5\ ft^2[/tex]
Step-by-step explanation:
Given
area of the rectangle is [tex]A=14\ ft^2[/tex]
Suppose width is given by w
So, length is [tex]l=2w-3[/tex]
The area of a rectangle is [tex]A=l\times w[/tex]
putting values
[tex]\Rightarrow A=(2w-3)w\\\Rightarrow 14=2w^2-3w\\\Rightarrow 2w^2-3w-14=0\\\Rightarrow 2w^2-7w+4w-14=0\\\Rightarrow w(2w-7)+2(2w-7)=0\\\Rightarrow (2w-7)(w+2)=0\\\\\Rightarrow w=\dfrac{7}{2}\ ft[/tex]
length is [tex]l=4\ ft[/tex]
A rod of length 30.0 cm has linear density (mass per length) given by
λ=50.0+20.0 x
where x is the distance from one end, measured in meters, and λ is in grams/ meter.
What is the mass of the rod?
The mass of the rod is 15.9 grams.
To find the mass of the rod, we need to integrate the linear density function over the entire length of the rod. The linear density function is given by λ = 50.0 + 20.0x, where x is the distance from one end measured in meters.
The mass of an infinitesimally small element of length dx is given by dm = λ*dx. Substituting the linear density function, we have dm = (50.0 + 20.0x)*dx.
Integrating both sides from x = 0 to x = 0.3 meters (corresponding to the length of the rod), we get:
∫dm = ∫(50.0 + 20.0x)dx
m = ∫(50.0 + 20.0x)dx
m = [50.0x + 10.0x^2] evaluated from x = 0 to x = 0.3
m = 50.00.3 + 10.0(0.3)^2
m = 15.0 + 0.9
m = 15.9 grams.
Therefore, the mass of the rod is 15.9 grams.
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ODE solutions using Laplace transforms
Solve the following initial value problem using Laplace transforms:
y" - 2y = e^x, y(0) = 0, y'(0) = 0.
The solution of the given differential equation using the Laplace transform is y(t) = 1 − [tex]e^{2t}[/tex] + 2 [tex]e^{-t}[/tex].
Given differential equation is y″−2y = eˣ, y(0) = 0, y′(0) = 0. We will solve it using Laplace transforms.
To solve the given differential equation using the Laplace transform, take the Laplace transform of each term of the differential equation. Then solve for Y(s) and find y(t) by taking the inverse Laplace transform.
Here, we take L{y} for Y(s) and L{y''} and L{y'} for s²Y(s) and sY(s) respectively.
L[y′′] − 2L[y] = L[eˣ] s²Y(s) − s⋅y(0) − y′(0) − 2Y(s)
= 1s − 1.0 − 0 − 2Y(s)
= 1s − 1
We have to rearrange the terms of the above equation in order to get Y(s) on one side.
Y(s) = 1s(s−2)(s−1) = As+Bs−2+Cs−1.
Using partial fraction expansion, A=1, B=1, and C=−2.
So,Y(s) = 1s+(1s−2)−2s−1.
We can write Y(s) as follows.Y(s) = 1s−(1s−2)+2s−1.
Then find the inverse Laplace transform of Y(s) to get the solution y(t).
y(t) = L − 1{Y(s)} = L − 1{1s} − L − 1{1s−2} + L − 1{2s−1}
= 1 − [tex]e^{2t}[/tex] + 2 [tex]e^{-t}[/tex].
Therefore, the solution of the given differential equation using Laplace transform is y(t) = 1 − [tex]e^{2t}[/tex] + 2 [tex]e^{-t}[/tex].
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Help me really please asp
Answer:
y1=4
x2=7
y2=-3
For this problem, type your answers directly into the provided text box. You may use the equation editor if you wish, but it is not required. Consider the following series. 3 Σa_₁n³e-n² n=1 Part 1 (2 points). State whether the series converges or diverges. Part II (3 points). Justify your result in part I by using an appropriate test (basic divergence test, integral test, basic comparison test, or limit comparison test). Make sure to briefly state how you applied the test.
By using the basic comparison test and limit comparison test for the series: 3Σn=1 to ∞ a₁n³e⁻ⁿ², The number π²/2 is a finite number, and the series 3Σn=1 to ∞ a₁n³e⁻ⁿ² is convergent.
Part 1: We need to identify whether the given series is converging or diverging.
Part II: We can use the basic comparison test to test the convergence of the given series. Let's compare the given series with another series that is easily recognizable and convergent. For
n ≥ 1, a₁n³e⁻ⁿ² > 0
We know that the series Σ a₁n³e⁻ⁿ² is convergent, which means that the sequence of terms must be converging to zero as well. This can be shown using the following limit statement.
lim n→∞ a₁n³e⁻ⁿ² = 0.
By comparing the series with a simpler convergent series, we can prove that the series is convergent. The simpler convergent series can be represented by the expression 3Σn=1 to ∞ 1/n². This series is known as the p-series with p = 2. We can use this series to perform a comparison test.
Let Sn be the nth partial sum of the given series. Thus,
3Σn=1 to ∞ a₁n³e⁻ⁿ² ≤ 3Σn=1 to ∞ 1/n²
Let's evaluate the second series using the formula for the sum of a p-series:
Σn=1 to ∞ 1/n² = π²/6
Thus, 3Σn=1 to ∞ a₁n³e⁻ⁿ² ≤ 3 x π²/6
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can anyone help me on this question but explain? What’s the difference between gross and net income
Answer & Explanation:
Gross income is the total pay you earn before any reduction and adjustments (taxes, mortgage, etc.).
Net income is the amount of pay you get after taxes and other reductions are taken out of your gross income; it's the money that you "actually get to take home in your pocket with" on the payday.