Answers
Answer:
g(x) = 4x² - x
General Formulas and Concepts:
Algebra II
Functions
Function NotationPiecewise FunctionsCalculus
Integration
IntegralsIntegral NotationIntegration Constant CIntegration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Step-by-step explanation:
*Note:
Integrating a piecewise function requires you to integrate both parts.
Step 1: Define
Identify.
[tex]\displaystyle f(x) = \left \{ {{8x - 1 ,\ x \leq 4} \atop {31 ,\ x \geq 4}} \right.[/tex]
[tex]\displaystyle \int {f(x)} \, dx = \left \{ {{g(x) + C ,\ x \leq 4} \atop {31x + C ,\ x \geq 4}} \right.[/tex]
Step 2: Find function g(x)
We can see that the 2nd part of the piecewise function already has been integrated:
[Integral] Set up: [tex]\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = \int {31} \, dx ,\ x \geq 4[/tex][Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = 31 \int {} \, dx ,\ x \geq 4[/tex][Integral] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = 31x + C ,\ x \geq 4[/tex]To find function g(x), we simply have the same setup:
[Integral] Set up: [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = \int {8x - 1} \, dx ,\ x \leq 4[/tex][Integral] Rewrite [Integration Rule - Addition/Subtraction]: [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = \int {8x} \, dx - \int {1} \, dx ,\ x \leq 4[/tex][Integrals] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 8 \int {x} \, dx - \int {} \, dx ,\ x \leq 4[/tex][Integrals] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 8 \bigg( \frac{x^2}{2} \bigg) - x + C ,\ x \leq 4[/tex]Simplify: [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 4x^2 - x + C ,\ x \leq 4[/tex]Redefine: [tex]\displaystyle g(x) = 4x^2 - x + C ,\ x \leq 4[/tex]The integration constant C is already included in the answer, so our answer is g(x) = 4x² - x.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration (Applications)
Related Questions
The sum of the angles of a triangle is 180 degrees. The second angle is 40 degrees more than the first
angle and the third angle is 5 times the measure of the first angle. What is the measure of the largest
angle?
Answers
Answer:
100⁰
Step-by-step explanation:
Let,
First angle = x
Second angle = x + 40
Third angle = 5x
=> x + x + 40 + 5x = 180
=> 7x + 40 = 180
=> 7x = 180 - 40
=> 7x = 140
=> x = 140/7
=> x = 20
Largest angle = 5x = 5(20) = 100⁰
[tex]\huge{\purple{\underline{\underline{\bf{\pink{ANSWER:-}}}}}}[/tex]
First angle = x Second angle = x + 40Third angle = 5x[tex]:\implies\bf{x + x + 40 + 5x = 180} \\ \\ :\implies\bf{7x + 40 = 180} \\ \\ :\implies\bf{7x = 180 - 40} \\ \\ :\implies\bf{7x = 140} \\ \\ :\implies\bf{x = \frac{140}{7} } \\ \\ :\implies\bf{x = 20}[/tex]
The largest angle above is 5x thus,
[tex]:\implies\bf{5x} \\ \\ :\implies\bf{5 \times 20} \\ \\ :\implies\bf{100^°}[/tex]
Can someone help me? Will make brainiest :)
Answers
Answer:
D
Step-by-step explanation:
BH is perpendicular to AC since an orthocenter is a point at which the three altitudes of a triangle intersect. The altitude of a triangle is that line that passes through its vertex and is perpendicular to the opposite side. Since you know that BH is an altitude due to Point H being an orthocenter, you know that BH is perpedendicular to AC.
Melllissa owns 4 sets of jewelry. Each set contains 2 earrings, 1 neckles, 1 bracelet, 1 ring. How many pieces of jewelry does Melissa have all together?
Answers
Answer:
The answer is to the problem is 18.
Sue has 100 dimes and quarters. If the total value of the coins is &21.40, how many of each kind of coin does she have?
Answers
Answer:
Let x be the number of dimes and y the number of quarters
x +y = 100
a dime is $0.10 and a quarter is $0.25
0.1x + 0.25y = 21.40 multiply by 100 to remove the decimals
10x + 25y = 2140, From the first equation y = 100 - x and plug it in to the second equation.
10x + 25(100 -x) = 2140
10x + 2500 - 25x =2140
2500 - 15x = 2140
360 = 15x x = 24 and y = 76
There are 24 dimes and 76 quarters
Answer:
76 Quarters = $19.00
24 Dimes = $2.40
$19.00 + $2.40 = $21.40
Step-by-step explanation:
Yaga
**WILL GIVE BRAINLIEST IF ANSWERED RIGHT** (this is not math but i just put it as math)
Choose all the answers that apply.
Which are benefits of cardiovascular exercise?
decreases the resting heart rate
helps the blood vessels dilate more easily
decreases the stroke volume
weakens the muscles surrounding the lungs
makes the lungs able to fill more completely
Answers
Answer:
Makes the lungs able to fill more
PLEASE HELLPPPPPP IDONTKNOWTHISSS
Solve this system of equations using the Substitution method. Don't forget to write your final answer as a coordinate point. You must show all your work.
3x + y = -2
y = 2x + 3
Answers
= 3x+y=-2
= Linear equations with two variables have multiple infinite solutions.
= Assuming x=0
then y=-2
= Assuming x=1
then y=-2-3
= y=-5
= Assuming x=2
then y=-2-6
= y=-8
Writing them as coordinate points -
(0,-2),(1,-5),(2,-8)...and so on.
__________________________.
= y=2x+3
Assuming x=0
then y=3
Assuming x=1
then y=2+3
= y=5
Assuming x=2
then y=4+3
= y=7
Writing them as coordinate points -
(0,3),(1,5),(2,7)...and so on.
Answer:
Point Form:
( − 1 , 1 )
Equation Form:
x = − 1 , y = 1
Step-by-step explanation: there you go
A jar contains black marbles and green marbles. Sixty percent of the marbles in the jar are green. A number generator simulates randomly selecting 10 marbles from the jar. The number generator is used 12 times and the number of green marbles in each trial is shown in the dot plot.
Which description is correct about the number generator being fair or not?
The number generator is not fair. The dot plot shows 80% of the marbles drawn are green.
The number generator is fair. The dot plot shows a distribution that is skewed right.
The number generator is not fair. In most of the experiments, considerably less than 60% of the selected marbles are green.
The number generator is fair. It shows that 60% of the marbles selected are green most of the time.
Answers
Answer:
The number generator is not fair. In most of the experiments, considerably less than 60% of the selected marbles are green.
Step-by-step explanation:
Run a significant number of times, a fair distribution should be a bell curve centered on the 6 for 60%
A furniture maker used 2/3 of a can of paint to paint some chairs. He used 1/6 of a can of paint for each chair. How many chairs did he paint?
Answers
Answer:
4 chairs
Step-by-step explanation:
Start with 2/3 can of paint: (2/3)
He used 1/6 of the can to paint each chair: (divided by 1/6)
Now use the 'keep, change, flip' method to get the expression: (2/3x6/1)
Then simplify your answer: (12/3)
Lastly simplify again: (4)
Thus, he painted 4 chairs.
help plsssssssssssssss
Answers
Answer:
Last answer, height is 150 ft
Step-by-step explanation:
from the graph, you can clearly see, when T=3, H=150
Which means The firework is at a height of 150 feet
Answer:
D
Step-by-step explanation:
At 3 seconds, the firework is at a height of 150 feet
Answer is D. third option
The firework is at a height of 150 feet
What are the steps in multiplying decimals?
Answers
Answer:
Ignore the decimal points then multiply the numbers that remain. Count the number of decimal places in all of the numbers of the question combined. Place the decimal point so that there are the same number of decimal places in the answer and the question.
Step-by-step explanation:
I need the correct answer to this problem please
Answers
Answer is ∫2x+1 = -3
2/7 divided by 3/4
Amanda
Answers
Answer:
2/7÷3/4=2/7×4/3=8/21
[tex] \rm \ \blue{ \overbrace{ \underbrace{ \tt \color{hotpink} \: \: \: \: \: \: \: \: \: \: answer \: \: \: \: \: \: \: \: \: }}}[/tex]
[tex]\large \color{pink} \frac{8}{21} [/tex]
Step-by-step explanation:
[tex] \huge \color{brown} \frac{2}{7} \div \frac{3}{4} [/tex]
divide by a fraction, multiply by the reciprocal of that fraction[tex] \large \color{hotpink} \frac{2}{7} \div \frac{4}{3} [/tex]
multiply the fraction[tex] \huge \color{darkblue} \frac{8}{21} [/tex]
hope it helpsHELLLLPPPP QUICKKLLLYYYY PLLZZZ VIEW THE QUESTION IN THE PHOTO ATTACHED!!!!
Answers
Answer:
[tex]\sqrt{18a}[/tex]
Step-by-step explanation:
To insert the factor, you must first find its squared value, in this case, 3's squared number is 9 so.....[tex]3=\sqrt{9}[/tex]
Now with square roots, they multiply like any other number
so you end up with[tex]\sqrt{9}*\sqrt{2a} \\\sqrt{9*2a}\\\\\sqrt{18a} <------ ANSWER[/tex]
Please leave a thanks!!!
Let |u| = 10 at an angle of 45° and |v| = 13 at an angle of 150°, and w = u + v. What is the magnitude and direction angle of w?
|w| = 9.4; θ = 72.9°
|w| = 9.4; θ = 107.1°
|w| = 14.2; θ = 72.9°
|w| = 14.2; θ = 107.1°
Answers
Recall that for two vector x and y making an angle θ with each,
x • y = ||x|| ||y|| cos(θ)
If we replace y with x, we see that
x • x = ||x|| ||x|| cos(0) = ||x||² ⇒ ||x|| = √(x • x)
Using the last identity, the magnitude of w is
||w|| = √(w • w)
but since w = u + v, we have
w • w = (u + v) • (u + v)
The dot product distributes over sums and is commutative, so
w • w = (u • u) + (u • v) + (v • u) + (v • v)
… = ||u||² + 2 (u • v) + ||v||²
… = ||u||² + 2 ||u|| ||v|| cos(θ) + ||v||²
If u has a direction of 45° with the positive x-axis, v has a direction of 150°, then the angle between u and v is |45° - 150°| = 105°. So,
||w|| = √(||u||² + 2 ||u|| ||v|| cos(150°) + ||v||²)
… = √(10² + 2 • 10 • 13 cos(150°) + 13²)
… ≈ 14.2
Using the parallelogram rule for vector addition (see attached sketch), the sum of the angle between w and u and 45° is equal to the direction of w.
If φ is the angle between w and u, then
w • u = ||w|| ||u|| cos(φ)
… = 14.2 • 10 • cos(φ)
but we also have
w • u = (u + v) • u
… = (u • u) + (v • u)
… = ||u||² + ||u|| ||v|| cos(105°)
… = 10² + 10 • 13 • cos(105°)
… ≈ 66.4
Then
14.2 • 10 • cos(φ) ≈ 66.4
cos(φ) ≈ 0.467
φ ≈ 62.1°
and so the direction of w is 62.1° + 45° ≈ 107.1°.
The students at Vienna Elementary sold donuts every day at school for 6 months. The table below shows the earnings for the first 6 weeks. If the pattern continues, how much will the students make in week 8?
Pics here
↓
Answers
The student made $100 in week 8.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
Amount earned from the first week to 6 weeks.
110, 100, 90, 110, 100, 90
We see that,
Every 3 weeks the amount gets repeated.
So,
7th week = 110
8th week = 100
Thus,
The amount earned in week 8 is $100.
Learn more about expressions here:
https://brainly.com/question/3118662
#SPJ2
use order of operations 7-2(-7-4)
Answers
Answer:
29
hope this helps
Which of the following graphs is described by the function given below?
y = 2x2 + 6x + 3
5
101
10
10
-20
20
- 10
10
-10
20-
101
A.
B.
C.
D.
A. Graph A
B. Graph B
O C. Graph C
D. Graph D
Answers
Answer:
Which of the following graphs is described by the function given below?
y = 2x2 + 6x + 3
-A. Graph A
The function y = 2x^2 + 6x + 3 is a quadratic function that has a vertex of (-1.5,-1.5)
How to determine the graph of the function?The equation of the function is given as:
y = 2x^2 + 6x + 3
Differentiate
y' = 4x + 6
Set to 0
4x + 6 = 0
This gives
4x = -6
Divide by 4
x = -1.5
Substitute x = -1.5 in y = 2x^2 + 6x + 3
y = 2(-1.5)^2 + 6(-1.5) + 3
Evaluate
y = -1.5
This means that the function y = 2x^2 + 6x + 3 has a vertex of (-1.5,-1.5)
See attachment for the graph of y = 2x^2 + 6x + 3
Read more about quadratic graphs at:
https://brainly.com/question/18797214
#SPJ5
Pauline wants to sell stainless steel water bottles as a school fundraiser.
She knows that she will maximize profits, and raise $1024, if she sells
the bottles for $28 each. She also knows that she will lose $4160 if she sells the bottles for only $10 each.
Write a quadratic relation to model her profit, P, in dollars, if she sells the bottles for x dollars each.
Answers
Answer:
P = -16(x -28)² +1024
Step-by-step explanation:
The equation can be written in vertex form as ...
P = -a(x -h)² +k
where (h, k) is the vertex, and 'a' is a scale factor. We are given the vertex (h, k) = (28, 1024) and another point (x, P) = (10, -4160). Using these points, we can find the value of 'a'.
-4160 = -a(10 -28)² +1024
-5184 = -324a . . . . simplilfy
a = 5184/324 = 16 . . . . divide by the coefficient of 'a'
A suitable quadratic relation is ...
P = -16(x -28)² +1024
[tex]prove that\ \textless \ br /\ \textgreater \ \frac { \tan \theta + \sec \theta - 1 } { \tan \theta - \sec \theta + 1 } = \frac { 1 + \sin \theta } { \cos \theta }[/tex]
pls i want the answwr as fast u can
ASAP !
MisterBrainly Answer Please :((((
0r any top user
who will answer correctly he will be a good user
plz answer
Answers
[tex] \large \bigstar \frak{ } \large\underline{\sf{Solution-}}[/tex]
Consider, LHS
[tex]\begin{gathered}\rm \: \dfrac { \tan \theta + \sec \theta - 1 } { \tan \theta - \sec \theta + 1 } \\ \end{gathered}[/tex]
We know,
[tex]\begin{gathered}\boxed{\sf{ \:\rm \: {sec}^{2}x - {tan}^{2}x = 1 \: \: }} \\ \end{gathered}
\\ \\ \text{So, using this identity, we get}
\\ \\ \begin{gathered}\rm \: = \:\dfrac { \tan \theta + \sec \theta - ( {sec}^{2}\theta - {tan}^{2}\theta )} { \tan \theta - \sec \theta + 1 } \\ \end{gathered}[/tex]
We know,
[tex]\begin{gathered}\boxed{\sf{ \:\rm \: {x}^{2} - {y}^{2} = (x + y)(x - y) \: \: }} \\ \end{gathered} \\ [/tex]
So, using this identity, we get
[tex]\begin{gathered}\rm \: = \:\dfrac { \tan \theta + \sec \theta - (sec\theta + tan\theta )(sec\theta - tan\theta )} { \tan \theta - \sec \theta + 1 } \\ \end{gathered}[/tex]
can be rewritten as
[tex]\begin{gathered}\rm\:=\:\dfrac {(\sec \theta + tan\theta ) - (sec\theta + tan\theta )(sec\theta -tan\theta )} { \tan \theta - \sec \theta + 1 } \\ \end{gathered} \\ \\ \\\begin{gathered}\rm \: = \:\dfrac {(\sec \theta + tan\theta ) \: \cancel{(1 - sec\theta + tan\theta )}} { \cancel{ \tan \theta - \sec \theta + 1} } \\ \end{gathered} \\ \\ \\\begin{gathered}\rm \: = \:sec\theta + tan\theta \\\end{gathered} \\ \\ \\\begin{gathered}\rm \: = \:\dfrac{1}{cos\theta } + \dfrac{sin\theta }{cos\theta } \\ \end{gathered} \\ \\ \\\begin{gathered}\rm \: = \:\dfrac{1 + sin\theta }{cos\theta } \\ \end{gathered}[/tex]
Hence,[tex]\begin{gathered} \\ \rm\implies \:\boxed{\sf{ \:\rm \: \dfrac { \tan \theta + \sec \theta - 1 } { \tan \theta - \sec \theta + 1 } = \:\dfrac{1 + sin\theta }{cos\theta } \: \: }} \\ \\ \end{gathered}[/tex]
[tex]\rule{190pt}{2pt}[/tex]
Some help please please
Answers
Answer:
[tex]8\sqrt{2}[/tex]
Step-by-step explanation:
To find the length of an isosceles triangle, you know that the angles of the triangle are 45-45-90, so let's use that triangle theorem.
Therefore, the length of the hypotenuse (BD) is [tex]x\sqrt{2}[/tex], and x is equal to 8 (length of BC).
[tex]x\sqrt{2}[/tex]
x=8
Therefore, [tex]8\sqrt{2}[/tex] would be the length of side BD.
Answer:
oh snap i have this question too sorry cant help :(
Step-by-step explanation:
but i can provide with advice bc is the side down and bd is across us this to your advantage
I am an odd number. Take away one letter and I become even. What number am I?
Answers
Answer:
seven
Step-by-step explanation:
Lol thxx for the points
Pls give brainliest! Also, plz sub to kgirl633 on yt.
How do you divide fractions? Can someone please explain step by step to get marked as brainliest!
Answers
When diving fractions, use KFC:
K eep the first fraction the same
F lip the second fraction upside down
C hange the divide sign to a multiply
After, just work it out multiplying the denominators and the numerators together getting your answer
Hope this helped you, have a good day bro cya)
K: Keep ( keep the first fraction the same)
C: Change (change your division sign to a multiplication sign)
F: Flip (flip your second faction numbers)
Ex.
Equation: 3/5 divided by 2/6
KCF: 3/5 multiplied by 6/2
Hope this helps!
Can someone help Pleasaseee
Answers
Answer:
i think it's 1.8.
Step-by-step explanation:
why? bc if u add 6/20 + 1.5 = 1.8
i think it's 1.8, im not sure, hopefully it's correct!!
Answer:
1.8
Step-by-step explanation:
you just simplify -6/20 into a decimal and you get .3 then you add it to 1.5.
One leg of an isosceles right triangle measures 5 inches. Rounded to the nearest tenth, what is the approximate length of the hypotenuse?
2.5 inches
5.0 inches
7.1 inches
9.8 inches
PLZ ANSWER ASAP
Answers
Answer:
C
Step-by-step explanation:
This means that there is another leg of an isosceles triangle that is also 5 inches.
a^2 + b^2 = c^2
a = 5
b = 5
c = ?
c^2 = 5^2 + 5^2 Combine
c^2 = 25 + 25
c^2 = 50 Take the square root of both sides.
sqrt(c^2) = sqrt(50^2)
c = sqrt(5 * 5 * 2) Take out 1 of two factors under the root sign.
c = 5 * sqrt(2) The 2 is left over because there is only 1 of them
sqrt(2) = 1.4142
c = 5 * 1.4142
c = 7.07 = 7.1
7/10 of guest were over 25 what is the percentage
Answers
line passes through the points (4,-2) and the slope is 5/4 write an equation in slope intercept form
Answers
Answer:
y=5/4x -7
Step-by-step explanation:
Hope this helps!
please help me and solve it
Answers
Answer:
Step-by-step explanation:
Find the equation that represents the proportional relationship in this graph, for y in terms of x.
Answers
Answer:
y = 0.4 x
Step-by-step explanation:
Espai
A house on the market was valued at $224,000. After several years, the value decreased by 9%. By how much did the house's value decrease in dollars? What
is the current value of the house?
Decrease in value:
Current house value:
Answers
Answer:
decrease in value: 20,160
current value: 203,840
Step-by-step explanation:
that is the 9% value decrease of the house
solve this please this equation
Answers
Answer:
x = 80 degrees
Step-by-step explanation:
Draw a line through DG and name the intersection of that line and line AB as K.
By the same side interior angle theorem, GKC is 40 degrees.
By the sum of interior angles of triangle theorem, CGK is 180-70=110 degrees.
By supplementary angle theorem, x = 80 degrees.