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
What are the coordinates of B
Answers
Answer: (-7,-5)
Step-by-step explanation:
Because when you mark point M and A if you mirror point M and A on the x Axis then you will get (-7,-5)
After sitting in a refrigerator for a while, a turkey at a temperature of 34^\circ34
∘
F is placed on the counter and slowly warms closer to room temperature (67^\circ67
∘
F). Newton's Law of Heating explains that the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the room, as given by the formula below:
T=T_a+(T_0-T_a)e^{-kt}
T=T
a
+(T
0
−T
a
)e
−kt
T_a=T
a
= the temperature surrounding the object
T_0=T
0
= the initial temperature of the object
t=t= the time in minutes
T=T= the temperature of the object after tt minutes
k=k= decay constant
The turkey reaches the temperature of 45^\circ45
∘
F after 15 minutes. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 60 minutes.
Enter only the final temperature into the input box.
Answers
The temperature of the body after 60 minutes is 60.5° F.
Using the Newton's law of cooling;
T(t) =[tex]Ts + (To - Ts)e^-kt[/tex]
T(t) = temperature at time t
Ts = temperature of the surrounding
To = Temperature of object
t = time taken
k = cooling constant
Now;
[tex]T(t) -Ts=(To - Ts)e^-kt\\45 - 67 = (34 - 67)e^-15k\\-22 = -33e^-15k\\-22/-33 = e^-15k\\0.667 = e^-15k\\ln(0.667) = ln( e^-15k)\\-0.405 = -15k\\k = -0.405/ -15\\k = 0.027[/tex]
At t = 60 min
[tex]T(t) = Ts + (To - Ts)e^-ktT(60) = 67 + (34 - 67)e^-( 0.027*60)\\T(60) = \\67 + (-33)e^-( 0.027 * 60)\\T(60) = \\60.5 F[/tex]
T(60) = 60.5° F
Learn more about Newton's law of cooling: https://brainly.com/question/13748261
Answer:
T≈169
Step-by-step explanation:
Enter only the final temperature into the input box.
T_a = 67\hspace{50px}T_0=210
T
a
=67T
0
=210
\text{Write Formula:}
Write Formula:
T=67+(210-67)e^{-kt}
T=67+(210−67)e
−kt
Plug in givens
\color{blue}{T}=67+143e^{-k\color{green}{t}}
T=67+143e
−kt
Simplify
\text{Plug in given time and temperature:}
Plug in given time and temperature:
\color{blue}{193}=67+143e^{-k(\color{green}{1.5})}
193=67+143e
−k(1.5)
Simplify
\text{Solve for }k\text{:}
Solve for k:
193=
193=
\,\,67+143e^{-1.5k}
67+143e
−1.5k
-67\phantom{=}
−67=
\,\,-67
−67
126=
126=
\,\,143e^{-1.5k}
143e
−1.5k
\frac{126}{143}=
143
126
=
\,\,\frac{143e^{-1.5k}}{143}
143
143e
−1.5k
Divide by 143
0.8811189=
0.8811189=
\,\,e^{-1.5k}
e
−1.5k
\ln\left(0.8811189\right)=
ln(0.8811189)=
\,\,\ln\left(e^{-1.5k}\right)
ln(e
−1.5k
)
Take ln of both sides
\ln\left(0.8811189\right)=
ln(0.8811189)=
\,\,-1.5k
−1.5k
\frac{\ln\left(0.8811189\right)}{-1.5}=
−1.5
ln(0.8811189)
=
\,\,\frac{-1.5k}{-1.5}
−1.5
−1.5k
Divide by -1.5
0.0843751=
0.0843751=
\,\,k
k
k\approx
k≈
\,\,\color{blue}{0.084}
0.084
Round k to nearest thousandth
\text{Complete Formula:}
Complete Formula:
T=67+143e^{-\color{blue}{0.084}t}
T=67+143e
−0.084t
Plug in k
\text{Find temperature after }\color{green}{4}\text{ minutes:}
Find temperature after 4 minutes:
T=67+143e^{-0.084(\color{green}{4})}
T=67+143e
−0.084(4)
Plug in time
T=67+143e^{-0.336}
T=67+143e
−0.336
Multiply
T=169.1911041
T=169.1911041
Plug into the calculator
T\approx 169
T≈169
One number is eight more than twice another. If their difference is 25, what is the larger
number?
35
42
17
Answers
Answer:
17
Step-by-step explanation:
Let the smaller number = x
Let the larger number = y
y = 2x + 8
y - x = 25 Add x to both sides
y = x + 25 Substitute this into the first equation
x + 25 = 2x + 8 Subtract 8 from both sides
x + 25 - 8 = 2x Combine
x + 17 = 2x Subtract x from both sides.
17 = 2x - x Combine
17 = x
Suppose that sin a= 15/17 and 90 degrees
Answers
Alfredo's Towing Service charges $53.27 for its hook-up fee plus $29.25 for every mile, m, they tow your car. The equation that models the total cost (T) is T(m) = 29.25m + 53.27
Which value in the equation represents the rate of change?
A - $82.52
B - $29.25
B - The number of miles towed
C - $53.27
Answers
~ Also I don’t know if there is supposed to be B twice if so it is both of them
how do I solve this?
Answers
Answer:
Step-by-step explanation:
total points obtained=141+87+81+91=400
total possible points=150+100+100+100=450
1.
% points obtained=(400/450)×100≈88.89
Jade did not secured more than 88.89 % marks.
2. can't read properly
The tree in Manuel’s backyard is 6.8 m high. How high is it in centimeters?
Answers
Sebastian said that if a number is a perfect square. then the number is even. Provide two examples that show that the statement is false.
Answers
Below!
Step-by-step explanation:You can take 25 as one example and 49 as the second example.
Let's factorize.
=> 25 = 5 x 5 = 5² (Which is odd)=> 49 = 7 x 7 = 7² (Which is odd)Conclusion:Hence, the statement is not true.
Hoped this helped.
Which relationships have the same constant of proportionality between y and x as the following table?
y
4
32
7
56
8 64
Choose 3 answers:
Answers
Answer:
A,C and D are the answers.
Step-by-step explanation:
I did it on Khan Academy and got it right.
A rectangular pool has a length of
4x + 7 feet and a width of 15x - 2 feet. Which
expression represents the perimeter of the
pool?
Answers
Please hurry!! This assignment is almost due!!
Answers
Answer: x = 17
Step-by-step explanation:
Hi there!
First, since we know that these two lines are parallel, 3x+14 is going to be congruent to the supplementary side of 7x-4. Given this, we are able to solve for x as we know that the sum of supplementary angles is 180. All you have to do is set up 3x+14+7x-4=180
Now, just use simple order of operations and combining like terms to isolate x. you will end with 17! :)
Hope this helps!
What is the solution to the equation 8b+32=104?
Answers
Answer: the answer is b=9
Step-by-step explanation:
Step 1: Subtract 32 from both sides.
104-32= 72
Step 2: Divide both sides by 8.
72 divided by 8 is 9
therefor the answer is 9
1 4 × √ 2 = 2 n Find N
Answers
Divide by 2 on both sides.
What is 2(n+7)+3n simplified and written in factored form?
Multiple choice question.
A)
5n+9
B)
5n+14
C)
3(2n+3)
D)
2(3n+7)
Answers
Answer:
B)
Step-by-step explanation:
2(n+7)+3n
now you multiply 2 though
2n+14+3n
Add common factors
5n+14
__________________________________________________________
Hope this helps!!
If I am wrong, please tell me, I enjoy learning from my mistakes:)
B.match illustrate by using rectangular regions then solve
C.complete each number sentence by writing the missing fraction then use the shaded regions to show the number sentence
WARNING:
I'LL DELETE YOUR ANSWER IF ITS NOT CORRECT
also theres a picture above
(WITH STEP BY STEP EXPLANATION )
Answers
Fractions can be represented by shading parts of a rectangular region
The simplified expressions are:
[tex]\frac 28 + \frac 24 = \frac 34[/tex].[tex]\frac 9{10} - \frac 35 =\frac{2}{5}[/tex].[tex]\frac 46 -\frac{2}{6}=\frac{2}{6}[/tex].[tex]-\frac{2}{12} + \frac 26 = \frac{2}{12}[/tex].[tex]\frac{9}{18} - \frac{3}{9} = \frac{1}{6}[/tex]Complete number sentenceTo start with, we simplify the fraction expressions as follows:
(1) [tex]\frac 28 + \frac 24 =[/tex]
Take LCM
[tex]\frac 28 + \frac 24 = \frac{2 + 4}8[/tex]
[tex]\frac 28 + \frac 24 = \frac{6}8[/tex]
Simplify
[tex]\frac 28 + \frac 24 = \frac 34[/tex]
(2) [tex]\frac 9{10} - \frac 35 =[/tex]
Take LCM
[tex]\frac 9{10} - \frac 35 =\frac{9 - 6}{10}[/tex]
[tex]\frac 9{10} - \frac 35 =\frac{4}{10}[/tex]
Simplify
[tex]\frac 9{10} - \frac 35 =\frac{2}{5}[/tex]
(3) [tex]\frac 46 - [\ ]=\frac{2}{6}[/tex]
Replace the blank with x
[tex]\frac 46 -x=\frac{2}{6}[/tex]
Collect like terms
[tex]x = \frac 46 -\frac{2}{6}[/tex]
Take LCM
[tex]x = \frac{4 - 2}{6}[/tex]
[tex]x = \frac{2}{6}[/tex]
Simplify
[tex]x = \frac{1}{3}[/tex]
(4) [tex]\frac{}{12} + \frac 26 = \frac{2}{12}[/tex]
Replace the blank with x
[tex]\frac{x}{12} + \frac 26 = \frac{2}{12}[/tex]
Collect like terms
[tex]\frac{x}{12} = \frac{2}{12} -\frac 26[/tex]
Take LCM
[tex]\frac{x}{12} = \frac{2 - 4}{12}[/tex]
[tex]\frac{x}{12} = -\frac{2}{12}[/tex]
Multiply both sides by 12
[tex]x=-2[/tex]
(5) [tex]\frac{9}{18} - \frac{3}{9}[/tex]
Take LCM
[tex]\frac{9}{18} - \frac{3}{9} = \frac{9 - 6}{18}[/tex]
[tex]\frac{9}{18} - \frac{3}{9} = \frac{3}{18}[/tex]
Reduce fraction
[tex]\frac{9}{18} - \frac{3}{9} = \frac{1}{6}[/tex]
Missing fractionsSo, the simplified expressions that contain the missing fractions are:
[tex]\frac 28 + \frac 24 = \frac 34[/tex].[tex]\frac 9{10} - \frac 35 =\frac{2}{5}[/tex].[tex]\frac 46 -\frac{2}{6}=\frac{2}{6}[/tex].[tex]-\frac{2}{12} + \frac 26 = \frac{2}{12}[/tex].[tex]\frac{9}{18} - \frac{3}{9} = \frac{1}{6}[/tex]Shaded regionsSee attachment for the shaded regions
Read more about fractions at:
https://brainly.com/question/5243429
You buy candy bars at 85 cents each plus one newspaper for 60 cents. You can spend no more than $4. How many candy bars can you buy? answer
Answers
Answer:
4
Step-by-step explanation:
Well first you want to take 0.60 from the total amount which is $4 leaving you with $3.40
then you want to divide the remaining amount of money by the cost of a candy bar ($0.85)
So you do 3.40/0.85 which is equal to 4
Curtis has started to bike every day. His goal is to ride 16.5 kilometers each day. He bikes of that distance on Monday. He bikes 3.5 fewer
kilometers on Tuesday than he did on Monday.
How many kilometers did Curtis bike on Monday and Tuesday combined? Enter the answer in the box.
kilometers
help
Answers
Answer:29.5 kilometers
Step-by-step explanation: 16.5-3.5=13 then you add 13 and 16.5
The solution is: Eric biked total 1.875 Km.
What is unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given:
On Monday, Eric biked = 1 Km 125 m
= 1.125 Km
On Tuesday, Eric biked = 1.125 Km - 375 m
= 1.125 - 0.375 Km
= 0.75 Km
So, Eric biked on both day
=1.125 + 0.75
= 1.875 Km
Hence, Eric biked total 1.875 Km.
Learn more about unitary method here:
brainly.com/question/22056199
#SPJ2
complete question:
Eric biked 1 kilometer 125 meters on Monday. on Tuesday, he biked 375 meters less than on Monday how far did he bike both days
Please help me solve this
Answers
Step-by-step explanation:
first line multiplied with first column. then with second column. and then third.
then the same for the second line and then the third line.
BA11 = 5×3 + 7×3 + 3×2 = 38
BA12 = 5×1 + 7×6 + 3×-1 = 44
BA13 = 5×2 + 7×-4 + 3×-3 = -27
BA21 = -2×3 + -1×3 + 9×2 = 9
BA22 = -2×1 + -1×6 + 9×-1 = -17
BA23 = -2×2 + -1×-4 + 9×-3 = -27
BA31 = 4×3 + 3×3 + 1×2 = 23
BA32 = 4×1 + 3×6 + 1×-1 = 21
BA33 = 4×2 + 3×-4 + 1×-3 = -7
If sin θ = 1/2 , then find the value of (sin 3 θ)/(1+ cos 2 θ)
Answers
EXPLANATION:
Given that
sin θ = 1/2
We know that
sin 3θ = 3 sin θ - 4 sin³ θ
⇛sin 3θ = 3(1/2)-4(1/2)³
⇛sin 3θ = (3/2)-4(1/8)
⇛sin 3θ = (3/2)-(4/8)
⇛sin 3θ = (3/2)-(1/2)
⇛sin 3θ = (3-1)/2
⇛sin 3θ = 2/2
⇛sin 3θ = 1
and
cos 2θ = cos² θ - sin² θ
⇛cos 2θ = 1 - sin² θ - sin² θ
⇛cos 2θ = 1 - 2 sin² θ
Now,
cos 2θ = 1-2(1/2)²
⇛cos 2θ = 1-2(1/4)
⇛cos 2θ = 1-(2/4)
⇛cos 2θ = 1-(1/2)
⇛cos 2θ = (2-1)/2
⇛cos 2θ = 1/2
Now,
The value of sin 3θ /(1+cos 2θ
⇛1/{1+(1/2)}
⇛1/{(2+1)/2}
⇛1/(3/2)
⇛1×(2/3)
⇛(1×2)/3
⇛2/3
Answer : Hence, the req value of sin 3θ /(1+cos 2θ) is 2/3.
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if cos θ -sin θ =1, find θ cos θ +sin θ?
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the inequality –1 + 6(–1 – 3x) > –39 – 2x.
Answers
Answer:
x > 2
Step-by-step explanation:
-1+6(-1-3x)>-39-2x
-1-6 - 18x > -39 - 2x
-7 - 18x + 2x > -39 - 2x + 2x
-7 - 16x + 7 > -39 + 7
16x / 16 > -32 / 16
x > 2
Ok done. Thank to me :>
If you had 10 questions on a test and you got 3 wrong what percentage is that?
Answers
Answer:
70% right, 30% wrong
Step-by-step explanation:
3 is 30% of 10
What is the equation for the parabola with a focus F(13/11,0)?
Answers
Answer:
B.ON THE EDGE
Step-by-step explanation:
for which value of x does each expression make sense:
(square root) -2x^2
(square root) (-2x)^3
Answers
The value of x that makes sense for the expression is the domain of the expression
The value of x makes sense for [tex]\sqrt{-2x^2}[/tex] is 0The value of x must not exceed 0 for [tex]\sqrt{(-2x)^3}[/tex] to make sense[tex]\sqrt{-2x^2}[/tex]
For the above expression to have a defined value, the expression in the bracket must be positive.
However, x^2 will always be positive, and -2 will ensure that the expression is always negative, except when x = 0
Hence, the value of x makes sense for [tex]\sqrt{-2x^2}[/tex] is 0
[tex]\sqrt{(-2x)^3}[/tex]
For the above expression to have a defined value, the expression in the square root must be positive.
However, the power of 3 will ensure that the expression is always positive only when x is less than or equal to 0
Hence, the value of x must not exceed 0 for [tex]\sqrt{(-2x)^3}[/tex] to make sense
Read more about domain at:
https://brainly.com/question/10197594
Answer: First answer = x<=0 and second answer is x<0
Step-by-step explanation: The other guy got the work correct but the answer wrong. The real answer is x≤0 for -2x^2 and the second answer is x < 0.
Which of the equations will be a true statement if p = 10 3 ? Select the two choices that apply.
A. 3.4 ÷ p = 0.034
B. 437 ÷ p = 0.437
C. 53.45 ÷ p = 53.45
D. 6,340 ÷ p = 6.34
E. 2,458.2 ÷ p = 24.582
Answers
Answer:A
Step-by-step explanation: make me brainlist if correct
Help me(((((((please
Answers
Answer:
I do not know English
Step-by-step explanation:
<3333
id like 75% of my order to be 9 chocolate and the rest vanilla. how many vanilla cupcakes are there
Answers
Answer:
3 vanilla
Step-by-step explanation:
If 75% of the order equates to 9 chocolate, then the remaining of the dozen would leave three.
Unless there is more than 12 in the order. The question is vague
which is the slope of the line that contains the point (-2,7) and (2,3)
Answers
Answer:
the slope of the line is -2.
Step-by-step explanation:
What is the volume of this rectangular prism?
1
cm
2
4 cm
ola
3
cm
cm3. Because this is hard
Answers
Step-by-step explanation:
Volume=length*width*height
V = (3/2)*(4)*(1/2)
= 12/4 = 3cm^3
Which triangle is a 30°-60°-90° triangle?
Answers
Answer:
-90
Step-by-step explanation:
it's right
Answer:
The first one.
Step-by-step explanation:
I took the quiz and got a 100, hope this helps :))
Suppose ΔDEF has an exterior angle at vertex D. The measure of the exterior angle is (8x−2)º, m∠E=(3x−8)°, and m∠F=(4x+13)°. What is the measure of the exterior angle at vertex D?
A. 7°
B. 13°
C. 41°
D. 54°
Answers
The measure of the exterior angle at vertex D is: D. 54°
Recall:
The exterior angle theorem of a triangle states that the measure of an exterior angle equals the sum of the measures of two opposite interior angles of the triangle.Thus:
In ΔDEF, (8x−2)º is an exterior angle at vertex D.
m∠E = (3x−8)° (interior angle)
m∠F = (4x+13)° (interior angle)
Therefore:
(8x−2)º = (3x−8)° + (4x+13)°
Solve for x8x - 2 = 3x - 8 + 4x + 13
Combine like terms8x - 2 = 7x + 5
8x - 7x = 2 + 5
x = 7
Exterior angle at vertex D = (8x−2)º
Plug in the value of x= 8(7) - 2
= 54º (option D)
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