Select the choice that translates the following verbal phrase correctly to algebra: (2 points)
the difference of m and 7 increased by 15
a.) m − (7 + 15)
b.) 7m + 15
c.) (m − 7) + 15
d.) m − 7 ÷ 15
The graph of f(x)= 3/1+x^2 is shown in the figure to the right. Use the second derivative of f to find the intervals on which f is concave upward or concave downward and to find the inflection points of f.
Answer:
Concave Up Interval: [tex](- \infty,\frac{-\sqrt{3} }{3} )U(\frac{\sqrt{3} }{3} , \infty)[/tex]
Concave Down Interval: [tex](\frac{-\sqrt{3} }{3}, \frac{\sqrt{3} }{3} )[/tex]
General Formulas and Concepts:
Calculus
Derivative of a Constant is 0.
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Quotient Rule: [tex]\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Chain Rule: [tex]\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Second Derivative Test:
Possible Points of Inflection (P.P.I) - Tells us the possible x-values where the graph f(x) may change concavity. Occurs when f"(x) = 0 or undefinedPoints of Inflection (P.I) - Actual x-values when the graph f(x) changes concavityNumber Line Test - Helps us determine whether a P.P.I is a P.IStep-by-step explanation:
Step 1: Define
[tex]f(x)=\frac{3}{1+x^2}[/tex]
Step 2: Find 2nd Derivative
1st Derivative [Quotient/Chain/Basic]: [tex]f'(x)=\frac{0(1+x^2)-2x \cdot 3}{(1+x^2)^2}[/tex]Simplify 1st Derivative: [tex]f'(x)=\frac{-6x}{(1+x^2)^2}[/tex]2nd Derivative [Quotient/Chain/Basic]: [tex]f"(x)=\frac{-6(1+x^2)^2-2(1+x^2) \cdot 2x \cdot -6x}{((1+x^2)^2)^2}[/tex]Simplify 2nd Derivative: [tex]f"(x)=\frac{6(3x^2-1)}{(1+x^2)^3}[/tex]Step 3: Find P.P.I
Set f"(x) equal to zero: [tex]0=\frac{6(3x^2-1)}{(1+x^2)^3}[/tex]Case 1: f" is 0
Solve Numerator: [tex]0=6(3x^2-1)[/tex]Divide 6: [tex]0=3x^2-1[/tex]Add 1: [tex]1=3x^2[/tex]Divide 3: [tex]\frac{1}{3} =x^2[/tex]Square root: [tex]\pm \sqrt{\frac{1}{3}} =x[/tex]Simplify: [tex]\pm \frac{\sqrt{3}}{3} =x[/tex]Rewrite: [tex]x= \pm \frac{\sqrt{3}}{3}[/tex]Case 2: f" is undefined
Solve Denominator: [tex]0=(1+x^2)^3[/tex]Cube root: [tex]0=1+x^2[/tex]Subtract 1: [tex]-1=x^2[/tex]We don't go into imaginary numbers when dealing with the 2nd Derivative Test, so our P.P.I is [tex]x= \pm \frac{\sqrt{3}}{3}[/tex] (x ≈ ±0.57735).
Step 4: Number Line Test
See Attachment.
We plug in the test points into the 2nd Derivative and see if the P.P.I is a P.I.
x = -1
Substitute: [tex]f"(x)=\frac{6(3(-1)^2-1)}{(1+(-1)^2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(3(1)-1)}{(1+1)^3}[/tex]Multiply: [tex]f"(x)=\frac{6(3-1)}{(1+1)^3}[/tex]Subtract/Add: [tex]f"(x)=\frac{6(2)}{(2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(2)}{8}[/tex]Multiply: [tex]f"(x)=\frac{12}{8}[/tex]Simplify: [tex]f"(x)=\frac{3}{2}[/tex]This means that the graph f(x) is concave up before [tex]x=\frac{-\sqrt{3}}{3}[/tex].
x = 0
Substitute: [tex]f"(x)=\frac{6(3(0)^2-1)}{(1+(0)^2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(3(0)-1)}{(1+0)^3}[/tex]Multiply: [tex]f"(x)=\frac{6(0-1)}{(1+0)^3}[/tex]Subtract/Add: [tex]f"(x)=\frac{6(-1)}{(1)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(-1)}{1}[/tex]Multiply: [tex]f"(x)=\frac{-6}{1}[/tex]Divide: [tex]f"(x)=-6[/tex]This means that the graph f(x) is concave down between and .
x = 1
Substitute: [tex]f"(x)=\frac{6(3(1)^2-1)}{(1+(1)^2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(3(1)-1)}{(1+1)^3}[/tex]Multiply: [tex]f"(x)=\frac{6(3-1)}{(1+1)^3}[/tex]Subtract/Add: [tex]f"(x)=\frac{6(2)}{(2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(2)}{8}[/tex]Multiply: [tex]f"(x)=\frac{12}{8}[/tex]Simplify: [tex]f"(x)=\frac{3}{2}[/tex]This means that the graph f(x) is concave up after [tex]x=\frac{\sqrt{3}}{3}[/tex].
Step 5: Identify
Since f"(x) changes concavity from positive to negative at [tex]x=\frac{-\sqrt{3}}{3}[/tex] and changes from negative to positive at [tex]x=\frac{\sqrt{3}}{3}[/tex], then we know that the P.P.I's [tex]x= \pm \frac{\sqrt{3}}{3}[/tex] are actually P.I's.
Let's find what actual point on f(x) when the concavity changes.
[tex]x=\frac{-\sqrt{3}}{3}[/tex]
Substitute in P.I into f(x): [tex]f(\frac{-\sqrt{3}}{3} )=\frac{3}{1+(\frac{-\sqrt{3} }{3} )^2}[/tex]Evaluate Exponents: [tex]f(\frac{-\sqrt{3}}{3} )=\frac{3}{1+\frac{1}{3} }[/tex]Add: [tex]f(\frac{-\sqrt{3}}{3} )=\frac{3}{\frac{4}{3} }[/tex]Divide: [tex]f(\frac{-\sqrt{3}}{3} )=\frac{9}{4}[/tex][tex]x=\frac{\sqrt{3}}{3}[/tex]
Substitute in P.I into f(x): [tex]f(\frac{\sqrt{3}}{3} )=\frac{3}{1+(\frac{\sqrt{3} }{3} )^2}[/tex]Evaluate Exponents: [tex]f(\frac{\sqrt{3}}{3} )=\frac{3}{1+\frac{1}{3} }[/tex]Add: [tex]f(\frac{\sqrt{3}}{3} )=\frac{3}{\frac{4}{3} }[/tex]Divide: [tex]f(\frac{\sqrt{3}}{3} )=\frac{9}{4}[/tex]Step 6: Define Intervals
We know that before f(x) reaches [tex]x=\frac{-\sqrt{3}}{3}[/tex], the graph is concave up. We used the 2nd Derivative Test to confirm this.
We know that after f(x) passes [tex]x=\frac{\sqrt{3}}{3}[/tex], the graph is concave up. We used the 2nd Derivative Test to confirm this.
Concave Up Interval: [tex](- \infty,\frac{-\sqrt{3} }{3} )U(\frac{\sqrt{3} }{3} , \infty)[/tex]
We know that after f(x) passes [tex]x=\frac{-\sqrt{3}}{3}[/tex] , the graph is concave up until [tex]x=\frac{\sqrt{3}}{3}[/tex]. We used the 2nd Derivative Test to confirm this.
Concave Down Interval: [tex](\frac{-\sqrt{3} }{3}, \frac{\sqrt{3} }{3} )[/tex]
10 1/8 - 3 5/6=.......?
Answer:
6 7/24
Step-by-step explanation:
7% of $582 is how much?
Answer:
$541.26
Step-by-step explanation:
Please let me know if this helps
Plz mark B R A I N L I E S T
help pls :)))))))))))))))))
Answer:
I think all measure can be used.
What fraction is equivalent to .9?
Answer:
9/100
Step-by-step explanation:
00.9=9/100
9%
000000
000000
000
3. A $5,000 principal is invested in two accounts, one earning 1% interest and another earning 6%
interest. If the total interest for the year is $170, then how much is invested in each account?
Let, amount invested in first account is x.
So, amount invested in second account is ( 5000 - x ).
Now,
Total interest = Interest from 1 + Interest from 2
170 = x × 0.01 × 1 + ( 5000 - x ) × 0.06 × 1
17000 = x + 6( 5000 - x )
17000 = x + 30000 - 6x
5x = 30000 - 17000
x = $2600
Therefore, money invested in first and second account is $2600 and $2400.
Hence, this is the required solution.
What is 19,998 divided by 1,000,000,000,000,000,000,000,000,000,000,000
Answer:
1.9998e-29
Step-by-step explanation:
please help!
does the circle have a radius or diameter?
Answer:
the circle has a radius
Step-by-step explanation:
Find the value of the variable, x
Group of answer choices
6
3
15
5
Answer:
I think you might have a mistake for the numbers you put there.
But the answer is about 15
Step-by-step explanation:
I did the calculation and got 14, but 15 is the closeest answer
On a number line, between which two consecutive whole numbers would the square root of 277 be located
9514 1404 393
Answer:
16 and 17
Step-by-step explanation:
16² = 256
(√277)² = 277
17² = 289
The root of 277 is between 16 and 17.
Write an ordered pair that is a solution to the equation of the line y=x+5
Answer:
(1,6)
(2,7)
(4,9)
(8,13)
Step-by-step explanation:
there are 3 arms for every 2 eyes. If there are 6 eyes, how many arms are there?
Answer:
There are 9 arms.
Step-by-step explanation:
Which two statements describe how saving $200 every month in her traditional 401(k) retirement account is different from saving $200 every month for her car? Select all the correct answers. Isabel’s retirement savings are made with after-tax dollars. Isabel’s retirement savings are made with pretax dollars. Isabel’s retirement savings can be used for other purposes without penalty. Isabel’s retirement savings have more liquidity than the money in her savings account. Isabel’s retirement savings have no contribution limits. Isabel’s retirement savings may include a contribution from her employer.
Answer:
Isabel’s retirement savings are made with pretax dollars.
Isabel’s retirement savings may include a contribution from her employer.
Step-by-step explanation:
When you contribute to an IRA account, your contribution is tax deductible. This means that all the money that you contribute to your account comes from pretax dollars. This contribution is limited to $19,500 per year (for 2020) or $26,000 if you are 50 or older.
Your employer can also contribute to your 401(k) account and employer's contribution may be even higher than the employee's. Total contributions including employer's and employee's cannot exceed $57,000 per year (or $63,500 if 50 or older) or the employee's total compensation per year, whichever is lower.
On Sunday, 69% of the tickets sold at a theme park were children tickets, the rest were adult tickets. If the theme park sold 260 tickets in all, about how many children tickets were sold on this particular day?
Answer:
182
Oh and btw, its funny because I'm in your 6th period class. And i was looking up the answer and couldn't find it so when i found your i decided to help you answer it too
-Mikey
Answer:182
Step-by-step explanation:
Select the functions that have identical graphs.
1 Y = cosx
2 cos(x-5pi/2)
3) sinx
Answer: 2 and 3 B
Step-by-step explanation:
Y=cos(x-5pi/2), y=sinx
The tree in Lisa’s backyard is 7.4 m high. How high is it in centimeters?
Answer:
740 cm
Step-by-step explanation:
Answer: 740 centimeters
Step-by-step explanation:
All you have to do is multiply the length value by 100 giving you 740.
plz i have no clue and this is due in 15 mins
Answer:
y = [tex]\frac{1}{2}[/tex] x
Step-by-step explanation:
Slope of perpendicular to y = - 2x - 5 line is [tex]\frac{1}{2}[/tex]
y - 2 = [tex]\frac{1}{2}[/tex] ( x - 4)
y = [tex]\frac{1}{2}[/tex] x
Answer:
[tex]y=\frac{1}{2} x[/tex]
Step-by-step explanation:
The process for rationalizing a denominator in a variable expression is the same as in a numeric expression. Here’s a real-world example. The kinetic energy of the car of a rollercoaster is given by the formula k = one-half m v squared where k is kinetic energy, m is the mass of the car, and v is the velocity of the car. Solving this formula for v, we get v = StartRoot StartFraction 2 k Over m EndFraction EndRoot Which formula gives the velocity of the car in simplest form?
Answer:
It's B
Step-by-step explanation:
Got it right on EDGE2020
The value of velocity of the car will be v = √(2km) / m. Then the correct option is B.
What is kinetic energy?If the object of mass m is moving with speed v. Then the kinetic energy of the object will be
KE = (1/2) mv²
The process for rationalizing a denominator in a variable expression is the same as in a numeric expression.
Here’s a real-world example.
The kinetic energy of the car of a rollercoaster is given by the formula
k = 1/2 m v²
Where k is kinetic energy, m is the mass of the car, and v is the velocity of the car.
Solving this formula for v, we get
[tex]\rm v = \sqrt{\dfrac{2k}{m}}[/tex]
Simplify the equation, we have
[tex]\rm v = \sqrt{\dfrac{2km}{m^2}}\\\rm v = \dfrac{\sqrt{2km}}{m}[/tex]
Then the correct option is B.
The complete question is attached below.
More about the kinetic energy link is given below.
https://brainly.com/question/12669551
#SPJ2
What is the mode?
1 4 4 1 1 0 6 6 6 1
The mode is 1.
It was repeated 4 times whereas 6 was only repeated 3 times, 4 twice, and 0 once.
Answer: 1
The mode is the number that occurs the most within the set of numbers. If the set of numbers is as follows: 1, 4, 4, 1, 1, 0, 6, 6, 6, 1, then the answer is one(1).
However, if the set of numbers appear as follows: 14, 41, 10, 66, 61, then there is no mode.
how tall is bill gates?
Answer:
Bill Gates is 5'10"
Hunter read 12 books in 2 months. If he reads at a constant rate, how many books did
he read in one month? Give your answer as a whole number or a FRACTION in
simplest form.
PLS ANSWER QUICK
Answer:Hunter read 6 months in a month
Step-by-step explanation:
12/2=6 books in a month
Answer:
6 Books a Month
Step-by-step explanation:
12 / 2 = 6
pls answer no is answering mine
Noel works as a driver for a transportation company. On average, he drives 152 hours to make 19 successful deliveries. Assume this rate is his constant rate for delivering goods. If you graph this relationship with time along the x-axis and number of deliveries along the y-axis, the slope of the line you get gives Noel's unit rate for a successful delivery. Which unit rate can be interpreted as the slope?
a: 8 deliveries per hour
b :2/17 of an hour per delivery
c: 1/8 of a delivery per hour
d: 1/8 of an hour per delivery
Answer:
A
Step-by-step explanation:
By doing the unit rate, dividing 19 on both the top and bottom, getting this 8.36.../1, you can just round it to 8/1, meaning he makes about 8 deliveries per hour. Hope this helps!
which branch of maths includes the most of other branches symbols/functions ect.
Answer:
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In elementary algebra, those symbols (today written as Latin and Greek letters) represent quantities without fixed values, known as variables.
Step-by-step explanation:
Every quadrilateral is a rhombus true or false
Answer:
false
Step-by-step explanation:
Answer:
false
Step-by-step explanation:
. 。 • ﾟ 。 .
. . 。 。 .
. 。 ඞ 。 . • •
ﾟ Josie was not An Impostor. 。 .
' 1 Impostor remains 。
ﾟ . . , . .
PLEASE HELP ME look at he file i put in please help
Answer:it would need to be 5
Juan is 1 1/4 feet shorter than maria.maria is 1/3 foot taller than Luis. if Luis is 62 inches tall, how tall are maria and Juan.
Answer:
Maria is 58 inches tall and Juan is 42 inches
Step-by-step explanation:
Bryce had a $25 gift card to use on songs and games at an online media store. Songs cost $2 each and games cost $5 each. Bryce spent all the money on the gift card to download 8 items. Solve the system to determine how many games he purchased. Let s represent the number of songs and g represent the number of games. s + g = 8 2s + 5g = 25 Bryce purchased games.
Answer:
Bryce purchased 3 games.
Step-by-step explanation:
To find the number of songs and games that Bryce downloaded, we need to solve the following system of equations:
s + g = 8
2s + 5g = 25
We know that:
s + g = 8 → 2s + 2g = 16 → 2s = 16 -2g
2s + 5g = 25 → 16 - 2g + 5g = 25
→3g = 25 - 16
→3g = 9
→ g = 3
Therefore, bryce downloaded 3 games and 5 songs!
Answer:
.
3x + 2y = 16,
Step-by-step explanation:
Pls help me in these questions i will mark u brainliest.
Answer:
each angle is 83
Step-by-step explanation:
166÷ 2 = 83
check
83+83=166
hope it helps!!
Answer:
sum of two vertically angle =166°
X=166°/2 = 83°.
hence, each angle is equal to 83°.
X=
Just for you :) happy thanksgiving
Answer:
thksss and happy thanksgiving to you 2
Step-by-step explanation: