I can't seem to figure this one out. Could someone show how to do it and the answer? Thanks in advance!
Answers
Answer:
[tex]\displaystyle (-\infty, -\frac{1}{6})[/tex]
Step-by-Step explanation:
We have the function:
[tex]\displaystyle y=\int_{1}^{x}\frac{1}{3+t+3t^2}\, dt[/tex]
And we want to find the interval for which y is concave upwards.
Therefore, we will need to find the second derivative of y, find its inflection points (where y''=0), and test for values.
So, let's take the derivative of both sides with respect to x. So:
[tex]\displaystyle y^\prime=\frac{d}{dx}\Bigg[\int_{1}^{x}\frac{1}{3+t+3t^2}\, dt\Bigg][/tex]
By the Fundamental Theorem of Calculus:
[tex]\displaystyle y^\prime=\frac{1}{3+x+3x^2}[/tex]
So, we will take the derivative again. Hence:
[tex]\displaystyle y^\prime^\prime=\frac{d}{dx}\Big[\frac{1}{3+x+3x^2}\Big]=\frac{d}{dx}\Big[(3+x+3x^2)^{-1}\Big][/tex]
We will use the chain rule. Let:
[tex]\displaystyle u=x^{-1}\text{ and } v=3+x+3x^2[/tex]
Differentiate:
[tex]\displaystyle y^\prime^\prime=-(3+x+3x^2)^{-2}(1+6x)[/tex]
Rewrite:
[tex]\displaystyle y^\prime^\prime=-\frac{6x+1}{(3+x+3x^2)^2}[/tex]
So, points of inflection, where the concavity changes, is whenever the second derivative is 0 or undefined.
We can see that the second derivative will never be undefined since the denominator can never equal 0.
So, our only possible inflection points are when it's equal to 0. Hence:
[tex]\displaystyle 0=-\frac{6x+1}{(3+x+3x^2)^2}[/tex]
Multiplying both sides by the denominator gives:
[tex]0=-(6x+1)[/tex]
Then it follows that:
[tex]\displaystyle x=-\frac{1}{6}[/tex]
So, our only possible point of inflection is at x=-1/6.
We will test for values less than and greater than this inflection point.
Testing for x=-1, we see that:
[tex]\displaystyle y^\prime^\prime=-\frac{6(-1)+1}{3+(-1)+3(-1)^2}=1>0[/tex]
Since the result is positive, y is concave up for all values less than -1/6.
And testing for x=0, we see that:
[tex]\displaystyle y^\prime^\prime=-\frac{6(0)+1}{3+(0)+3(0)^2}=-\frac{1}{3}[/tex]
Since the result is negative, y is concave down for all values greater than -1/6.
Therefore, the interval for which y is concave up is:
[tex]\displaystyle (-\infty, -\frac{1}{6})[/tex]
Note that we use parentheses instead of brackets since at exactly x=-1/6, our graph is neither concave up nor concave down.
Related Questions
PLS HELP Solve 4 • (−6). (20 points)
a
−42
b
−24
c
10
d
24
Answers
How I found the answer:
4x(-6)
-24x
Given m| n, find the value of x.
Answers
Answer:
45°
Step-by-step explanation:
x° = (180 - 135)°
= 45°
Hope this helps
With interior alternate angles and adjacent angles concept, the value of x from the given parallel lines is 45°.
Given that, straight line m parallel to straight line n and t is the transversal.
When a transversal connects two coplanar lines, alternate interior angles are created. They are located on the transverse sides of the parallel lines, but on the inner side of the parallel lines. At two different locations, the transversal passes through the two lines that are coplanar.
Here, y=135° (Interior alternate angles are equal)
Now, y°+x°=180° (Adjacent angles on straight line)
135°+x°=180°
x°=45°
Therefore, the value of x from the given parallel lines is 45°.
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1/sin10°-√3/cos10°=4
Answers
Answer:
[tex]\frac{1}{\sin \left(10^{\circ \:\:}\right)}\:\:\frac{1}{\sin \:\left(10^{\circ \:\:}\right)}=4[/tex]
Step-by-step explanation:
Given the expression
[tex]\frac{1}{\sin \left(10^{\circ \:}\right)}-\frac{\sqrt{3}}{\cos \left(10^{\circ \:}\right)}=4[/tex]
Taking the left-hand side and solving it
[tex]\frac{1}{\sin \left(10^{\circ \:}\right)}-\frac{\sqrt{3}}{\cos \left(10^{\circ \:}\right)}[/tex]
as
[tex]\frac{1}{\sin \:\left(10^{\circ \:\:}\right)}=5.76[/tex]
and
[tex]\frac{\sqrt{3}}{\cos \left(10^{\circ \:\:}\right)}=1.76[/tex]
so the expression becomes
[tex]\frac{1}{\sin \left(10^{\circ \:\:}\right)}\:\:\frac{1}{\sin \:\left(10^{\circ \:\:}\right)}=5.76-1.76[/tex]
[tex]=4[/tex]
Thus, we conclude that
[tex]\frac{1}{\sin \left(10^{\circ \:\:}\right)}\:\:\frac{1}{\sin \:\left(10^{\circ \:\:}\right)}=4[/tex]
No I need 54 inches of wood to build a picture frame. if the wood cost $2.50 a foot, how much will be pay?
Answers
Answer:
135
Step-by-step explanation:
2 x 54 = 108 and .50 x 54 = 27 so 108 + 27 = 135
Allen went out to dinner and hls total bill
came to $22.14. The tax on his meal was
8%, or $1.64. What was the subtotal of
his dinner?
Answers
PLEASE HELP!!!!!!!!!!!!!!!!!!!!
Answers
X= 6-4
X = 8-6
Is this what u want?
Find the equation of the line that contains a slope of 3/4 and a 'y'-intercept of (0,3). Write the answer in slope-intercept form.
Answers
Answer:
Step-by-step explanation:
y=-3/4 x-3
The half-life of Palladium-100 is 4 days. After 20 days a sample of Palladium-100 has been reduced to a
mass of 5 mg.
What was the initial mass (in mg) of the sample?
What is the mass (in mg) 4 weeks after the start?
You may enter the exact value or round to 4 decimal places
Answers
Answer:
A = (Ao)(0.5^(t/t1.2))
1 = (Ao)(0.5^(20/4)) = Ao(0.03125)
Ao = 1/0.03125 = 32mg
5 weeks = 5(7) = 35 days
A = 32(0.5^(35/4)) = 32(0.0232) = 0.074mg
Step-by-step explanation:
One year Roger had the lowest ERA (earned-run average, mean number of runs yielded per nine
innings pitched) of any male pitcher at his school, with an ERA of 3.49. Also, Kate had the lowest ERA of any female pitcher at the school with an ERA of 3.44. For the males, the mean ERA was 3.805 and the standard deviation was 0.671. For the females, the mean ERA was 3.967 and the standard
deviation was 0.604. Find their respective z-scores. Which player had the better year relative to their
peers, Roger or Kate? (Note: In general, the lower the ERA, the better the pitcher.)
Answers
Answer:
id. k
Step-by-step explanation:
I. dk
Sheila buys two concert tickets from her friend.
She pays $90 for the two tickets. She looks at the
tickets and sees that each ticket has a face value
of $52.50.
a. How much of a markdown did her friend give
Sheila? Explain how you know.
B. What was the perecent markdown, rounded to the nearst whole percent?
Answers
Answer:
15 because the 2 tickets were originally 105
Step-by-step explanation:
52.50 x 2 = 105 because 52 + 52 = 104 and .50 + .50 = 1 so 104 + 1 = 105
PLEASE HELP!!! ALGEBRA 2A
Answers
300 feet above sea level??
Answers
Answer:
300
Step-by-step explanation:
if it is 300 feet ABOVE sea level, then it is a positive answer.
you would want to look for the words 'above' or 'below' if it is talking about sea level.
Evaluate the given expression. Subscript 10 Baseline C Subscript 4 Baseline times Subscript 4 Baseline C Subscript 2 a. 1,260 c. 1,275 b. 1,278 d. 1,270
Answers
Answer:
The answer is A: 1,260
Step-by-step explanation:
I took the test on Edge
Answer:
A-1,260
Step-by-step explanation:
Omar’s watch shows that he takes 642 steps to walk a lap around his school’s track. If he takes a total of 12,198 steps while walking the track, how many laps does Omar walk? It is one of these.
a)18
b)19
c)20
d)21
Answers
Answer:
B. 19
12,198÷642 so the final answer will be 19.
while in Manchester, england, chelsea bought a sweater for bitirsh pound 28 . when she returned, she saw the same sweater for C$38. which sweater cost more by how much in canadian dollar show your work pls
Answers
The sweater that cost more is the sweater bought in Manchester. It is more expensive by C$7.40.
Which sweater is more expensive?The exchange rate is the price at which one currency is exchanged for another currency. For example, if the exchange rate of the pound and Canadian dollars is 1.60. It means that 1 unit of the pound would buy 1.60 of the Canadian dollars.
The first step is to convert the cost of the sweater that is priced in the British pound to Canadian dollars.
The exchange rate : 1 British pound = 1.6215 CAD
28 X 1.6215 = 45.402
Difference in price = 45.402 - 38 = C$7.402
Based on the above calculation, the sweater when converted from the British pound to the Canadian dollars, it is more expensive to buy the sweater in Manchester than it is to buy it in Canada.
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A tree measures 40.5 feet. Over the next 7 1/2 year, it grows to a height of 78 feet. During the 7 1/2 years, what was the average yearly growth rate of the height of the tree?
Will give brainlist.
Answers
Answer:
10.4 feet per year
Step-by-step explanation:
Divide the tree growth by the number of years.
78/7.5=10.4
What is the greatest common factor of 12a and 9a2
Answers
Answer:
For these types of problems, You must look at the coefficients and variables separately and find the GCF for each. Then, combine the two
First, take 12 and 9
Factors of both
9- 1, 3, 9
12- 1, 2, 3, 4, 6, 12
So the gcf of 12 and 9 is 3
Now do the variables a and a^2
a^2 is just a twice
The GCF is a
Now combine a and 3
Answer: GCF is 3a
Answer:
GCF is 3a
Step-by-step explanation:
I just know LOLLL
Bentley wants to ride his bicycle 39.6 miles this week. He has already ridden 8 miles.
If he rides for 4 more days, write and solve an equation which can be used to
determine x, the average number of miles he would have to ride each day to meet his
goal.
Answers
Answer:
Step-by-step explanation:
Using a linear function, it is found that he would have to ride his bicycle 7.9 miles a day to meet his goal.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.For this problem, we have that:
The 8 miles he has already ridden is the intercept.The daily average he needs is the slope.When x = 4, y = 39.6, hence:
39.6 = 4m + 8
4m = 31.6
m = 31.6/4
m = 7.9.
He would have to ride his bicycle 7.9 miles a day to meet his goal.
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The Martin family and the Kelly family each used their sprinklers last summer. The water output rate for the Martin family's sprinkler was 30L per hour. The water output rate for the Kelly family's sprinkler was 35L per hour. The families used their sprinklers for a combined total of 60 hours resulting in a total water output of 1925L. How long was each sprinkler used?
Answers
Answer:
The Martins used their sprinkler for 35 hours, while the Kellys used their sprinkler for 25 hours.
Step-by-step explanation:
As the Martins and the Kellys used their sprinklers for 60 hours combined last summer, and the water output rate of the Martins' sprinkler is 30 liters per hour, while the Kelly's is 35 liters per hour To determine how many hours each family used their sprinklers, knowing that in total 1925 liters of water were used, the following logical reasoning must be carried out:
There is a 5 liter difference between the Kelly sprinkler and the Martin sprinkler. If both used 30 liters of water per hour, for 60 hours 1800 liters (30 x 60) would have been used. Therefore, there is an excess of 125 liters that corresponds to an excess of consumption, attributable to the Kelly sprinkler, which consumes 5 liters more of water per hour than the Martin's. Thus, since 125/5 equals 25, in total the Kellys used their sprinkler for 25 hours, while the Martins did it for 35 hours.
This reasoning is proved with the following calculation:
25 x 35 = 875
35 x 30 = 1,050
875 + 1,050 = 1,925
3 2/4(15×5/6 .........................................
Answers
Challenge A bag contains pennies, nickels, dimes, and quarters. There are 50 coins in all. Of the
coins, 12% are pennies and 40% are dimes. There are 5 more nickels than pennies. How much
money does the bag contain?
The bag contains $
Answers
Answer:
$5.86
Step-by-step explanation:
12% of 50=6 pennies
40% of 50= 20 dimes
6+5= 11 nickels
11+20+6= 37 coins
50 total coins-37= 13 quarters
13 quarters = $3.25
20 dimes = $2.00
11 nickels = $0.55
6 pennies = $0.06
$3.25+$2.00+$0.55+$0.06=$5.86
Answer:
$5.86
Step-by-step explanation:
Absolute value of 13+14
Answers
Answer:
27
Step-by-step explanation:
13 + 14 = 27
Answer: -13+-14
Step-by-step explanation: the Absolute value is the opposite value of something like 24-57 the absolute value of that would be -24-(-57)
Find the midpoint of GH
Answers
Answer:
2.3 is the mid point of gh
Tyler and his children went into a bakery and will buy cupcakes and donuts. He must buy a maximum of 11 cupcakes and donuts altogether. Write an inequality that would represent the possible values for the number of cupcakes purchased, cc, and the number of donuts purchased, d.d.
Answers
Answer:
0 < cc + dd ≤ 11
Step-by-step explanation:
≤ means equal to or less than.
It has to be more than zero, obviously.
Answer:
c+d≤11
Step-by-step explanation:
welcome
Jack wants to model a situation where the perimeter of the rectangle to the right is 6 feet plus or minus 1.5 feet. Because he is modeling a length "plus or minus" another length, he decides to use an absolute value equation for his model. Do you agree with his decision? Explain your reasoning.
Answers
The absolute value of the equation will not work because the value of x would have to be negative for the perimeter to be 6 ft plus or minus 1.5 ft.
What is the perimeter of the rectangle?The perimeter of the rectangle is given as:
Perimeter = 2(Length + Breadth)
The perimeter of a rectangle is 6 feet plus or minus 1.5 feet.
So, we have:
Perimeter = 2(Length + Breadth) = 6 feet ± 1.5 feet.
Then Substitute values for length and width
2(Length + Breadth) = 6 feet ± 1.5 feet.
2(4 + x) = 6 feet ± 1.5 feet.
Now,
2(4 + x) = 6 feet + 1.5 feet.
8 + 2x = 6.5 ft
2x = 6.5 - 8
x = -0.25 ft
2(4 + x) = 6 feet - 1.5 feet.
8 + 2x = 4.5 ft
2x = 4.5 - 8
x = -1.75ft
Here Both values of x are negative.
This means that the absolute value model can't work, because a rectangle cannot have a negative dimension.
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Did ancient mathematicians use a ruler to determine the values of sine and cosine in a circle?
No answer needed! Just take points.
Answers
Answer and Explanation:
No, they did not use a ruler to compute the values of trig functions. They used geometry to prove theorems about trigonometry, then used those results and ordinary computations involving addition, subtraction, multiplication, division, and square roots to determine the entries in trigonometric tables.
Claudius Ptolemy (85 C.E.–165) created the first trig tables that we know of, and he included proofs of the theorems he used. One of the main theorems he used is what is now called Ptolemy’s theorem.
In order to prove his sum and difference forumlas, Ptolemy first proved what we now call Ptolemy’s theorem.
Ptolemy’s theorem. The product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of the opposite sides.
A cyclic quadrilateral is a quadrilateral inscribed in a circle as ABCD. Ptolemy’s theorem says:
AC ⋅ BD = AB ⋅ CD + AD ⋅ BC
When one of the diagonals (like ) is a diameter of the circle, that gives you a way to determine the trig functions for sum and differences of angles.
Sometimes, the sum and difference formulas for sine and cosine are still called Ptolemy’s formulas.
sin(α + β) = sin α cos β + cos α sin β
cos(α + β) = cos α cos β − sin α sin β
sin(α − β) = sin α cos β − cos α sin β
cos(α − β) = cos α cos β + sin α sin β
Ptolemy started with angles he knew the trig functions for. These included angles in various isosceles triangles: 45°-45°-90°, 60°-60°-60°, and 36°-72°-72°, the last one described by Euclid in Proposition IV.10. That gave him the trig functions for 36°, 60°, and 90°. Using half-angle formulas (mentioned by Hipparchus (190–120 B.C.E.), he could compute trig functions for half of those angles and a quarter of those angles. Then using the sum and difference formulas he computed the rest of the numbers in his trig table.
Hope this helps! :)
Jocelyn has read 160 pages in a book. This is of the entire book. How many pages
does the book have?
Answers
you need more information to finish this question. You can't get an answer for a math question like this one with that little information.
Answer:
Step-by-step explanation:
lol it means 4/5,
your in rsm i aleready know
Instructions: Find the missing side lengths. Leave your answers as radicals in simplest form.
Help quick.
Answers
Answer: y=8 and x=8 radical 3
Suppose you and 10 friends are taking a road trip for Spring Break. You plan on visiting multiple museums, national parks, and shopping locales along this trip. Your
road trip will a total of 10 days. You drive a total distance of 5000 miles. Some of the drive your average speed was 45 mph. While the rest of the trip was 70 mph.
1. How many hours were you driving each speed? Round 3 decimals
2. What distance did you drive for each speed? Round 3 decimals
Answers
Using the relation between velocity, distance and time, and supposing the group travels for 8 hours a day, it is found that:
1. You were driving 24 hours at 45 mph and 56 hours at 70 mph.
2. You drove 1080 miles at 45 mph and 3920 miles at 70 mph.
What is the relation between velocity, distance and time?Velocity is distance divided by time, that is:
v = d/t.
For the first part, we have that:
The velocity is of 45 mph.The distance is of d.The time is of t.Hence:
v = d/t
45 = d/t
d = 45t.
For the second part, we have that:
The velocity is of 70 mph.The distance is of 5000 - d.The time is of 80 - t. (10 days of 8 hours driving = 80 hours).Hence:
70 = (5000 - d)/(80 - t).
d = 45t, hence:
70 = (5000 - 45t)/(240 - t).
5000 - 45t = 5600 - 70t
25t = 600.
t = 600/25
t = 24. -> (80 - 24) = 56.
d = 45t = 45 x 24 = 1080 -> (5000 - 1080) = 3920.
Hence:
1. You were driving 24 hours at 45 mph and 56 hours at 70 mph.
2. You drove 1080 miles at 45 mph and 3920 miles at 70 mph.
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The acceleration of gravity on Mars is about 3.7 meters per second squared. Suppose a rock falls from a tall cliff on Mars. Which of the following equations indicates how fast the rock will be falling after 8 seconds?
Answers
An equation that indicates how fast (velocity) the rock will be falling after 8 seconds is V = (3.7 × 8) m/s.
Given the following data:
Time, t = 8 seconds.Acceleration of gravity on Mars, a = 3.7 meters per second squared (m/s²).How to determine the velocity of this rock?In order to determine the velocity at which this rock would fall after 8 seconds, we would apply the first equation of motion.
Mathematically, the first equation of motion is given by this formula:
V = u + at
Where:
V represents the final velocity.u represents the initial velocity.a represents the acceleration.t represents the time.Substituting the given parameters into the formula, we have;
V = 0 + 3.7 × 8
V = (3.7 × 8) m/s.
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Complete Question:
The acceleration of gravity on Mars is about 3.7 meters per second squared. Suppose a rock falls from a tall cliff on Mars. Write the equation that indicates how fast the rock will be falling after 8 seconds.