find the poin on the graph 0f f(x)=1-x^2 that are closest to0(0,0)
Answers
[tex] \sqrt{(x-0)^2+(1-x^2-0)^2}=\sqrt{x^2+(1-x^2)^2}=\sqrt{x^2+1-2x^2+x^4} [/tex]
To make this easier, let's minimize the SQUARE of this quantity because when the square root is minimal, its square will be minimal.
So minimize [tex]L=x^4-x^2+1[/tex]
Find the derivative of L and set it equal to zero.
[tex]\frac{d}{dx}(L)=4x^3-2x \\ 4x^3-2x=0 \\ 2x(2x^2-1)=0[/tex]
This gives you [tex]x=0[/tex] or [tex]x^2=\frac{1}{2} \\ x=\pm\sqrt{2}/2[/tex]
You can use the Second Derivative Test to figure out which value(s) produce the MINIMUM distance.
[tex]\frac{d^2}{dx}=12x^2-2[/tex]
When x = 0, the second derivative is negative, indicating a relative maximum. When [tex]x=\pm\frac{\sqrt{2}}{2}[/tex], the second derivative is positive, indicating a relative MINIMUM.
The two points on the curve closest to the origin are [tex]\left( \pm\frac{\sqrt{2}}{2},\frac{1}{2} \right)[/tex]
Related Questions
Ann is grouping 38 rocks. She can put them into groups of 10 rocks or as single rocks.what are different ways Ann can group the rocks?
Answers
To answer this question, you need to make a list based on how much "10 group" of rock can be made.
The possible way to group it should be:
0 pcs of 10 group + 38 pcs of 1 group
1 pcs of 10 group + 28 pcs of 1 group
2 pcs of 10 group + 18 pcs of 1 group
3 pcs of 10 group + 8 pcs of 1 group
You are trying to determine the half-life of a new radioactive element you have isolated. You start with 1 gram, and 5 days later you determine that it has decayed down to 0.7 grams. What is its half-life? (Round your answer to three significant digits.)
Answers
t(1/2)=[tln2]/[lnN-lnN(t)]
where;
t=time
N=initial amount
N(t)=final amount
thus the half-life of our material will be as follows:
t=5 days
N=1 g
N(t)=0.7 g
N(1/2)=[5*ln2]/[ln1-ln0.7]
=9.717
Which of the following gives all of the sets that contain -1/2?
A.the set of all rational numbers and the set of all real numbers
B.the set of all natural numbers and the set of all irrational numbers
C.the set of all integers and the set of rational numbers
D.the set of all natural numbers, the set of rational numbers, and the set of all real numbers
Answers
Answer:
Option A is Correct Option.
Step-by-step explanation:
Given: [tex]\frac{-1}{2}[/tex]
[tex]\frac{-1}{2}[/tex] is a rational no.
Option A:
Given no is a rational no.
⇒ Both Set of rational no and real nos. contain rational nos..
Thus, Correct Option.
Option B:
set of natural nos and irrational nos does not contain rational nos.
Thus, Not correct option.
Option C:
Set of integers does not contain rational numbers.
Thus, Incorrect Option.
Option D:
Set of all Natural Nos. does not contain rational nos.
Thus, This option is not correct option
Therefore, Option A is Correct Option.
Atmospheric pressure decreases by about 11.8% for every 1000 meters you climb. The pressure at sea level is 1013 millibars (a unit of pressure). Construct an exponential model to represent the atmospheric pressure at a given altitude in thousands of meters. Use your model to determine the atmospheric pressure at an altitude of 4000 meters.
(Express your answer in millibars rounded correctly to the nearest tenth.)
Answers
[tex]\bf \textit{Periodic Exponential Decay}\\\\ A=I(1 - r)^{\frac{t}{p}}\qquad \begin{cases} A=\textit{accumulated amount}\to &1013\\ I=\textit{initial amount}\\ r=rate\to 11.8\%\to \frac{11.8}{100}\to &0.118\\ t=\textit{meters climbed}\to &0\\ p=period\to &1000 \end{cases} \\\\\\ 1013=I(1-0.118)^{\frac{0}{1000}}\implies 1013=I\cdot 1\implies 1013=I[/tex]
so, the inital amount is 1013, when t = 0,
[tex]\bf \textit{Periodic Exponential Decay}\\\\ A=I(1- r)^{\frac{t}{p}}\qquad \begin{cases} A=\textit{accumulated amount}\\ I=\textit{initial amount}\to &1013\\ r=rate\to 11.8\%\to \frac{11.8}{100}\to &0.118\\ t=\textit{meters climbed}\to &t\\ p=period\to &1000 \end{cases} \\\\\\ A=1013(1-0.118)^{\frac{t}{1000}}\implies A=1013(0.882)^{\frac{t}{1000}}[/tex]
now, to check the atmospheric pressure at 4000, simply set t = 4000, to get A.
A rug is shaped like a rectangle who's per Perimeter there's 192 feet the length is five times longer with find the length and the width
Answers
Plug these values into the Perimeter equation P = 2(length) + 2(width):
P = 2(5x) + 2(x), or P = 12x, and this is known to have the value 192 units. Solve for the width, x:
(192 units) = 12x gives us x = 16 units. Then the length is 5(16 units) = 80 units.
Check: Does 2(80 units) + 2(16 units) = 192 units? If so, then our width (16) and length (80) are correct.
If a wheel with a radius of 80 inches spins at a rate of 50 revolutions per minute, find the approximate linear velocity in miles per hour.
Answers
The extreme rock climbing club planned a climbing expedition. The total cost was $5000, which was to be divided equally among the members going. Prior to the trip, 3 members decided not to go. If the cost per person increases by $375, how many people went on the expedition?
Answers
What's the answer to the equation;
√2y+3=11
Answers
Find the common ratio of the geometric sequence: −320,80,−20,5,…−320,80,−20,5,… A. −14−14 B. 1414 C. -4 D. 4
Answers
Divide each term by the previous term to get the common ratio r
r = (term 2)/(term 1) = 80/(-320) = -1/4
r = (term 3)/(term 2) = -20/80 = -1/4
r = (term 4)/(term 3) = 5/(-20) = -1/4
No matter which term you pick (as long as it's not the first one) dividing it by the previous term leads to -1/4 or -0.25 as the common ratio
Answer: -1/4
It seems like none of the answer choices have -1/4 but it could be choice A was supposed to show "-1/4" instead of "-14"? Perhaps the division sign didn't show up for some reason?
a triangular prism has height of 10cm and a base area of 12cm. a rectangular prism has height of 10cm and base dimensions of 3 cm and 4 cm. do the solids have the same volume?
Answers
notice, the base area of the rectangular prism is just 3x4.
The volume of the Moon is about 2.18×10102 cubic kilometers. The volume of Earth is about 1.09×10121 cubic kilometers. The number of Moons that can fit inside Earth can be found by dividing Earth's volume by the Moon's volume. About how many Moons can fit inside Earth?
Answers
Volume of Earth is 1.09×1012 cubic kilometers
Volume of Moon is 2.18×1010 cubic kilometers
Hence number of Moons that can fit in Earth is
1.09×10122.18×1010
= 1012−102 as 2.18=1.09×2
= 1022
= 1002=50
Answer:
50 moons
Step-by-step explanation:
cause i got it right on the test
A wheelchair access ramp has an angle of elevation of 20°. If the ramp reaches to the top of a 33 inch high porch, how long is the ramp?
Answers
length=height/sinα, we are given that α=20° and h=33 in
length=33/sin20° in
length≈96.49 in (to nearest hundredth of an inch)
Sin(20) = 33/x
x*Sin(20) = 33
x = 33/Sin(20)
x = 96.49in long.
*This answer is the length of the SLOPE of the ramp.
Find a_1 and d for an arithmetic sequence with these terms. a_4=9and a_7=18
Answers
[tex]\bf n^{th}\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ n=7\\ d=3 \end{cases} \\\\\\ a_7=a_1+(7-1)d\implies 18=a_1+(7-1)3\implies 18=a_1+18 \\\\\\ 18-18=a_1\implies \boxed{0=a_1}[/tex]
What is the greatest common factor of 4k, 18k4, and 12?
Answers
4k=2*2*k
18k⁴=2*3*3*k*k*k*k
12=2*2*3
GCF=2
the greatest common factor is 2
You deposit $5000 in an account earning 2% interest compounded continuously. How much will you have in the account in 15 years?
Answers
A=p(1+r/100)^n
where:
A=future amount
p=principle
r=rate
n=time in years
Therefore given that, p=$5000, r=2% and n=15 years, the amount after 15 years will be:
A=5000(1+2/100)^15
A=5000(1+0.02)^15
A=5000(1.02)^15
A=6,729.34
We conclude that the amount after 15 years will be $6,729.34
What property describes the number sentence? 6+0=6
Answers
the property describes the number sentence 6+0=6 is : " 0 Neutral element for addtion"
Out of 30 students surveyed 17 have a dog based on these results predict how many of the 300 students in the school have a dog
Answers
17/30 = 0.5666 so about 57% of the sample had dogs
300 x 0.57 = 171 would have dogs
A math teacher gave her class two tests. 76% of the class passed the first test. 58% of the class passed both tests. What is the approximate probability that a student who passed the second test, given that he passed the first test.
Answers
P(test 2 | test 1) = 0.58 ÷ 0.76
P(test 2 | test 1) = 0.76 ≈ rounded to two decimal places
The perimeter of a rectangular field is 168 feet. The field is 36 feet wide. How long is the field
Answers
Length= Peremiter/2 - width
L = P/2 - W
L = 168/2 - 36
L = 84 - 36
L = 48
The length is 48
A cone has a volume of 12 cubic inches. What is the volume of a cylinder that the cone fits exactly inside of?
Answers
notice, the volume of a cylinder which is the same height(h) and radius(r) as a cone, is three times as much of that of the cone's.
The length of a shadow of a building is
29 m
. The distance from the top of the building to the tip of the shadow is
35 m
. Find the height of the building
Answers
35^2=29^2+y^2
y^2=35^2-29^2
y^2=384
y=√384
y≈19.60 m (to the nearest hundredth of a meter, or nearest centimeter)
The total ages of 2 buildings is 201 years. if one building is twice as old as the other, what are the ages of 2 buildings?
Answers
In this case, you are given the buildings sum ages ( building A ages+ building B ages = 201 years) and the building ratio ( Building A ages= 2x Building B ages).
To answer the question, you simply need to insert the second equation into the first. It would be:
building A ages + building B ages= 201
2x building B ages + building B ages= 201
building B ages= 67 years.
Then building A ages= 2x 67 years= 134 years.
The ages of the two buildings are 67 and 134 years. This is found by setting up an algebraic equation where the sum of their ages is 201, and one is twice as old as the other.
Finding the Ages of Two Buildings
If the total ages of two buildings is 201 years and one building is twice as old as the other, then we can express the ages of the buildings using algebra. Let's denote the age of the younger building as y years. The older building is twice as old, so its age would be 2y years. The equation based on their total ages would be:
y + 2y = 201 years
Solving this equation:
Combine like terms: 3y = 201.
Divide both sides by 3 to isolate y: y = 201 / 3.
Calculate the exact value: y = 67 years.
So, the younger building is 67 years old, and the older building, being twice as old, is 134 years old.
Suppose an isosceles triangle ABC has A = 45° and b = c = 4. What is the length of a2?
Answers
a^2=4^2+4^2-2*4*4*cos45°
a^2=32-32cos45°
a^2=32(1-cos45°)
a=√(32(1-cos45°))
a≈3.06 units (to nearest hundredth of a unit)
what is the remainder when 3 is synthetically divided into the polynomial -x^2+5x-9
Answers
Answer:
REmainder is -3
Step-by-step explanation:
find remainder when 3 is synthetically divided into the polynomial -x^2+5x-9
In synthetic division we write all the coefficient of the polynomial and divide by 3
3 -1 5 -9
0 -3 6
-------------------------------
-1 2 -3 ----> Remainder
So the remainder is -3
The table below represents the distance of a train from its destination as a function of time: Time (hours) x Distance (miles) y 0 665 1 570 2 475 3 380 4 285 Part A: What is the y-intercept of the function, and what does this tell you about the train? (4 points) Part B: Calculate the average rate of change of the function represented by the table between x = 1 to x = 4 hours, and tell what the average rate represents. (4 points) Part C: What would be the domain of the function if the train continued to travel at this rate until it reached its destination? (2 points)
Answers
B. Ave rate of change is -95
C. y=0 so x=7 Domain is [0,7]
A rectangular greenhouse has an area of 6480 square meters and a length of 120 meters. How many meters is it in width?
Answers
120 x width = 6480
So, to solve, you would divide 6480 by 120 to find the width. 6480/120 = 54.
Therefore, the width is 54.
Comment under this post or message me for further help!
W=A/L, A=6480 and L=120 so
W=6480/120 m
W=54 m
The course for a walkathon is 6 miles long. If Chloe wants to walk 50 miles during the walkathon, how many times must she walk the course?
Answers
divide 50 by 6
50/6 = 8.333
so she has to walk it 8.33 times
round answer as needed.
Evaluate the limit using the appropriate limit law(s). (if an answer does not exist, enter dne.) lim x → 1 2x2 + 4 x2 + 4x − 4
Answers
[tex]\displaystyle\lim_{x\to1}\frac{2x^2+4}{x^2+4x-4}[/tex]
Since the given function is continuous at [tex]x=1[/tex], the limit of function is equal to the value of the function at [tex]x=1[/tex]:
[tex]\displaystyle\lim_{x\to1}\underbrace{\frac{2x^2+4}{x^2+4x-4}}_{f(x)}=f(1)=\frac{2+4}{1+4-4}=6[/tex]
an airline ramp with a base of 50 makes an angle of 27 with the ground. find the distance a person will walk on a ramp round to the nearest foot
Answers
make sure your calculator is in Degree mode.
Kathy walked 3 1/2 miles per hour for 1 1/4 hours. During 1/2 of her walking time it rained. How many miles did she walk while it was raining?
Answers
1 1/4 = 5/4 * 1/2 = 5/8
3 1/2 = 7/2 * 5/8 = 35/16 = 2 3/16 miles