Calculus HW , will someone please teach me how to do this problem. I'm struggling, thanks! 10 Points
Answers
Compute the first two derivative.
\(f(x) = x \sqrt{16 - x^2} = x (16 - x^2)^{1/2}\)
\(f'(x) = x \dfrac{d}{dx} (16 - x^2)^{1/2} + (16 - x^2)^{1/2} \dfrac{d}{dx} x \\\\ ~~~~ = \dfrac x2 (16 - x^2)^{-1/2} \dfrac{d}{dx} (16-x^2) + (16 - x^2)^{1/2} \\\\ ~~~~ = \dfrac x2 (16 - x^2)^{-1/2} (-2x) + (16 - x^2)^{1/2} \\\\ ~~~~ = (16-x^2)^{-1/2} \left(-x^2 + (16 - x^2)\right) \\\\ ~~~~ = \dfrac{16 - 2x^2}{(16 - x^2)^{1/2}}\)
\(f''(x) = \dfrac{(16-x^2)^{1/2} \frac d{dx} (16-2x^2) - (16-2x^2) \frac{d}{dx} (16-x^2)^{1/2}}{\left((16-x^2)^{1/2}\right)^2} \\\\ ~~~~ = \dfrac{(16-x^2)^{1/2} (-4x) - (8-x^2) (16 - x^2)^{-1/2} \frac{d}{dx} (16-x^2)}{16 - x^2} \\\\ ~~~~ = \dfrac{-4x (16-x^2) - (8-x^2) (-2x)}{(16 - x^2)^{3/2}} \\\\ ~~~~ = \dfrac{2x^3-48x}{(16 - x^2)^{3/2}}\)
Take note of the domain of \(f(x)\). We must have \(16-x^2\ge0\) for the square root to be defined, so
\(16 - x^2 \ge 0 \implies x^2 \le 16 \implies |x| \le 4\)
and \(f(x)\) exists only for \(-4 \le x \le 4\).
Intercepts
Set \(x=0\) to find the \(y\)-intercepts. The only one is (0, 0), since
\(f(0) = 0 \sqrt{16-0^2} = 0\)
The first intercept you listed is not a valid intercept. One or both coordinates must be 0.
Set \(f(x)=0\) and solve for \(x\) to find \(x\)-intercepts. We already found (0, 0); there are two others at (-4, 0) and (4, 0).
\(f(x) = x \sqrt{16 - x^2} = 0 \\\\ x = 0 \text{ or } \sqrt{16 - x^2} = 0 \\\\ x = 0 \text{ or } 16 - x^2 = 0 \\\\ x = 0 \text{ or } x^2 = 16 \\\\ x = 0 \text{ or } x = \pm4\)
The instructions say to list these in order from smallest to largest by \(x\)-coordinate first, then by \(y\)-coordinate. So the proper order of the intercepts would be (-4, 0), (0, 0), and (4, 0).
Relative minima/maxima
Find the critical points. We have \(f'(x) = 0\) when
\(f'(x) = \dfrac{16 - 2x^2}{(16 - x^2)^{1/2}} = 0 \\\\ 16 - 2x^2 = 0 \\\\ x^2 = 8 \\\\ x = \pm2\sqrt2 \approx \pm 2.83\)
and \(f'(x)\) is undefined when
\((16 - x^2)^{1/2} = 0 \\\\ 16 - x^2 = 0 \\\\ x^2 = 16 \\\\ x = \pm4\)
Check the sign of the second derivative at the first two critical points.
\(f''(-2\sqrt2) = 4 > 0 \implies \text{rel. min. at } f(-2\sqrt2) = -8\)
\(f''(2\sqrt2) = -4 < 0 \implies \text{rel. max. at } f(2\sqrt2) = 8\)
For posterity, we should also check the value of \(f(x)\) at the endpoints of the domain.
\(f(-4) = f(4) = 0\)
So \(f(x)\) has a relative minimum at (-2.83, -8) and a relative maximum at (2.83, 8).
Inflection points
We have \(f''(x) = 0\) for
\(f''(x) = \dfrac{2x^3-48x}{(16 - x^2)^{3/2}} = 0 \\\\ 2x^3 - 48x = 0 \\\\ 2x (x^2 - 24) = 0 \\\\ 2x = 0 \text{ or } x^2 - 24 = 0 \\\\ x = 0 \text{ or } x = \pm2\sqrt6\approx\pm4.90\)
The two non-zero solutions fall outside the domain, so the only inflection point is (0, 0).
Related Questions
Please help me solve this.
Answers
\(\boxed{A}\\\\ U=5(2n+22)+2\left( n+\cfrac{3}{2} \right) \\\\\\ U=10n+110+2n+3 \implies U=12n+113 \\\\[-0.35em] ~\dotfill\\\\ \boxed{B}\hspace{5em}\textit{in 2009, that's 9 years after 2000, n = 9}\\\\ U(9)=12(9)+113\implies U(9)=221 ~~ millions\)
What is the approximate length of side GF in triangle EFG?
Answers
Answer:
41.93 degrees
Step-by-step explanation:
Find the coefficient of the given term when the expression is expanded by the binomial theorem.
x7 in (3x + 4)10
Can someone help me out with this problem i need step by step help
Discrete mathematics
Answers
The coefficient of \(x^{7}\) is 16796160.
For this, we have to find the coefficient of \(x^{7}\) in \((3x+4)^{10}\)
General Term: This term symbolizes all of the terms in the binomial expansion of (x + y)^n.
The general term in the binomial expansion of (x + y)^n is
T(r+1) = nCr x^(n-r)y^r.
Here the r-value is one less than the number of the term of the binomial expansion. Also, nCr is the coefficient, and the sum of the exponents of the variables x and y is equal to n.
We can use the general term for this.
\(T_{r+1} = (nCr) a^{n-r} x^{r}\)
Since we have to find \(x^{7}\),
\(T_{3+1} = (10C3)(3x)^{7} 4^{3} \\T_{3+1} = [(10C3)3^{7} 4^{3} ]x^{7}\)
This is the coefficients of \(x^{7}\)
Calculating,
10C3 = 120 (using combinations)
\(3^{7} = 2187\\4^{3} = 64\)
Therefore, multiplying the 3,
Coefficients of \(x^{7}\) = 120 * 2187 *64
= 16796160
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In a certain Algebra 2 class of 28 students, 7 of them play basketball and 5 of them play baseball. There are 18 students who play neither sport. What is the probability that a student chosen randomly from the class plays both basketball and baseball?
Answers
Answer:
2 students play both
Step-by-step explanation:n(u)=28
n(b)=7,n(b)=5,n(who play neither sport)=18
HERE,
n(u)=n(b)+n(b)-n(BnB)+n(who play neither sport)
28=7+5-n(BnB)+18
28=30-n(BnB)
n(BnB)=30-28
=2 answer
PLEASE HELP ME I NEED HELP PLEASE HELP
PLEASE GIVE THE ANSWER UNDER THE QUESTION AND RATIO
Answers
Answer:
6. 225 feet in 3 minutes = 75 feet in 1 minute (slower)
360 feet in 5 minutes = 72 feet in 1 minute
7. $4 for 5 ounces = $0.80 for 1 ounce (most expensive)
$14 for 20 ounces = $0.70 for 1 ounce
8. 72 people in 6 hours = 12 people in 1 hour
117 people in 9 hours = 13 people in 1 hour (greater rate per hour)
Answer:
A = 369 feet in 5 minutes B = $4 for 5 ounces C = 117 people in 9 hours
5-x=12 two step equations
Answers
Answer:
x=-7
Step-by-step explanation:
ur welcome
Answer: x = -7
5-x=12
-5. -5
-x=7
-x/-1=7/-1
x=-7
Hope this helps !
Reflect the figure across the y- axis, then state the ordered pairs of the image.
May someone please explain how I would reflect it?
Answers
R’ (2, -3)
X’ (-2, 0)
Z’ (-2, -2)
EXPLANATION:
Reflecting a figure across the y-axis basically means flipping all of the x-values of your coordinates while retaining its y-value.
In other words, flip the signs of x and keep y the exact same for each coordinate.
B (-3, 2)
=
B’ (3, 2)
Find the mode of the data set 6, 8, 9, 4, 9, 6, 6, 9, 4, 8, 6 plz help asap
A. 6
B. 4
C. 8.5
D.9
Answers
Answer:
9
Step-by-step explanation:
6 members of the Benton family are going to their school's Community Day. They have a coupon for $4.50 off their total. If they pay $40.50 for all their tickets, how much does one ticket cost without the coupon?
Answers
Answer:
$7.50
Step-by-step explanation:
you add 40.50 + 4.50 which then equals 45 and you divide that by 6 and you get 7.50
One ticket costs $7.50 without the coupon this we obtained by dividing total cost before discount by Number of tickets
Let us find the total cost of the tickets before the coupon is applied. We can do this by adding the discount amount ($4.50) to the final cost ($40.50):
Total cost before discount = Final cost + Discount amount
Total cost before discount = $40.50 + $4.50
Total cost before discount = $45.00
Now we can divide the total cost before discount by the number of tickets
(6) to find the cost of one ticket:
Cost of one ticket = Total cost before discount / Number of tickets
Cost of one ticket = $45.00 / 6
Cost of one ticket = $7.50
Therefore, one ticket costs $7.50 without the coupon.
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O
14) Determine which of the following functions is
an example of exponential decay.
A) f(x) = 6.7e*
C) f(x)=3.2*
2
4·(9)
(³)
D) f(x) = -7 8 -x
3
B) f(x) = 1
Answers
The functions which represent the exponential decay is option D. f(x) = -7 (\(\frac{8}{3}\))⁻ˣ.
What is Exponential Decay?Exponential decay is defined as the decrease in the population or a group of something and the decreasing amount is directly proportional to the population size.
Given are four functions. We have to find out whether the function represents exponential decay.
A) f(x) = 6.7 eˣ
e is a number greater than 1.
So with increasing value of x, the value of f(x) also increases.
So it is not exponential decay.
B) f(x) = \(\frac{1}{4}\) × (\(\frac{5}{2}\))ˣ
When x = 0, f(x) = 1/4
When x = 1, f(x) = 5/8
Here also, as x increases, f(x) also increases.
So this is not a exponential decay.
C) f(x) = 3 × 2ˣ
When x = 0, f(x) = 3
When x = 1, f(x) = 6
This is also not a exponential decay.
D) f(x) = -7 (\(\frac{8}{3}\))⁻ˣ
When x = 0, f(x) = -7
When x = 1, f(x) = -2.625
This is decreasing, so it is an exponential decay function.
Hence option D represents the exponential decay function.
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HI PLEASE HELP ON QUESTION ASAP USING AVERAGE (MEAN) TO ANSWER QUESTION! IF UR ANSWER AND EXPLAINATION IS CORRECT ILL RATE YOU FIVE STARS, A THANKS AND MAYBE EVEN BRAINLIEST. PLEASE MAKE SURE YOU ANSWER MY QUESTION USING AVERAGES.
1) a meal for 6 cost £12 per person. as it is one of the diners birthday , the other 5 decided to pay for his meal. how much do each of the five friends need to pay?
Answers
Each of the five friends needs to pay £14.40 to cover the cost of the birthday person's meal.
To calculate how much each of the five friends needs to pay, we can use the concept of averages or mean.
The total cost of the meal for 6 people is £12 per person. This means that the total cost of the meal is 6 * £12 = £72.
Since the other five friends have decided to pay for the birthday person's meal, they will evenly divide the total cost of £72 among themselves.
To find the average amount each friend needs to pay, we divide the total cost by the number of friends paying, which is 5:
£72 / 5 = £14.40
Using the concept of averaging or finding the mean allows us to distribute the cost equally among the friends, ensuring fairness in sharing the expenses.
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444444r54drtfjfftddtcghgcgvvvvbvgiygygvvvvuygyv8uu8yut
Answers
Answer:
l
Step-by-step explanation:
l
2 x 3 2 x 5 x 7 is the prime factorization of what number
Answers
Answer: 2240
Step-by-step explanation: Simplify the expression.
I hope this helps you out.
Here, 2 x 3 x 2 x 5 x 7 is the prime factorization of 420.
Given that,
To determine the number that is this prime factorization 2 x 3 2 x 5 x 7 belongs to.
Factors can be defined by splitting the value into multipliable values.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Prime factors of the number are the factors that consist of only prime numbers,
i.e.
2 x 3 x 2 x 5 x 7 = 420
Thus, 2 x 3 x 2 x 5 x 7 is the prime factorization of 420.
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2.1, 1.4, 0.7, 0, ...
Answers
pls help me thank you
Answers
The probability that a student chosen at random is in the choir or band or both is 0.911
What is probability?Probability is the likelihood that an event will happen. This can range from an event being impossible to some likelihood to being absolutely certain. In mathematics terms, probability is on a scale from 0 to 1. Zero means the event is impossible, like rolling a seven on a die that only has digits from 1 to 6. One is an event that will happen, it is certain.
Total number of students in the school = 450
number of students in choir = 110
number of students in band = 240
number of students in both = 60
Probability a student chosen at random is in the choir or band or both is
110/450 + 60/450 + 240/450
= 410/450 which is 0.911
In conclusion, the probability of a student being in music, or band or both is 0.911
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Will give brainliest to whoever solves this properly, Please help me
Answers
The positions will be filled in 56,280 ways
How to find the number of waysThere are 11 qualified candidates applying for 7 positions. The order of the positions does not matter. after choosing a position the number of persons drop so we solve as follows
For the position of president
11 combination 1 = 11
For the position of secretary
10 combination 1 = 10
For the position of treasurer
10 combination 1 = 10
For the position of three directors
8 combination 3 = 56 ways
the total number of ways to fill these positions is
11 x 10 x 9 x 56 = 56,280.
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please help only 10 minutes
Answers
Answer:
1 gallon = 4 quarts
20 gallon = (4×20) quarts=80quarts
C. is the right answer.In a sale the price of a bike is reduced by 40% the sale price of the bike is £192 how much did the bike cost before the sale?
Answers
the three data sets show the number of text messages sent by jada, and diego, and lin over 6 days. one of the data sets has a mean of 4, one has a mean of 5, and one has a mean of 6.
1. which data set has which mean? what does this tell you about the text messages sent by the three students?
2. which data set has the greatest variabillity? Explain
Answers
1. Jada's data set has a mean of 5, Diego's data set has a mean of 6, and Lin's data set has a mean of 4.
2. The data set that has the greatest variability is Lin's data set because it has the highest standard deviation of 3.6.
How to calculate the mean for the set of data?In Mathematics and Geometry, the mean for this set of data can be calculated by using the following formula:
Mean = [F(x)]/n
Question 1.
In this scenario and exercise, we would determine the mean of the three data sets based on the number of text messages sent by Jada, and Diego, and Lin over 6 days as follows;
Mean number of text for Jada = (4 + 4 + 4 + 6 + 6 + 6)/6
Mean number of text for Jada = 30/6
Mean number of text for Jada = 5.
Mean number of text for Diego = (4 + 5 + 5 + 6 + 8 + 8)/6
Mean number of text for Diego = 36/6
Mean number of text for Diego = 6.
Mean number of text for Lin = (1 + 1 + 2 + 2 + 9 + 9)/6
Mean number of text for Lin = 24/6
Mean number of text for Lin = 4.
Question 2.
In Statistics, variability is a measure of the spread of a data set. Also, we would use standard deviation to determine the variability in each data set as follows;
Standard deviation = √(1/n × ∑(xi - u₁)²)
Standard deviation for Jada = √(1/6 × [(4 - 5)² + (4 - 5)² + (4 - 5)² + (6 - 5)² + (6 - 5)² + (6 - 5)²]
Standard deviation for Jada = √(6/6) = 1
Standard deviation for Diego = √(1/6 × [(4 - 6)² + (5 - 6)² + (5 - 6)² + (6 - 6)² + (8 - 6)² + (8 - 6)²]
Standard deviation for Diego = √(14/6) = 1.53
Standard deviation for Lin = √(1/6 × [(1 - 4)² + (1 - 4)² + (2 - 4)² + (2 - 4)² + (9 - 4)² + (9 - 4)²]
Standard deviation for Lin = √(76/6) = 3.6
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Find two irrational numbers and two rational numbers between 0.5 and 0.552.
Answers
Answer:
Rational: 0.51 and 0.52
Irrational: sqrt(.26) and sqrt(.27)
according to money magazine, maryland had the highest mean annual household income of any state in at (time website). assume that annual household income in maryland follows a normal distribution with a mean of and standard deviation of .
Answers
The probability of getting a household income of $75,000 in Maryland is 0.3989.
The normal distribution formula is given by:
P(x) = 1/√(2πσ^2) * e^[-(x-μ)^2 / (2σ^2)]
Where,
P(x) - Probability of getting a household income of x
μ - Mean annual household income in Maryland
σ - Standard deviation of annual household income in Maryland
Assuming that the mean annual household income in Maryland is $75,000 and the standard deviation is $10,000, the probability of getting a household income of $75,000 can be calculated using the formula given above.
P(75000) = 1/√(2π10000^2) * e^[-(75000-75000)^2 / (2*10000^2)]
= 0.3989 * e^(0)
= 0.3989
Therefore, the probability of getting a household income of $75,000 in Maryland is 0.3989.
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X=91° and y=42° what is z?
Answers
Here, we just need to add x and y and subtract it from 180.
x + y = 91 + 42 = 133.
180 - 133 = 47.
Hence, z = 47 degrees.
Answer:
47°
Step-by-step explanation:
straight line = 180° we remove the angles x and y and find z
180 - 91 - 42 =
47°
Teresa is maintaining a campfire.She has kept the fire steadily burning for 4 hours using 6 logs. The number of logs needed to keep the fire burning is proportional to the number of hours it burns. She wants to know how many logs she needs to keep the fire burning for 18 hours.
Select the equationis Teresa can use to determine the number of logs she needs, x, to maintain the fire for 18 hours.
A. x/6 = 4/18
B. x/6 = 18/4
C. x/18 = 6/4
D. x/4 = 18/6
E. 4/6 = 18/x
Answers
Answer:
E. 4/6 = 18/x
Step-by-step explanation:
4 hour using 6 logs
18 hours using x logs
what is the quotient of 14,875÷35
Answers
Answer:
424.8571428571429
Step-by-step explanation:
Answer:
Step-by-step explanation:
Sally was driving her own car and collided with a pick-up truck. Sally sustained $150,000 in injuries and her passenger sustained $1,000 in injuries. If her coverage was 100/300/50, what are the total medical expenses the insurance company would pay for this accident?
Answers
The insurance company would pay a total of $101,000 for the medical expenses resulting from the accident.
In this scenario, Sally's car insurance policy has coverage limits of 100/300/50. These numbers represent the maximum amount the insurance company will pay for different types of damages in an accident. Let's break down what these numbers mean in the context of the collision.
The first number, 100, represents the maximum amount (in thousands of dollars) that Sally's insurance company will pay for bodily injury per person. Since Sally sustained $150,000 in injuries and her passenger sustained $1,000 in injuries, the insurance company will cover Sally's medical expenses up to the policy limit, which is $100,000.
The second number, 300, represents the maximum amount (in thousands of dollars) the insurance company will pay for bodily injury per accident. In this case, both Sally and her passenger were injured, so the insurance company will cover their combined medical expenses up to the policy limit, which is $300,000.
The third number, 50, represents the maximum amount (in thousands of dollars) the insurance company will pay for property damage per accident. However, since the question only mentions injuries and not property damage, we can disregard this number for now.
Therefore, the total medical expenses the insurance company would pay for this accident would be $100,000 for Sally's injuries and $1,000 for her passenger's injuries, totaling $101,000.
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Which angle number is a vertical angle to ZGKJ? B C/3 \E A 2 5\6 O. G ZA H K M J
Answers
Answer:
∠NKL = angle 12
Explanation:
Vertical angles are opposite angles that are formed when two lines cut each other.
Therefore, a vertical angle to ∠GKJ is ∠NKL because they are formed when lines GL and OJ cross.
So, the answer is angle 12
it can be proven that BD=CD if it is known that
Answers
If it is known that ∠BAD ≅ ∠CAD, BD ≅ CD can be proved.
Given, that In ΔABC
AB = AC
we have to prove that, BD≅CD
Now we have,
AB = AC (Given)
AD = AD (Common)
if it can be known that
∠BAD ≅ ∠CAD
then, by SAS congruence property we can say
that, ΔABD ≅ ΔACD
as, AB = AC (Given)
∠BAD ≅ ∠CAD
AD = AD (Common)
Hence, By CPCT it can be proved that BD ≅ CD.
If it is known that ∠BAD ≅ ∠CAD, BD ≅ CD can be proved.
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What is 7/11 x 1/4 in simplest form
Answers
Answer:
\(\frac{7}{11} * \frac{1}{4}\) = \(\frac{7}{44}\).
Cannot reduce it. Then it will make the numerator a decimal
Which cannot happen.
\(\frac{7}{44}\) is the answer then
Answer:
7/11 x 1/4 = 0.015
I hoped it helped you.
PLSSS ANSWER
\(\sqrt{28-3(4)}\)
Answers
Answer:
\(\boxed{\sf{4}}\)Step-by-step explanation:
Use the order of operations to solve this problem.
\(\Longrightarrow: \sf{\sqrt{28-3(4)} }\)
Remove parentheses.
→ (4)=4
Rewrite the problem down.
\(\Longrightarrow:\sf{\sqrt{28-3*4} }\)
PEMDAS stands for:
ParenthesesExponentsMultiplyDivideAddSubtract→ 28-3*4
Multiply.
→ 3*4=12
→ 28-12
Subtract.
→ 28-12=16
\(\Longrightarrow:\sf{\sqrt{16} }\)
\(\Longrightarrow\sf{\sqrt{16}=4^2 }\)
Use the radical rule.
\(\Longrightarrow: \sf{4^2=\boxed{\sf{4}}\)
Therefore, the correct answer is 4.I hope this helps you! Let me know if my answer is wrong or not.
Answer:
\(\±4\)
Step-by-step explanation:
Given expression: \(\sqrt{28 - 3(4)}\)
Here, we can see that the terms, 28 and 3(4) are inside the root. Since the two terms are inside the root, we can subtract them. Therefore:
\(\implies\sqrt{28 - 3(4)}\)\(\implies\sqrt{28 - 12}\)\(\implies\sqrt{16}\)Usually, \(\sqrt{16}\) can also be written as \(\sqrt[2]{16}\). Therefore:
\(\implies\sqrt[2]{16} = \sqrt[2]{4 \times 4} = \± 4\)Therefore, the simplified expression is ±4.
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Parábola donde su directriz eje x
Answers
Answer:
umm realmente no lo sé
Step-by-step explanation: