The first hand derivative of the function is [tex]6x^5 - \frac{35}{63}x^4 - \frac{1}{x^2} - \frac{4}{63x^3}[/tex] and the second derivative is [tex]30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]. To find the first derivative, apply the product rule to the given function. Then, differentiate the first derivative to obtain the second derivative. Simplify each step carefully.
To find the first derivative of the given function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we'll use the product rule, which states that if [tex]y = u(x) \cdot v(x)[/tex], then [tex]y' = u' \cdot v + u \cdot v'[/tex].
First, define u(x) and v(x) as following:
[tex]u(x) = x^2 - \frac{7}{63}x = x^2 - \frac{1}{9}x[/tex][tex]v(x) = x^4 + \frac{1}{x^3}[/tex]Compute u'(x):
[tex]u'(x) = 2x - \frac{1}{9}[/tex]Compute v'(x):
[tex]v'(x) = 4x^3 + (-3)x^{-4} = 4x^3 - \frac{3}{x^4}[/tex]Apply the product rule: [tex]y' = u' \cdot v + u \cdot v'[/tex]
Thus,
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex]Simplify this expression step-by-step to find the first derivative.
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= \left(2x \cdot x^4 + 2x \cdot \frac{1}{x^3} - \frac{1}{9} \cdot x^4 - \frac{1}{9} \cdot \frac{1}{x^3} \right)[/tex][tex]&\quad + \ \left( x^2 \cdot 4x^3 - x^2 \cdot \frac{3}{x^4} - \frac{1}{9}x \cdot 4x^3 + \frac{1}{9}x \cdot \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= 2x^5 + \frac{2}{x^2} - \frac{1}{9}x^4 - \frac{1}{9x^3} \\[/tex] [tex]&\quad + \ 4x^5 - \frac{3}{x^2} - \frac{4}{9}x^4 + \frac{1}{3x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{1}{x^2} - \frac{5}{9}x^4 + \frac{2}{9x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]To find the second derivative, differentiate the first derivative, carefully differentiating each term:
[tex]y'' &= \frac{d}{dx}\left( 6x^5 \right) - \frac{d}{dx}\left( \frac{5}{9}x^4 \right) - \frac{d}{dx}\left( \frac{1}{x^2} \right) + \frac{d}{dx}\left( \frac{2}{9x^3} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{5}{9} \cdot 4x^3 - \left( -2x^{-3} \right) + \left( -\frac{2}{9} \cdot 3x^{-4} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]So, for the function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we have:
First derivative [tex](y')[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]Second derivative [tex](y'')[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]A sample is selected from a population with m = 50 and s = 12. if the sample mean of m = 56 produces a z-score of z = +1.00, then how many scores are in the sample? 2 4 9 16
Final answer:
The number of scores in the sample is 4.
Explanation:
The question provided asks us to determine the number of scores in the sample based on the z-score of +1.00, given a population with a mean (μ) of 50 and a standard deviation (σ) of 12.
The z-score formula for a sample is
z = (M - μ) / (σ / √n),
where
M is the sample mean, μ is the population mean, and σ is the population standard deviation, n representing the sample size.We are given that M = 56, z = 1, μ = 50, and σ = 12.
To find the sample size (n), we use the formula and rearrange to solve for n:
z = (M - μ) / (σ / √n)1 = (56 - 50) / (12 / √n)1 = 6 / (12 / √n)√n = 12 / 6n = (2)^2n = 4Therefore, the number of scores in the sample is 4.
Which point does NOT lie on the graph of the line -2x + 5y = 20?
A. (-10, 0)
B. (0, 4)
C. (1, 4)
D. (5, 6)
Isosceles triangles may be obtuse or acute but never right
The statement, "Isosceles triangle may be obtuse or acute but not right", is a false statement.
Isosceles triangles can be defined as triangles that have two sides (angles) having equal measures.
Obtuse triangles have at least one angle greater than 90°.
Acute triangles have all three angles less than 90°.
Right triangles have one of the angles 90°.
Consider a right triangle ABC.
So, one of the angles, let it be A, will be right.
So, m ∠A = 90°.
By the angle sum property of triangles,
90° + m∠B + m∠C = 180°
m∠B + m∠C = 90°
Let both angles are having equal measures, let it be x.
x + x = 90°
2x = 90°
x = 45°
So, each of the angles has a measure of 45°, and so is an isosceles triangle since the two angles are equal.
So, a right triangle can be isosceles.
Hence the statement is false.
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When a number decreased by 10% the result in 63. What is the number?
Final answer:
To find the number when decreased by 10%, set up an equation and solve for the number.
Explanation:
When a number decreased by 10% the result in 63. What is the number?
Let the number be x.
Equation: x - 0.10x = 63
Solve for x: 0.90x = 63 => x = 63 / 0.90 = 70
Find the value of x (7x) (2x+27)
A. 7
B. 49
C. 61
D. 17
The value of x is 17.
To find the value of x when (7x) and (2x+27) are the two angles made on a straight line, we can use the fact that the sum of the angles on a straight line is 180 degrees.
Let's set up an equation:
(7x) + (2x+27) = 180
First, we simplify the equation by combining like terms:
9x + 27 = 180
Next, we isolate the variable x by subtracting 27 from both sides of the equation:
9x = 180 - 27
9x = 153
Finally, we solve for x by dividing both sides of the equation by 9:
x = 153/9
Simplifying the division gives us:
x = 17
Therefore, the value of x is 17.
Complete question :-
Find the value of x, if (7x) and (2x+27) are the two angles made on a straight line?
A. 7
B. 49
C. 61
D. 17
So, the value of x is not one of the given options A, B, C, or D.
To find the value of x in the expression (7x)(2x+27), we need to simplify the expression and solve for x.
Step 1: Distribute the 7x to both terms inside the parentheses:
(7x)(2x+27) = 14x^2 + 189x
Step 2: Set the expression equal to 0 and factor out common terms, if possible:
14x^2 + 189x = 0
Step 3: Factor out x:
x(14x + 189) = 0
Step 4: Set each factor equal to 0 and solve for x:
x = 0 (from the first factor)
14x + 189 = 0
14x = -189
x = -189/14
So, the value of x is not one of the given options A, B, C, or D.
Item 2
Find −1 1/5+(−3/5). Write your answer as a fraction in simplest form.
The scores of adults on an iq test are approximately normal with mean 100 and standard deviation 15. clara scores 118 on such a test. she scores higher than what percent of all adults
Franco's Pizza is selling their pizzas 35% cheaper than usual if a pizza normally cost $12 how much is it now
Answer:
New or current cost = 12 - 4.2 = $7.8
Step-by-step explanation:
Franco's Pizza is selling their pizzas 35% cheaper than usual. This is a discount on the normal cost. To get the amount it costs now, we will find the percentage of the discount on the original cost and subtract what we get from the original cost.
Discount = 35℅
Original cost of pizza is $12 (if a pizza normally cost $12).
So the discount in terms of dollars
= ℅ discount / 100 × normal cost of pizza
= 35 / 100 × 12 =0.35 × 12 = $4.2
New or current cost of pizza would be normal or original cost - the discount in dollars.
New or current cost = 12 - 4.2 = $7.8
The pizza is now $7.80.
Explanation:If Franco's Pizza is selling their pizzas 35% cheaper than usual, we can calculate the new price by multiplying the original price by (1 - 0.35).
So, the new price of the pizza would be $12 * (1 - 0.35) = $7.80.
Therefore, the pizza is now $7.80.
Geometry Help!!
For triangle TRI the following facts are given:
Segment AN || Segment RI
AT = 8cm
AR = 2cm
TN = 12cm
(a) use a two-column or paragraph format to prove triangle TAN ~ triangle TRI
(b) use the side splitting theorem to find NI.
A rectangular room is twice as long as it is wide, and its perimeter is 36 meters. Find the width of the room.
The width of the rectangular room is 6 meters.
What is a rectangle?A rectangle is one of the types of quadrilaterals in which all four angles are right angles or equal to 90 degrees. It is four-sided polygon in which the opposite sides are parallel and equal to each other.
For the given situation,
Let the width of the rectangular room, w = x
A rectangular room is twice as long as it is wide,
length of the rectangular room, l = 2x
The perimeter of the rectangle = 36 meters.
The formula of perimeter of rectangle is
[tex]P=2(w+l)[/tex]
⇒ [tex]36=2(x+2x)[/tex]
⇒ [tex]36=2(3x)[/tex]
⇒ [tex]x=\frac{36}{6}[/tex]
⇒ [tex]x=6[/tex]
Hence we can conclude that the width of the rectangular room is 6 meters.
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Explain why the hypotenuses of the triangles below have the same slope.
What are the next three numbers in this pattern? 729, 243, 81, 27, __, __, __
Your Bank account balance was $235 and 24. After two checks were cashed (each for the same amount) your balance is now -45.58. What was the amount of each of those checks?
235.24 + 45.58 = 280.82
280.82/2 = 140.41
each check was 140.41
What ratio correctly compares the measurements 10,000 cm to 1000 m? Drag and drop the appropriate numbers into the boxes to show the ratio in the simplest form.
The standard form of the equation of a circle with center (h, k) and radius r is __________________.
Answer:
The center is at (4, -6), and the length of the radius is 5.
Step-by-step explanation:
(x − 4)2 + (y + 6)2 = 25
(x − 4)2 + (y − (-6))2 = 52
When I compare my equation with the standard form, (x − h)2 + (y − k)2 = r2, I get h = 4, k = -6, and r = 5. The center is at (4, -6), and the length of the radius is 5.
plato answer
In Jim's class, 5 times as many kids take the bus as thos who walk. A total of 24 kids walk or take the bus. How many more kids take the bus than walk?
At 1 P.M., ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 5 P.M.?
The reason that ship A is moving -35km/h is because ship B acts as an "origin" and as things move closer to the "origin" they become negative and as things move away from the "origin" they become positive.
We know that the rate of Ship A is expressed as dAdt=−35 and the rate of Ship B is expressed as dBdt=25
Step 2: We want to figure out how many km Ship A and Ship B traveled in 4 hours
ShipA:−35kmh⋅4hr=−140km
Now if we take this −140km and add it to the 150km we can see that Ship A is now only 10km away from where Ship B began
ShipB:25kmh⋅4hr=100km
This means that Ship B is now 100km from where it originally started
We can now redraw our information
Step 3: We must use the Pythagorean Theorem to find the third side of the triangle
x2+y2=z2→102+1002=z2
z=√102+1002
Step 4: Now that we know z, we must differentiate the original equation in order to find the rate at which z in changing
x2+y2=z2→2xdxdt+2ydydt=2zdzdt
We can simplify by canceling out the 2's
xdxdt+ydydt=zdzdt
Step 5: Write down all the information and then plug into equation
x=10anddxdt=−35
y=100anddydt=25
z=√102+1002anddzdt=?
Now plug in the information
xdxdt+ydydt=zdzdt
10(−35)+100(25)=√102+1002dzdt
−350+2500=√102+1002dzdt
2150=√102+1002dzdt
21.393=dzdt
The distance between the ships are increasing at a rate of 21.393 km/h
Eight times the sum of 4 and some number is -56. What is the number?
Using the quadratic formula to solve 2x2 = 4x – 7, what are the values of x
Answer:
C. 2+_ square root 10i / 2
Step-by-step explanation:
What us the equivalent ratio for 4:3
What will be the result of this formula =if(a1<100000,a1*5%,a1*7.5%) , if the cell a1 has a value of 90000?
A loan of $12,500 at 9% is to be repaid with n level payments. if in = 128.04, what is the value of n?
present value of annuity = annual payment * [ 1 - (1+i)^-n ]/i
=>
12500 = 128.04 * [1-(1+9%/12^-n]/9%/12
=>n = 42 payments
Present value of annuity formula yields approximately 42 payments for $12,500, with $128.04 monthly payments at 9% interest.
Let's break it down:
[tex]\[PV = Pmt \times \left[1 - \frac{{(1 + i)^{-n}}}{i}\right]\][/tex]
Where:
[tex]- \(PV\) is the present value of the annuity.\\- \(Pmt\) is the amount of each payment in the annuity.\\- \(i\) is the interest rate per period, expressed as a decimal.\\- \(n\) is the total number of payments.[/tex]
Given your values:
- [tex]\(PV\)[/tex] is $12,500.
- [tex]\(Pmt\)[/tex] is $128.04.
- [tex]\(i\)[/tex] is the monthly interest rate, which is [tex]\(9\% / 12 = 0.09 / 12\)[/tex] since the annual rate is divided by 12 for monthly payments.
- [tex]\(n\)[/tex] is what we're solving for.
Substituting these values into the formula, we have:
[tex]\[12,500 = 128.04 \times \left[1 - \frac{{(1 + 0.09/12)^{-n}}}{0.09/12}\right]\][/tex]
Now let's solve for [tex]\(n\):[/tex]
[tex]\[12,500 = 128.04 \times \left[1 - \frac{{(1 + 0.0075)^{-n}}}{0.0075}\right]\]\[1 - \frac{{(1 + 0.0075)^{-n}}}{0.0075} = \frac{{12,500}}{{128.04}}\]\[\frac{{(1 + 0.0075)^{-n}}}{0.0075} = 1 - \frac{{12,500}}{{128.04}}\]\[ (1 + 0.0075)^{-n} = 0.0075 \times \left(1 - \frac{{12,500}}{{128.04}}\right)\]\[ (1 + 0.0075)^{-n} = 0.0075 \times \left(1 - \frac{{12,500}}{{128.04}}\right)\]\[ (1 + 0.0075)^{-n} = 0.9920\]Taking the natural logarithm of both sides:\[ -n \ln(1 + 0.0075) = \ln(0.9920)\][/tex]
Using a calculator:
n ≈ [tex]\frac{{-0.008025}}{{-0.007472}}\][/tex]
n ≈ [tex]41.961\][/tex]
So, rounding up, we get , n ≈ 42
Therefore, it would take approximately 42 payments to reach a present value of $12,500 given the annuity with an annual interest rate of 9% compounded monthly and a payment of $128.04 per month.
Sara drives 117 miles on 5.2 gallons of gas. She uses this information to calculate how many miles per gallon she can drive. Using this result, how many miles can Sara drive on 12 gallons of gas?
a. 187.5
b. 1.875 c. 270 d. 27Final answer:
To find the number of miles Sara can drive on 12 gallons of gas, divide the total miles she drove by the total gallons of gas she used. Multiply this fuel efficiency by the number of gallons to get the result. (Option C)
Explanation:
To calculate the number of miles Sara can drive on 12 gallons of gas, we need to first find her fuel efficiency in terms of miles per gallon. To do this, we divide the total miles she drove (117 miles) by the total gallons of gas she used (5.2 gallons).
117 miles / 5.2 gallons = 22.5 miles per gallon
Next, we can use this fuel efficiency to calculate the number of miles Sara can drive on 12 gallons of gas. We multiply her fuel efficiency (22.5 miles per gallon) by the number of gallons (12 gallons).
22.5 miles per gallon * 12 gallons = 270 miles
Therefore, Sara can drive 270 miles on 12 gallons of gas.
The table shows the mass and density of some substances. Density of Substances Mass (g) Density of Iron (g/cm3) Density of Silver (g/cm3) 100 7.8 Q 200 P 19.3 Part 1: Is the value of Q less than, greater than, or equal to the value of P? Part 2: Explain your answer for Part 1.
Answer:
Step-by-step explanation:
greder
Part 1: Q is less than P.
Part 2: The density of silver (Q) is higher than that of iron (P), indicating that silver has a greater mass per unit volume, making P greater than Q.
Part 1: The value of Q is less than the value of P.
Part 2: This conclusion can be drawn by comparing the densities of the two substances, iron, and silver. Density is calculated by dividing mass by volume (D = m/V), and since the mass of iron (100g) is equal to the mass of silver (100g), we can simplify the comparison by focusing on their densities.
The density of iron (7.8 g/cm^3) is significantly less than the density of silver (19.3 g/cm^3). Density is a measure of how much mass is packed into a given volume, so the higher the density, the more mass is concentrated in a smaller space. In this case, silver is much denser than iron, which means that for an equal mass, silver occupies a smaller volume compared to iron.
Since Q represents the density of silver and P represents the density of iron, and the density of silver (Q) is greater than the density of iron (P), we can conclude that Q is indeed less than P. The higher density of silver implies that it has a greater mass concentrated in the same volume. The value of Q is less than the value of p.
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what is 6 3/4 - 2 3/12=
Answer:
4.5
Step-by-step explanation:
A regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°. How many sides does the polygon have? 10 12 18 20
Answer:
18
Step-by-step explanation:
Edge 2021 (^Just to back up the answer above, so nobody feels doubts^.)
We have that the sides of the polygon is mathematically given as
n=20
Option C
From the question we are told
A regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°.
How many sides does the polygon have? 10 12 18 20
ArithmeticGenerally the equation for the Polygon exterior angle is mathematically given as
[tex]\theta=\frac{360}{n}\\\\18=\frac{390}{18}[/tex]
n=20
OptionC
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indicate the number of significant figures in 3x 10^6
[tex] \sqrt{26x+13} = x+7[/tex]
The divison of the whole bumber N by 13 gives a quotient of 15 and a remainder of 12. Find N
Carlos bought a new Mastercraft boat for $17,000. He made a $2,500 down payment on it. The bank's loan was for 60 months, and the finance charges totaled $4,900. What's his monthly payment?
A. $323.33
B. $232.33
C. $313.33
D. $332.33
Answer:
Carlos bought a new Mastercraft boat for $17,000. He made a $2,500 down payment on it. The bank's loan was for 60 months, and the finance charges totaled $4,900. What's his monthly payment?
A. $323.33
B. $232.33
C. $313.33
D. $332.33
Step-by-step explanation:
1.- Carlos paid $ 2,500 of the $ 17,000 that Mastercraft costs and would be equal to: $ 17,000 - $ 2,500 = $ 14,500
2.- We have financial charges amounting to $ 4,900 to be added to the previous amount, and would be equal to: $ 14,500 + $ 4,900 = $ 19,400
3.- The bank loan is for 60 months, which would result in: $ 19,400 / 60 = $ 323.33
The answer is: A. $ 323.33