Answer:
[tex]\displaystyle y'' = \frac{\sqrt{x} + \sqrt{y}}{2x^\big{\frac{3}{2}}}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Implicit Differentiation
Step-by-step explanation:
*Note:
Allow a to be defined as an arbitrary constant.
Step 1: Define
[tex]\displaystyle x^\big{\frac{1}{2}} + y^\big{\frac{1}{2}} = a^\big{\frac{1}{2}}[/tex]
Step 2: Differentiate
Basic Power Rule [Addition/Subtraction, Chain Rule]: [tex]\displaystyle \frac{1}{2\sqrt{x}} + \frac{y'}{2\sqrt{y}} = 0[/tex]Isolate y' term: [tex]\displaystyle \frac{y'}{2\sqrt{y}} = -\frac{1}{2\sqrt{x}}[/tex]Isolate y': [tex]\displaystyle y' = -\frac{\sqrt{y}}{\sqrt{x}}[/tex]Derivative Rule [Quotient Rule]: [tex]\displaystyle y'' = -\frac{(\sqrt{y})'(\sqrt{x}) - \sqrt{y}(\sqrt{x})'}{(\sqrt{x})^2}[/tex]Basic Power Rule [Derivative Rule - Chain Rule]: [tex]\displaystyle y'' = -\frac{\frac{y'\sqrt{x}}{2\sqrt{y}} - \frac{\sqrt{y}}{2\sqrt{x}}}{(\sqrt{x})^2}[/tex]Simplify: [tex]\displaystyle y'' = -\frac{\frac{y'\sqrt{x}}{2\sqrt{y}} - \frac{\sqrt{y}}{2\sqrt{x}}}{x}[/tex]Rewrite: [tex]\displaystyle y'' = \frac{-(y'x - y)}{2x^\big{\frac{3}{2}}\sqrt{y}}[/tex]Substitute in y': [tex]\displaystyle y'' = \frac{-(-\frac{\sqrt{y}}{\sqrt{x}}x - y)}{2x^\big{\frac{3}{2}}\sqrt{y}}[/tex]Simplify: [tex]\displaystyle y'' = \frac{\sqrt{x} + \sqrt{y}}{2x^\big{\frac{3}{2}}}[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
A textbook store sold a combined total of 255 math and sociology textbooks in a week. The number
of math textbooks sold was 55 more than the number of sociology textbooks sold. How many textbooks
of each type were sold?
Answer:
c + s = 258
s = c - 60
---
c + s = 258
c + c - 60 = 258
2c - 60 = 258
2c = 318
---
c = 159
s = 99
Step-by-step explanation:
Question is below
A. x + 2
B. x + 3
C. x - 2
D. x + 1
Answer:
D. x + 1
Step-by-step explanation:
The expression[tex] \frac{x^2+3x+2}{x+2} [/tex] can be simplified as follows:
[tex]x^2 + 3x + 2 [/tex]
Doing middle term factorisation
[tex]= x^2+(2+1)x+2[/tex]
[tex]=x^2+2x+1x+2[/tex]
[tex]=(x + 2)(x + 1)[/tex]
Again
Dividing both side by x+2.
[tex]\frac{x^2 + 3x + 2}{x + 2} = \frac{(x + 2)(x + 1)}{x + 2}[/tex]
we get
[tex]\frac{x^2 + 3x + 2}{x + 2}= x + 1[/tex]
Therefore, the expression
[tex]\frac{x^2+3x+2}{x+2}[/tex] is equivalent to D. x + 1.
Note that the expression is only equivalent to x + 1 when x is not equal to -2. If x = -2, then the expression is undefined.
Jorge runs 12 yards 6 seconds. What is the rate
of how fast Jorge runs?
If $120 doubles in approximately 13 years, what is the interest rate?
Answer:
33.3%Step-by-step explanation:
Simple Interest = Principal * Rate * Time/100
Amount = Principal + Interest
If $120 doubles in approximately 13 years, then the amount after 13years = 2*$120 = $240
Interest = $240-$120
Interest = $120
120 = 120*Rate * 13/100
1 = 13Rate/100
13rate = 100
rate = 100/3
rate = 33.3%
Hence the interest rate is 33.3%
find the airthmetic mean
x= 10-20, 20-30,30-40,40-50,50-60,60-70,70-80
F= 4,10,5,7,3,4,2
Answer:
x
Step-by-step explanation:
You can finish 9 math problems in 20/3 minutes. What is your unit rate of problems per minute? (Please help now!)
Answer:
You can finish 1 problem (rounded) or 1.35 problems
Step-by-step explanation:
6 2/3 minutes=9 math problems
1 minute=1.35
What is added to2/4 added to1/3?
Answer:
5/6, multiply the denominators so that they are the same number. multiply their numerators by the same number you did for the denominator. 2/4 becomes 6/12 and 1/3 becomes 4/12. then you add them and simplify.
HURRY NEED HELP WITH THIS MATH
How do you write 8,300 in scientific notation?
Answer:
8.3 x 10³
Step-by-step explanation:
8,300 can be written as
8.3 x 1000 (re-writing 1000 in terms of powers of 10, 1000 = 10³)
= 8.3 x 10³
A package of 16 individual bags of Takis cost $12.80. If each bag contains 12 ounces, what is the unit cost per ounce?
Answer:
$0.067
Step-by-step explanation:
Total cost of 16 bags = $12.80
weight in 1 bag = 12 ounce
weight in 16 bag = 16 * weight in 1 bag
weight in 16 bag = 16* 12 ounce = 192 ounce
Thus,
192 ounce cost = Total cost of 16 bags= $12.80
cost of 182 ounce = $12.80
to find cost of 1 ounce , we divide both side by 182
cost of 1 ounce = 12.80/192 = $0.067
Thus,
the unit cost per ounce is $0.067
Remove the parentheses from the following expression, and combine like terms: (a + b - c) + 3a - 2c
Answer:
4a + b - 3c
Step-by-step explanation:
you combine a and 3 a, there is nothing to add to be, and since both c's are negative you get - 3c
Answer:
4a + b - 3c
Step-by-step explanation:
How to solve 4x-10>x/3-12 step by step showing to to get rid of fractions
Answer:
x > -6/11
Step-by-step explanation:
4x - 10 > x/3 - 12
~Simplify
4x - 10 > 1/3x - 12
~Add 10 to both sides
4x > 1/3x - 2
~Subtract 1/3x to both sides
11/3x > -2
~Multiply 3/11 to both sides
x > -6/11
Best of Luck!
What is the greatest common factor of 30b^2, 80b^2, and 20b^2
Answer:
10b²
Step-by-step explanation:
First, let's look at the coefficients. The greatest common factor of all three numbers 30, 80, and 20, is 10. Now, let's look at the variables. All three terms are divisible by b² (with no remainder). We can combine 10 and b² so they become one term, and that is the GCF. 10b².
PLEASE HELP ME PLSSSSSSSSZZ
Answer:
The answer is 5.
Step-by-step explanation:
Use order of operations. Parentheses come first, so I will compute what is in parentheses first. 11-3 is 8. Then, division comes next. I will divide 8 by 2 to get 4. Lastly, add 1 to get 5.
Question is in the picture
The formula that describes the average revenue per person in terms of weeks since the movie debuted is R = 1.4 / w².
How to calculate the valueGiven:.M = 350w
T = 250w³
In order to find the average revenue per person (R), we divide M by T:
R = M / T
Substituting the given formulas:
R = (350w) / (250w³)
Now, let's simplify this expression:
R = 350w / 250w³
R = (350 / 250) * (w / w³)
R = 1.4 * (1 / w²)
R = 1.4 / w²
Therefore, the formula that describes the average revenue per person in terms of weeks since the movie debuted is R = 1.4 / w².
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an object accelerates 16m/s2 when a force of 4N is applied what is the mass
Answer:
0.25kg
Step-by-step explanation:
Newton's second law states that Force=Mass x Acceleration.
Since we are given two of the three values, we can plug them into this equation to solve for the third: Mass.
Force = 4N
Acceleration = 16m/s/s
4 = Mass x 16
Divide both sides by 16 so Mass is by itself on one side:
4/16 = Mass x 16/16
0.25 = Mass
The mass of the object is 0.25 kg (SI Units).
Hope this helped!
Name the sides of the following angle.
Answer:
your answers c. v t and t w
Answer:
TV and TW
Step-by-step explanation:
If the complement of an angle is 32 degrees, what is the measure of the supplement
Answer:
122
Step-by-step explanation:
The complement of an angle adds up to 90.
The supplement of an angle adds up to 180.
So if the unknown angle has a complement of 32, then that angle has to be 58. 32 + 58 = 90.
You now know the angle is 58 so the supplement is
180 - 58 which is 122
A 2020-inch candle burns down in 10 hours. At what rate does the candle burn, in inches per hour?
Answer:
202 inches per hour
Step-by-step explanation:
2020 ÷ 10 = 202
Hope this helps..
Answer:
202
Step-by-step explanation:
2020/10=202
A project team from 2 to 6 people is being formed from a staff of 22. How many such teams are possible?
Answer:
110,033 ways :)
Step-by-step explanation:
Alright, so, there can be between 2 and 6 people in the team.
For a team of 2, there are C(22, 2) = 231 ways to form a team.
For a team of 3, there are C(22, 3) = 1540 ways
For a team of 4, there are C(22, 4) = 7315 ways
For a team of 5, there are C(22, 5) = 26334 ways
For a team of 6, there are C(22, 6) = 74613 ways, mind boggling! :)
So, the total is 231 + 1540 + 7315 + 26334 + 74613 = 110,033 ways!
a ___ inequality in which each term is either a constant or the product of a constant and a single variable
A linear inequality is an inequality in which each term is either a constant or the product of a constant and a single variable.
In mathematics, an inequality expresses a relationship between two quantities using symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to). A linear inequality specifically refers to inequalities where the terms involved are either constants or the product of a constant and a single variable.
For example, consider the linear inequality 3x - 2y < 5.
Here, the terms 3x, -2y, and 5 satisfy the criteria of a linear inequality as they involve constants (3 and -2) and a single variable (x and y).
The multiplication of constants and variables is allowed in linear inequalities as long as each term follows the specified conditions.
Linear inequalities are commonly used in various fields of mathematics, economics, and sciences to represent relationships between quantities, such as constraints in optimization problems or conditions for feasible solutions.
They provide a framework for analyzing and modeling real-world situations where variables are subject to certain limitations or conditions.
In summary, a linear inequality is an inequality where each term is either a constant or the product of a constant and a single variable, allowing for the representation of relationships between quantities in various mathematical and real-world contexts.
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If u (x) = negative 2 x squared and v (x) = 1/x, what is the range of (u circle v) (x)? (1/3, 0) (3, infinity) (negative infinity, 3) (negative infinity, positive infinity)
The range of (u ∘ v) (x) is (negative infinity, positive infinity).
To find the range of (u ∘ v) (x), we need to determine the possible output values of the composite function (u ∘ v) (x), which is formed by applying function u to the output of function v.
1. Let's start by evaluating the composite function (u ∘ v) (x):
(u ∘ v) (x) = u(v(x))
2. Substitute v(x) into function u:
u(v(x)) = u(1/x)
3. Replace u(x) with its given expression:
u(1/x) = -2(1/x)^2
4. Simplify the expression:
u(1/x) = -2/x²
5. Now, we need to determine the possible values of -2/x². Since x can take any real value except 0 (as 1/0 is undefined), the expression -2/x² will vary based on x.
6. As x approaches negative infinity, -2/x² approaches 0 from the negative side.
As x approaches positive infinity, -2/x² approaches 0 from the negative side as well.
7. Therefore, the range of (u ∘ v) (x) is (negative infinity, positive infinity), which means that the composite function can take any real value.
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The paragraphs below come from the rental agreement signed by a customer when they set up an account at Super Video.
“All rentals are due back by midnight of the due date as printed on the transaction receipt. Any rental not received by midnight on the day it is due is subject to a late charge of $1.50 for each day it is late. Any rental not returned by the fifth day after the due date will be transferred to a sale. The Customer will then be required to pay the purchase price of the item in addition to five (5) days of late fees. The Customer will not be required to return the product once the total balance is paid.”
According the rental agreement, which of the following is not a responsibility of the customer?
a.
Pay a $1.50 late fee per movie per day past the due date.
b.
Get his or her rentals back by midnight the day they are due.
c.
Return a rental item kept out thirteen days past the due date.
d.
Pay the purchase price and five days of late fees for any item kept more than five days past the due date.
Answer:
C) Return a rental item kept out thirteen days past the due date.
Step-by-step explanation:
Right on edge2020
All rentals are due back by midnight of the due date as printed on the transaction receipt. Any rental not received by midnight on the day it is due is subject to a late charge of $1.50 for each day it is late. Any rental not returned by the fifth day after the due date will be transferred to a sale. The Customer will then be required to pay the purchase price of the item in addition to five (5) days of late fees. The Customer will not be required to return the product once the total balance is paid.
According the rental agreement, Returning a rental item kept out thirteen days past the due date is not a responsibility of the customer.
The responsibilities of the customerReturn the rentals by the due date in the receipt.To pay a late service charge once due by each day.Buy the item after the 5th day following the failure to return it.These are all the responsibilities expected of the consumer when they do not follow due process. Returning the rental after 13 days is not a responsibility here.
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a tax accountant requires 2 1/2 hours to prepare a personal tax return and 4 7/8 hours to prepare a small business tax return. how much longer does it take the accountant to prepare the small business tax return
Answer:
2 3/8 hours longer
Step-by-step explanation:
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Find the total area of the shape shown below in square feet
Answer:
Step-by-step explanation:
256
The required solution for the total area of the shape is 129 square feet.
What is rectangle?A rectangle is a part of a quadrilateral, whose sides are parallel to each other and equal.
In the given diagram, there are two shapes,
One is rectangle whose sides are 16 feet and 5 feet.
And other is square whose side is 7 feet.
Since, the area of rectangle = length x width = 16 x 5 = 80 square feet.
And area of square = (side)² = (7)² = 49 square feet.
Total area of shape = area of rectangle + area of square
= 80 + 49
= 129 square feet.
The total area of the shape is 129 square feet.
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Which addition does the model below represent?
3 + (negative 1)
3 + 4
3 + (negative 4)
3 + 1
Answer:
The correct answer is 3 + -4.
Step-by-step explanation:
First the arrow begins at 0 and moves to the right 3 spaces positive, ending at 3. Then it moves backwards, which is subtracting left 4 spaces towards -1 past zero. When adding a negative to a positive, you convert it to subtraction. 3 + -4 is the same as 3 - 4. We know 4 - 3 normally is 1, but the other way around, since 4 is more than 3, we go one extra space negative to -1. Therefore, 3 + -4 is the answer.
What is the image point of (3, 1) after a translation left 3 units and up 3 units?
Answer:
0,4
Step-by-step explanation:
Write an equation in point-slope form of the line through point J(4,1) with slope -4. Help
Answer:
for example...
Step-by-step explanation:
dame un ejemplo!!
Answer:
(y-1)= -4(x-4)
Step-by-step explanation:
point slope form is y - y1= m (x - x1) so just plug in the slope and the (x,y).
1. Name the property the equation illustrates.
a + C + 2b = a + 2b + c
Commutative Property of Addition
Associative Property of Addition
Distributive Property
Identity Property of Addition
Answer:
communtative propperty of addition
Step-by-step explanation:
search up what the communitve property is
I need math people, please
ok so my parents have a small bouncy house company and I work with them and I get paid 5 dollars an hour which is fine because I'm literally 13 but anyway
These past 2 weeks I've worked 6 hours and 48 minutes so since I get paid 5 dollars an hour how much money would that be?
Answer:
it's 60 dollars if I'm wrong in sorry
