If x2 + xy + y3 = 1, find the value of y''' at the point where x = 1.
The third derivative of the function is [tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{\left(3y^2+x\right)^2}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{\left(3y^2+x\right)^3}[/tex]and the value at the point x = 1 is 42
How to determine the third derivative at the point x = 1
From the question, we have the following parameters that can be used in our computation:
[tex]x^2 + xy + y^3 = 1[/tex]
Differentiate implicitly
So, we have
[tex]3y^2y' + xy'+y+2x=0[/tex]
Make y' the subject of formula
So, we get
[tex]y'=-\dfrac{y+2x}{3y^2+x}[/tex]
Differentiate the second time
Using a graphing tool, we have
[tex]y''=\dfrac{\left(3y^2+12xy-x\right)y'-6y^2+y}{\left(3y^2+x\right)^2}[/tex]
Differentiate the third time to get the third derivative
Using a graphing tool, we have
[tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{\left(3y^2+x\right)^2}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{\left(3y^2+x\right)^3}[/tex]
Recall that
x = 1
Calculating y, we have
[tex]1^2 + (1)y + y^3 = 1[/tex]
[tex]1 + y + y^3 = 1[/tex]
[tex]y^3 + y = 0[/tex]
Factorize
[tex]y(y^2 + 1) = 0[/tex]
So, we have
y = 0 or [tex]y^2 + 1 = 0[/tex]
The equation [tex]y^2 + 1 = 0[/tex] will give a complex solution
So, we have
x = 1 and y = 0
Calculating y', we have
[tex]y'=-\dfrac{0+2(1)}{3 * 0^2+1}[/tex]
[tex]y'=-\dfrac{2}{1}[/tex]
y' = -2
Calculating y", we have
[tex]y''=\dfrac{\left(3y^2+12y-1\right)y'-6y^2+y}{\left(3y^2+1\right)^2}[/tex]
[tex]y''=\dfrac{\left(3(0)^2+12(1)(0)-1\right)(-2)-6(0)^2+0}{\left(3(0)^2+1\right)^2}[/tex]
[tex]y''=\dfrac{\left2}{1}[/tex]
y" = 2
Calculating y", we have
[tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{\left(3y^2+x\right)^2}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{\left(3y^2+x\right)^3}[/tex]
Simplifying the denominators, we have
[tex](3y^2 + x)^2 = (3(0)^2 + 1)^2 = 1[/tex]
[tex](3y^2 + x)^3 = (3(0)^2 + 1)^3 = 1[/tex]
So, we have
[tex]y^{'''}=\dfrac{\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy'}{1}-\dfrac{2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)}{1}[/tex]
Divide
[tex]y^{'''}=[\left(3y^2+12xy-x\right)y''+y'\left(6yy'+12xy'+12y-1\right)-\left(6y-1\right)y'-6yy']-[2\left(6yy'+1\right)\left(\left(3y^2+12xy-x\right)y'-y\cdot\left(6y-1\right)\right)][/tex]
Simplifying each term:
[tex](3y^2+12xy-x)y''+y'(6yy'+12xy'+12y-1)-(6y-1)y'-6yy' = (3(0)^2+12(1)(0)-(1))(2) + (-2)(6(0)(-2) +12(1)(-2) + 12(0) - 1) - (6(0) - 1)(-2) - 6(0)(-2)[/tex]
[tex](3y^2+12xy-x)y''+y'(6yy'+12xy'+12y-1)-(6y-1)y'-6yy' = 46[/tex]
Also, we have
[tex]2(6yy'+1)((3y^2+12xy-x)y'-y(6y-1)) = 2(6(0)(-2) + 1)((3(0)^2 + 12(1)(0) - 1)(-2) - 0(6(0)-1))[/tex]
[tex]2(6yy'+1)((3y^2+12xy-x)y'-y(6y-1)) = 4[/tex]
So, the expression becomes (by substitution)
[tex]y^{'''}= 46 -4[/tex]
This gives
[tex]y^{'''}= 42[/tex]
Hence, the third derivative at the point x = 1 is 42
-2(4g-3)= 30 how do i solve this
what is 1,627,187 rounded to the nearest ten?
If y=m^4=n^3 and y is greater than 1, then mn=
If we equate the two given equations y = m^4 and y = n^3, we can express mn as m^(7/3) or as n^(7/4). This is done by taking the cube root and square root on both sides of the equated equation to isolate m and n.
Explanation:If we have two equations, y = m^4 and y = n^3, where y > 1, we can equate these two equations since they are both equal to same value 'y'. So the equation will be m^4 = n^3. To find the value of mn, we want to express it in terms of 'y'. So we take cube root on both sides, it will be m = (n^3)^(1/4) = n^(3/4). And square root on both sides will give n = (m^4)^(1/3) = m^(4/3). Now mn = m * m^(4/3) = m^(1 + 4/3) = m^(7/3). Similarly, mn = n * n^(3/4) = n^(1 + 3/4) = n^(7/4). Hence, mn can be expressed either as m^(7/3) or n^(7/4).
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factors of 3x^2y^2+6x^2+12y^2+24
the base of an exponential function cannot be a negative number true or false
Answer:
True
Step-by-step explanation:
Given statement : The base of an exponential function cannot be a negative number.
We need to check whether the given statement is true or not.
The general form of an exponential function is
[tex]f(x)=ab^x[/tex]
where, a is the initial value and b is the base of exponential function.
The value of base b is always greater than 0 because
1. The terms like [tex](-3)^\frac{1}{2}[/tex] is an imaginary number and it make no sense. So, the base of an exponential function cannot be a negative number.
2. It b=0, then [tex](0)^x=0[/tex], which is a constant function. So, the base of an exponential function cannot be 0.
It means b>0.
Therefore, the given the given statement is true.
Final answer:
The base of an exponential function must be positive, as a negative base can lead to undefined values when raised to non-integer exponents. The rate of growth of an exponential function with a positive base near one can be approximated by 1+x for small x. Bases for exponential functions can often be connected to Euler's number e (approximately 2.71818).
Explanation:
The statement that the base of an exponential function cannot be a negative number is true. The base of an exponential function needs to be positive because a negative base can lead to undefined or complex values when raised to a fractional or irrational exponent. For example, the exponential function with base e (where e is the Euler's number, approximately equal to 2.71818) demonstrates natural growth, and its rate of growth for small x values approximates to 1+x. This approximation indicates how for a small change in x, the change in the value of the function is nearly proportional when the base is a positive number close to one.
Considering other bases, they can be related to base e using the identity ax = eln(a)x, where a is a positive real number, and ln(a) is the natural logarithm of a. Thus, whether for common or natural bases, the exponential function is well-defined only for positive bases.
are all semi circles simular
A rectangle is placed around a semicircle as shown below. the width of the rectangle is 6ft . find the area of the shaded region. use the value 3.14 for π , and do not round your answer. be sure to include the correct unit in your answer.
The shaded region is by assumption the region which is not covered by the semicircle in in given rectangles.
The area of the shaded region is given by 15.48 sq. ft.
What is a semicircle?
A semicircle is a circle cut in half. Thus, one circle produces two semicircle.
Firstly we will find the area of the rectangle and then subtract the area of the semicircle to find the are of the shaded region.
Since the radius of the semicircle is equal to width of the rectangle(6 ft), thus the length of the diameter of the circle( twice the radius which is 12 ft) serves as length of the considered rectangle.
Thus, we have:
[tex]\text{Area of the given rectangle\:} = 6 \times 12 = 72 \: \rm ft^2[/tex]
Since the semicircle is having radius of 6 ft, thus:
[tex]\text{Area of semicircle} = \dfrac{\pi r^2}{2} = \dfrac{3.14 \times 6^2}{2} = 56.52 \: \rm ft^2[/tex]
Thus, area of the shaded region will be equal to area of rectangle - area of semicircle = 72 - 56.52 = 15.48 sq. ft.
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What is 10(x+7) if x=6 ?
Nancy is a telemarketer who calls 96 people each day. How many people will she call in a 5-day workweek?
A. 430
B. 460
C. 480
D. 380
2x − y = 3 4x = 6 + 2y
How many real solutions does the equation: x2-7x+10=0 have?
Solve for F. Let m=5, a=3 F=ma
Simplify 6 to the fifth power over 7 cubed all raised to the second power. 6 to the seventh power over 7 to the tenth power 6 to the tenth power over 7 to the sixth power 6 cubed over 7 12 to the fifth power over 14 cubed
Answer:
6 to the tenth power over 7 to the sixth power
Step-by-step explanation:
Given phrase,
6 to the fifth power over 7 cubed all raised to the second power,
[tex]\implies (\frac{6^5}{7^3})^2[/tex]
By using [tex](a^m)^n=a^{mn}[/tex]
[tex]=\frac{6^{5\times 2}}{7^{3\times 2}}[/tex]
[tex]=\frac{6^{10}}{7^6}[/tex]
= 6 to the tenth power over 7 to the sixth power
Simplify the expressions
(6⁵/7³)² = 2143588816/117649
(6⁷/7¹⁰) = 279936/282475249
(6¹⁰/7⁶) = 60466176/117649
(6³/7) = 216/7
(12⁵/14³) = 90855/1001
To simplify the given expressions, we can calculate the numerical values and perform the necessary operations. Let's evaluate each expression:
(6⁵/7³)²:
First, calculate the numerator and denominator:
Numerator: 6⁵ = 6 × 6 × 6 × 6 × 6 = 7776
Denominator: 7³ = 7 × 7 × 7 = 343
Now, substitute the values into the expression and square the result:
(7776/343)² = (7776/343) × (7776/343) = 2143588816/117649
The simplified form is 2143588816/117649.
(6⁷/7¹⁰):
Calculate the numerator and denominator:
Numerator: 6⁷ = 6 × 6 × 6 × 6 × 6 × 6 × 6 = 279936
Denominator: 7¹⁰ = 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 = 282475249
Substitute the values into the expression:
279936/282475249
This expression cannot be simplified further.
(6¹⁰/7⁶):
Calculate the numerator and denominator:
Numerator: 6¹⁰ = 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 = 60466176
Denominator: 7⁶ = 7 × 7 × 7 × 7 × 7 × 7 = 117649
Substitute the values into the expression:
60466176/117649
This expression cannot be simplified further.
(6³/7):
Calculate the numerator and denominator:
Numerator: 6³ = 6 × 6 × 6 = 216
Denominator: 7
Substitute the values into the expression:
216/7
This expression cannot be simplified further.
(12⁵/14³):
Calculate the numerator and denominator:
Numerator: 12⁵ = 12 × 12 × 12 × 12 × 12 = 248832
Denominator: 14³ = 14 × 14 × 14 = 2744
Substitute the values into the expression:
248832/2744 = 90855/1001
The simplified form is 90855/1001.
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Simplify 6 to the fifth power over 7 cubed all raised to the second power. 6 to the seventh power over 7 to the tenth power 6 to the tenth power over 7 to the sixth power 6 cubed over 7 12 to the fifth power over 14 cubed
(6⁵/7³)²
(6⁷/7¹⁰)
(6¹⁰/7⁶)
(6³/7)
(12⁵/14³)
What is the length of the hypotenuse, x, if (12, 35, x) is a Pythagorean triple?
Answer:37
Step-by-step explanation:
12•12=144
35•35=1225
1225+144=1369
Square root 1369=37
Does arkansas lie south of 40 degrees latitude
Find the derivative of f(x) = 8 divided by x at x = -1.
A. 4
B. 0
C. 8
D. -8
The derivative of f(x) at x = -1 is -8.
What is a derivative?Derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations.
According to the given problem,
f(x) = [tex]\frac{8}{x}[/tex]
Derivative of f(x),
⇒ f'(x) = [tex]-\frac{8}{x^{2} }[/tex]
At x = -1,
⇒ f'(-1) = [tex]-\frac{8}{(-1)^{2} }[/tex]
= -8
Hence, we can conclude, the derivative of f(x) at x = -1 is -8.
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How much would $125 invested at 8% interest compounded continuously be worth after 16 years? Round your answer to the nearest cent. Do not include units in your answer.
Charlie wants to order lunch for his friends. He'll order 6 sandwiches and a $3 kid's meal for his little brother. Charlie has $27. How much can he spend on each sandwich if they are all the same price? Choose two answers: one for the inequality that models this situation and one for the correct answer. A. Inequality: 3x + 6 < 27 B. Inequality: 6x + 3 ≤ 27 C. Inequality: 6x + 3 ≥ 27 D. Inequality: 3x + 6 ≥ 27 E. Answer: $7 or less F. Answer: $4 or less
Answer-
The inequality that models this situation is,
[tex]\boxed{\boxed{B.\ 6x+3\leq 27}}[/tex]
Solution-
The total money Charlie has = $27
He wants to order 6 sandwiches and a kid's meal of $3
Let us assume the price of each sandwiches is x.
So, the amount he will be spending for sandwiches is $6x
The total amount he will be spending is $(6x+3)
As he has only $27, so he can spend any amount less than or equal to 27
So the inequality becomes,
[tex]\Rightarrow 6x+3\leq 27[/tex]
Therefore, the inequality that models this situation is [tex]6x+3\leq 27[/tex]
A car travels at an average speed of 52 miles per hour. how many miles does it travel in 3 hours and 30 minutes?
How do units help you to better understand a problem? How do units sometimes get in the way of solving a problem?
Which of the following is the number 0.008005 written in scientific notation?
A. 8.005 x 10(squared)
B. 8.500 x 10(to the negative third power)
C. 8.005 x 10(to the negative fourth power)
D. 8.005 x 10(to the negative third power
If you multiply me by 3 and increase that value by 3 I am 20. What's my number?
Which of the following are vertical asymptotes of the function y = 2cot(3x) + 4? Check all that apply. A.x = pi/3 B.x = +/- pi/2 C.x = 2pi D.x = 0
The vertical asymptotes of the function y = 2cot(3x) + 4 are A.x = π/3 C. x = 2π D.x = 0
How to determine the vertical asymptote?The function is given as:
y = 2cot(3x) + 4
The above function is a cotangent function, represented as:
y = Acot(Bx +C) + D
By comparison, we have:
B = 3
The vertical asymptotes are then calculated using:
[tex]x = \frac{\pi}{B}n[/tex], where n are integers
Substitute 3 for B
[tex]x = \frac{\pi}{3}n[/tex]
Using the above format, the vertical asymptotes in the options are A.x = π/3 C. x = 2π D.x = 0
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A local supermarket sells chicken fir $2.49/lb and pork for $3.19/lb.Todd buys "c" pounds of chicken and "p" pounds ofpork which of the following inequalities represents that Todd only has $40 to spend?
Give an equation for a horizontal line and a vertical line. what are the slopes of each line
Answer:
1 + 1 = 2
2 - 1 = 1
The line's slopes are cute
Step-by-step explanation:
I hope your lines enjoy their equations, you are a great parent to your lines. =D
Find the three arithmetic means between 7 and 21
solve the following equation -2x + 4 = 2 (4x - 3) -3 (-8 + 4x)
a.7
b.2
c.-7
d.3
Answer:
[tex]A) 7[/tex]
Step-by-step explanation:
[tex]-2x+4=2\left(4x-3\right)-3\left(-8+4x\right)[/tex]
[tex]2\left(4x-3\right)-3\left(-8+4x\right)[/tex]
[tex]2 (4x - 3)[/tex]
[tex]\longrightarrow8x-6[/tex]
[tex]\longrightarrow 8x-6-3\left(-8+4x\right)[/tex]
[tex]-3 (-8 + 4x)[/tex]
[tex]\longrightarrow 24-12x[/tex]
[tex]8x-6+24-12x[/tex]
Combine like terms:
[tex]8x-12x-6+24[/tex]
Add: [tex]8x-12x=-4x[/tex]
Add: [tex]-6+24=18[/tex]
[tex]\longrightarrow-2x+4=-4x+18[/tex]
Subtract 4 from both sides:
[tex]\longrightarrow-2x+4-4=-4x+18-4[/tex]
[tex]\longrightarrow-2x=-4x+14[/tex]
Add 4x to both sides:
[tex]\longrightarrow-2x+4x=-4x+14+4x[/tex]
[tex]\longrightarrow2x=14[/tex]
Divide both sides by 2:
[tex]\longrightarrow\frac{2x}{2}=\frac{14}{2}[/tex]
[tex]\longrightarrow x=7[/tex]
____________________________
OAmalOHopeO
Oil is poured on a flat surface and it spreads out forming a circle. the area of this circle is increasing at a constant rate of 10cm2/s. determine the rate of change of the radius of the circle when the radius is 10cm
The rate of change of radius of the circle when the radius is 10 cm is
dr / dt = 1/2π cm / sec
What is implicit differentiation?
In mathematics, differentiation is the process of determining the derivative, or rate of change, of a function. Differentiation is a technique for determining a function's derivative. Differentiation is a mathematical procedure that determines the instantaneous rate of change of a function based on one of its variables. The process of finding the derivative of dependent variable in an implicit function by differentiating each term separately
Given data ,
Oil is poured on a flat surface and it spreads out forming a circle
The area of the circle is increasing at a rate of 10 cm²/sec
Let the area of the circle be = A
So , the equation is dA² / dt² = 10 cm²/sec
The area of the circle = πr²
So , dA² / dt² = 2π dr / dt
2πr dr / dt = 10 cm²/sec
Substituting the value of r in the equation , we get
2π x 10 dr / dt = 10 cm²/sec
20π dr / dt = 10 cm²/sec
Divide both sides of the equation be 20π
dr / dt = ( 1/2π ) cm / sec
Therefore , the rate of change of radius of the circle is ( 1/2π ) cm / sec
Hence , The rate of change of radius of the circle when the radius is 10 cm is dr / dt = 1/2π cm / sec
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A rectangular yard measuring 41ft by 60ft is bordered (and surrounded) by a fence. Inside, a walk that is 4ft wide goes all the way along the fence. Find the area of this walk. Be sure to include the correct unit in your answer.
Anyone wanna help? I'd really appreciate it.
area of yard = 41 x 60 = 2460 square feet
walk is 4 feet wide
41-4 =37, 60-4 = 56
37*56 = 2072
2460-2072 = 388
area of walk is 388 square feet