The curve C has parametric equations x = t², y = (2 - t)^1/2, for 0 ≤ t ≤2.
Find d2y/dx2 in terms of t.
Can I have the full workings also please?
Answers
Answer:
y''=(4-3t)/[16 t^3 (2-t)^(3/2)]
Step-by-step explanation:
x = t², y = (2 - t)^1/2
dy/dx=dy/dt×dt/dx by chain rule
dy/dt=1/2 (2-t)^(1/2-1) × (-1)
dy/dt=-1/2 (2-t)^(-1/2)
dy/dt=-1/[2(2-t)^(1/2) ]
dx/dt=2t
dy/dx=-1/[2(2-t)^(1/2) ] × 1/[2t]
dy/dx=-1/[4t (2-t)^(1/2) ]
We need to find the second derivative now.
That is we calculate d/dt(dy/dx in terms of t) then divide by derivative of x in terms of t).
dy/dx=-1/[4t (2-t)^(1/2) ]
Let's find derivative of this with respect to t.
d/dt(dy/dx)=
[0[4t (2-t)^(1/2)]-(-1)(4(2-t)^(1/2)+-4t(1/2)(2-t)^(-1/2))]/ [4t (2-t)^(1/2) ]^2
Let's simplify
d/dt(dy/dx)=
[(4(2-t)^(1/2)+-4t(1/2)(2-t)^(-1/2))]/ [4t (2-t)^(1/2) ]^2
Continuing to simplify
Apply the power in the denominator
d/dt(dy/dx)=
[(4(2-t)^(1/2)+-4t(1/2)(2-t)^(-1/2))]/ [16t^2 (2-t) ]
Multiply by (2-t)^(1/2)/(2-t)^(1/2):
d/dt(dy/dx)=
[(4(2-t)+-4t(1/2)]/ [16t^2 (2-t)^(3/2)]
Distribute/multiply:
d/dt(dy/dx)=
[(8-4t+-2t)]/ [16t^2 (2-t)^(3/2)]
Combine like terms:
d/dt(dy/dx)=
[(8-6t)]/ [16t^2 (2-t)^(3/2)]
Reducing fraction by dividing top and bottom by 2:
d/dt(dy/dx)=
[(4-3t)]/ [8t^2 (2-t)^(3/2)]
Now finally the d^2 y/dx^2 in terms of t is
d/dt(dy/dx) ÷ dx/dt=
[(4-3t)]/ [8t^2 (2-t)^(3/2)] ÷ 2t
d/dt(dy/dx) ÷ dx/dt=
[(4-3t)]/ [8t^2 (2-t)^(3/2)] × 1/( 2t)
d/dt(dy/dx) ÷ dx/dt=
[(4-3t)]/ [16t^3 (2-t)^(3/2)]
Or!!!!!!
x = t², y = (2 - t)^1/2
Since t>0, then t=sqrt(x) or x^(1/2).
Make this substitution into the equation explicitly solved for y:
y = (2 - x^1/2)^1/2
Differentiate:
y' =(1/2) (2 - x^1/2)^(-1/2) × -1/2x^(-1/2)
y'=-1/4(2 - x^1/2)^(-1/2)x^(-1/2)
y'=-1/4(2x-x^3/2)^(-1/2)
Differentiate:
y''=1/8(2x-x^3/2)^(-3/2)×(2-3/2x^1/2)
y''=(2-3/2x^1/2)/[8 (2x-x^3/2)^(3/2)]
Replace x with t^2
y''=(2-3/2t)/[8 (2t^2-t^3)^(3/2)]
Multiply top and bottom by 2
y''=(4-3t)/[16 (2t^2-t^3)^(3/2)]
Factor out t^2 inside the 3/2 power factor:
y''=(4-3t)/[16 (t^2)^(3/2) (2-t)^(3/2)]
y''=(4-3t)/[16 t^3 (2-t)^(3/2)]
Related Questions
What is the length of the line?
Answers
Answer:
5.4
Step-by-step explanation:
You need to use the Pythagorean theorem and if you know that the base is 2 and the height 5 then you got 5.4
HELPPPPPP WIL GIVE BRANELISTTTT
Answers
Answer: i think the answer is B
Step-by-step explanation:
Please help me with this
Answers
Step-by-step explanation:
the only reason you would need help with this is, if you don't understand what exponents are and how they work.
because once you understand that, you can do the whole question in 30 seconds yourself. way faster than the posting here and then copying answers back into your system.
so, let me explain this important element :
when you have something like 5³, then this 3 is the exponent, and the 5 is the base.
when we talk about such a thing we typically say "5 to the power of 3".
now what did it mean ?
well, it is a multiplication. a multiplication of the base by itself (like when we square a number).
and the exponent tells is how often the base is multiplied by itself.
so, back to my example :
5³ = 5×5×5
3² = 3×3
13⁴ = 13×13×13×13
once you understand that, you can solve the question yourself in seconds.
e.g. the first is looking for the equivalent of 7⁴.
we need to find the answer option that dissolves this like the examples above to 7×7×7×7.
and we find B to be the right answer.
for 3⁶ = 3×3×3×3×3×3
and E is right. and so on.
what is x and what is y?
Answers
Step-by-step explanation:
i hope the illustrations are clear
CAN SOMEONE HELP ME PLEASE ASAP!?
Answers
Answer:
21.6 ft
Step-by-step explanation:
Similar Triangles
6/10=B/36
B=21.6
Can someone please give me the (Answers) to this? ... please ...
Answers
Answer:
A.
1 ) 1/2
2 ) - 1
3 ) 1/2
4 ) 5/4
B.
5 ) not defined
6 ) 0
7 ) 15/14
8 ) - 2/5
C )
9. - 1
10. - 4/3
11. 2
12. 14/3
Answer:
A.
1) 1/2
2) -1
3) 1/2
4) 5/4
B.
5) not defined
6) 0
7) 15/14
8) - 2/5
C.
9) -1
10) -4/3
11) 2
12) 14/3
Help! 100 points to correct!
Answers
Answer:
B is correct
Step-by-step explanation:
adding the amount taken away and then subtracting it is the same as subtracting twice
Answer:
either b or c
Step-by-step explanation:
Put the following equation of a line into slope-intercept form, simplifying all
fractions.
2y - 2 = -14
Answers
Step by step explanation:
2y-2=-14
2y=-12 (move the negative 2 to the right)
y= -6 (divide both sides by 2)
Your equation is then y=-6
Victor has put 1000 hours into playing an online video game he has paid 671.34 over the course of the entire game how much did much did he pay per hour played?
Answers
Answer:
Give the person above me brainliest plz
Step-by-step explanation:
connor took out a 4-year loan at 4% If he has to pay $240 how much principle did he borrow
Answers
Answer:
The principal he borrowed is $1500
Step-by-step explanation:
Let's assume principal he borrowed as P
now, we can use formula
where
SI is simple interest
r is interest rate
t is time in years
we are given
r=4%
SI=240
t=4
now, we can plug this value
now, we can solve for P
So,
The principal he borrowed is $1500. Hope this helps, and mark me as brainlest!
mr. jessup and mr allen i ready mathematics pages 77 and 78
Answers
Suppose that $2000 is loaned at a rate of 12%, compounded semiannually. Assuming that no payments are made, find the amount owed after 3 years
Do not round any intermediate computations, and round your answer to the nearest cent.
Answers
The total amount owed after three years is $2,837.04.
The formula that can be used to determine the total amount owed after 3 years is:
FV = A(1 + r/m)^nm
FV = Future value of debt A = amount borrowed R = interest rate = 12/2 = 6% m = number of compounding = 2 N = number of years = 3$2000 x (1.06)^6 = $2,837.04
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Which set of ordered pairs represents a function?
{(2, –2), (1, 5), (–2, 2), (1, –3), (8, –1)}
{(3, –1), (7, 1), (–6, –1), (9, 1), (2, –1)}
{(6, 8), (5, 2), (–2, –5), (1, –3), (–2, 9)}
{(–3, 1), (6, 3), (–3, 2), (–3, –3), (1, –1)}
Answers
Answer: {(2,-2), (1,5), (-2,2), (1,-3), (8,-1)}, and {(3,-1), (7,1), (-6,-1), (9,1), (2,-1)
Step-by-step explanation: In order for a given relation to be a function, there can only be 1 x-value that can repeat. The y-values on the other hand can repeat an infinite amount of times. In the last 2 relations, you have (-2,-5), (-2,9) and (-3,1), (-3,2), and (-3,-3). You can also graph these relations and do a vertical line test with your pencil drawing a straight line downward and you'll notice that you will touch the graph more than 1 time. The first 2 relations if you were to graph them and do a vertical line test by using a pencil or pen, you'll notice that you will touch the graph once which shows you that you have a relation that is indeed a function.
Hope this explanation helps.
Answer:
{(3, –1), (7, 1), (–6, –1), (9, 1), (2, –1)}
1-w+w2 what is complex cube
Answers
[tex] \: \: \: \: \: \: [/tex]
[tex]1 + w[/tex]
Step-by-step explanation:
[tex]1 - w + w2[/tex]
use the commutative property to reorder the terms[tex]1 - w + 2w[/tex]
collect like terms[tex] = \: \: 1 + w[/tex]
hope it helpsThe line 2x + 3y = 8 meets the curve 2x^2 + 3y^2 =110 at the points A and B. Find the coordinates of A and B.
extra: This is the first time I’ve seen this type of question without a diagram. Pls help!
Answers
Answer:The cordinates of A and B are (-3.8, 5.2) and (7, -2).
Step-by-step explanation:From (1), we get x = (8 - 3y) / 2 ––(3)
Substitute (3) into (2),
2 [(8 - 3y) / 2]² + 3y² = 110
(8 - 3y)² + 3y² = 220
15y² - 48y - 156 = 0
Therefore y = 5.2, then x = (8 - 3×5.2) / 2 = -3.8
OR y = -2, then x = [8 - 3×(-2)] / 2 = 7
Answer:
The coordinates are A. (-3.8, 5.2) and B. (7, -2).
Step-by-step explanation:
At A and B the coordinates satisfy both equations so we solve simultaneously using substitution:
2x^2 + 3y^2 = 110
2x + 3y = 8
From the above equation
x = (8 - 3y)/2
So substituting for x in the first equation:
2 [(8-3y)/2]^2 + 3y^2 = 110
(8-3y)^2 / 2 + 3y^2 = 110
Multiply through by 2:
(8 - 3y)^2 + 6y^2 = 220
64 + 9y^2 - 48y + 6y^2 - 220 = 0
15y^2 - 48y - 156 = 0
3(5y^2 - 16y - 52) = 0
(5y - 26)(y + 2) = 0
y = 26/5, 2
y = 5.2, -2.
So substituting in 2x + 3y = 8
When y = -2:
2x - 6 = 8
2x = 14
x = 7.
When y = 5.2
2x + 3(5.2) = 8
2x = 8 - 15.6 = -7.6
x = -3.8.
An appropriate tip for a waiter at a restaurant is at least 15%. For a tab of $20, how much money should be left, including tip?
Answers
$23, a $20 bill plus $3 tip
explain why 256 + 4 is equivalent to (200 + 4) + (40+4) + (164)
Answers
The expression 256 + 4 is equivalent to (200 + 4) + (40+4) + (16+4) according to distributive and commutative property of addition.
Given the expression 256 + 4. This can be solved using the partial sum expressed as:
256 + 4
256 + 4
= (200 + 40 + 16) + 4
According to the commutative property, A+B = B+A
The arrangement does not affect the result. Hence;
(200 + 40 + 16) + 4 = 4 + (200 + 40 + 16)Using the distributive law;
4 +(200 + 40 + 16) = (200 + 4) + (40 + 4) + (16 + 4)
Hence the expression 256 + 4 is equivalent to (200 + 4) + (40+4) + (16+4) according to distributive and commutative property of addition.
Learn more on partial sum here: https://brainly.com/question/6958503
Solve for x
log2(5x-4)=4
Please show steps
Answers
Answer:
x=1.2
Step-by-step explanation:
2(5x-4)=4
distribute 2 to 5x and -4
10x-8=4
add 8 to negative 8 and 4
10x=12
now divide both sides by 10
10x/10= 12/10
x=1.2
The value of x for the logarithmic expression will be x=1.2.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given expression is log2(5x-4)=4. The value of x will be calculated as,
2(5x-4)=4
Distribute 2 to 5x and -4,
10x-8=4
Add 8 to negative 8 and 4,
10x=12
Now divide both sides by 10,
10x/10= 12/10
x=1.2
Therefore, the value of x for the logarithmic expression will be x=1.2.
To know more about an expression follow
https://brainly.com/question/28041634
#SPJ2
PLEASEE HELP !!! 30 POINTSSS
What are the exact measures of the other two sides of the triangle? Use special right triangles ratios and
show your work
Answers
Answer:
30-60-90 is a special right triangle
if the side opposite to the 30 degree angle = x,
then the side opposite to the 90 degree angle = 2x
and the side opposite to the 60 degree angle = x*sqr3
2x=14
x=7
side BC is opposite to the 30 degree angle, so it is 7 cm
side AB is opposite to the 60 degree angle, so it is 7sqr3 cm
Answer:
clearer pic?
Step-by-step explanation:
Evaluate: 4.3 - (-2.7)
7
1.6
-1.6
-7
Answers
Answer:
7
Step-by-step explanation:
That means 4.3 (2.7) = 7
The negative signs get removed because negative × negative = positive
64PT AND BRAINLIEST! Which of the following would produce a graph with no solution?
| x | = − 5
| x | < 5
| x | < 5
| x | = 0
Answers
Answer:
| x | = − 5
Step-by-step explanation:
Let's solve your equation step-by-step.
| x | = − 5
Solve Absolute Value.
| x | = − 5
No solutions. (Absolute value cannot be less than 0.)
Answer:No solutions.
I hope this helps :)
Answer:
| x | = -5
Step-by-step explanation:
Let's think about absolute value, the number we stick in side of these magical type "brackets" (so to speak) is always going to end up being positive because absolute value is the distance of a number from 0. Therefore, we know that because "x" cannot be negative in this scenario, the absolute value of x cannot be equal to -5 and therefore would produce a graph with no solution!
Determine whether each quadrilateral is a parallelogram. Justify your answer.
Answers
Answer:
4. Parallelogram
5. Parallelogram
6. Parallelogram
Step-by-step explanation:
4. This is a parallelogram because the longer sides are parallel to each other and the shorter sides are also parallel to each other. Hence, this is a parallelogram.5. This is a parallelogram because it's angles are the same to the opposite angles. Hence, this is a parallelogram.6. This is a parallelogram because the longer sides are parallel to each other and the shorter sides are also parallel to each other. Hence, this is a parallelogram.Hoped this helped.
Name three equal ratios from the graph
Answers
250:10
125:5
100:4
Step-by-step explanation:
ratio of y to x
Why is it a shame that parallel lines have so much in common riddle answer
Answers
Answer:
Bc they will never meet (intersect, touch, etc)
Step-by-step explanation:
Parallel do not intersect.
Mark me the Brainliest! Ty!
what is 3 3/5 - 6 1/7
Answers
3 3/5 - 6 1/7
18/5 - 43/7
= -89/35
= -2 19/35
Helpppppppppp plssssssssss
Answers
Answer:
12 in.
Step-by-step explanation:
We already know that the bottom line all together is 19 inches, and the line for the top of the square box at the end is also 7in. We can then infer that the top line of 7 cm is the same as the bottom line, so we therefore do 19 - 7 which equals to 12. Hope this helps.
a
Fitness Club A has a membership fee of $35 plus $3 each time you swim. Fitness Club B has a membership fee of
$27.50 plus $4.50 each time you swim. After how many swims will the cost be the same?
A
4
B) 7
6
D
8
E 5
Answers
Write an equation for each one set the equal to each other and solve.
A: 35 + 3x
B: 27.50 + 4.50x
35 + 3x = 27.50 + 4.50x
Subtract 27.50 from both sides:
7.50 + 3x = 4.50x
Subtract 3x from both sides
7.50 = 1.50x
Divide both sides by 1.50
X = 5
Answer: 5
Given the function below find f(0).
f(x)=x2−3x+5
Group of answer choices
-5
0
5
10
Answers
Answer:
5
Step-by-step explanation:
f(x) = x² - 3x + 5
f(0)
putting x as 0 in the equation
f(0) = 0² - 3(0) + 5
= 0 - 0 + 5
= 5
Which account option is more convenient for frequent check or debit card withdrawals and purchases but gains no interest?
A. Checking
B. Certificate of deposit
C. Savings
D. Money market
Answers
This is because it’s not a savings so it just goes in and out
how do you find the ratio of something
Answers
Answer:
To calculate the ratio of an amount we divide the amount by the total number of parts in the ratio and then multiply this answer by the original ratio. We want to work out $20 shared in the ratio of 1:3. Step 1 is to work out the total number of parts in the ratio. 1 + 3 = 4, so the ratio 1:3 contains 4 parts in total.
Step-by-step explanation:
Step-by-step explanation:
hope it will help you a lot.......
