Answers
First let's compute dx/dt
[tex]x = t - \frac{1}{t}\\\\x = t - t^{-1}\\\\\frac{dx}{dt} = \frac{d}{dt}\left(t - t^{-1}\right)\\\\\frac{dx}{dt} = 1-(-1)t^{-2}\\\\\frac{dx}{dt} = 1+\frac{1}{t^{2}}\\\\\frac{dx}{dt} = \frac{t^2}{t^{2}}+\frac{1}{t^{2}}\\\\\frac{dx}{dt} = \frac{t^2+1}{t^{2}}\\\\[/tex]
Now compute dy/dt
[tex]y = 2t + \frac{1}{t}\\\\y = 2t + t^{-1}\\\\\frac{dy}{dt} = \frac{d}{dt}\left(2t + t^{-1}\right)\\\\\frac{dy}{dt} = 2 - t^{-2}\\\\\frac{dy}{dt} = 2 - \frac{1}{t^2}\\\\\frac{dy}{dt} = \frac{2t^2}{t^2}-\frac{1}{t^2}\\\\\frac{dy}{dt} = \frac{2t^2-1}{t^2}\\\\[/tex]
From here, apply the chain rule to say
[tex]\frac{dy}{dx} = \frac{dy*dt}{dx*dt}\\\\\frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx}\\\\\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\\\\\frac{dy}{dx} = \frac{2t^2-1}{t^2} \div \frac{t^2+1}{t^{2}}\\\\\frac{dy}{dx} = \frac{2t^2-1}{t^2} \times \frac{t^{2}}{t^2+1}\\\\\frac{dy}{dx} = \frac{2t^2-1}{t^2+1}\\\\[/tex]
We could use polynomial long division, or we could add 2 and subtract 2 from the numerator and do a bit of algebra like so
[tex]\frac{dy}{dx} = \frac{2t^2-1}{t^2+1}\\\\\frac{dy}{dx} = \frac{2t^2-1+2-2}{t^2+1}\\\\\frac{dy}{dx} = \frac{(2t^2+2)-1-2}{t^2+1}\\\\\frac{dy}{dx} = \frac{2(t^2+1)-3}{t^2+1}\\\\\frac{dy}{dx} = \frac{2(t^2+1)}{t^2+1}-\frac{3}{t^2+1}\\\\\frac{dy}{dx} = 2-\frac{3}{t^2+1}\\\\[/tex]
This concludes the first part of 4b
=======================================================
Now onto the second part.
Since t is nonzero, this means either t > 0 or t < 0.
If t > 0, then,
[tex]t > 0\\\\t^2 > 0\\\\t^2+1 > 1\\\\\frac{1}{t^2+1} < 1 \ \text{ ... inequality sign flip}\\\\\frac{3}{t^2+1} < 3\\\\-\frac{3}{t^2+1} > -3 \ \text{ ... inequality sign flip}\\\\-\frac{3}{t^2+1}+2 > -3 + 2\\\\2-\frac{3}{t^2+1} > -1\\\\-1 < 2-\frac{3}{t^2+1}\\\\-1 < \frac{dy}{dx}\\\\[/tex]
note the inequality signs flipping when we apply the reciprocal to both sides, and when we multiply both sides by a negative value.
You should find that the same conclusion happens when we consider t < 0. Why? Because t < 0 becomes t^2 > 0 after we square both sides. The steps are the same as shown above.
So both t > 0 and t < 0 lead to [tex]-1 < \frac{dy}{dx}[/tex]
We can say that -1 is the lower bound of dy/dx. It never reaches -1 itself because t = 0 is not allowed.
We could say that
[tex]\displaystyle \lim_{t\to0}\left(2-\frac{3}{t^2+1}\right)=-1\\\\[/tex]
---------------------------------------
To establish the upper bound, we consider what happens when t approaches either infinity.
If t approaches positive infinity, then,
[tex]\displaystyle L = \lim_{t\to\infty}\left(2-\frac{3}{t^2+1}\right)\\\\\\\displaystyle L = \lim_{t\to\infty}\left(\frac{2t^2-1}{t^2+1}\right)\\\\\\\displaystyle L = \lim_{t\to\infty}\left(\frac{2-\frac{1}{t^2}}{1+\frac{1}{t^2}}\right)\\\\\\\displaystyle L = \frac{2-0}{1+0}\\\\\\\displaystyle L = 2\\\\[/tex]
As t approaches infinity, the dy/dx value approaches L = 2 from below.
The same applies when t approaches negative infinity.
So we see that [tex]\frac{dy}{dx} < 2[/tex]
---------------------------------------
Since [tex]-1 < \frac{dy}{dx} \text{ and } \frac{dy}{dx} < 2[/tex], those two inequalities combine into the compound inequality [tex]-1 < \frac{dy}{dx} < 2[/tex]
So dy/dx is bounded between -1 and 2, exclusive of either endpoint.
Related Questions
Find the missing side lengths. Leave your answers as radicals in simplest form.
Answers
Answer:
Step-by-step explanation:
For the question 1:
The given is a special right triangle with angle measures of
90-60-30 and side lengths represented by :
a - a[tex]\sqrt{3}[/tex] and 2a
The side length that sees 90 degrees is represented with a
The side length that sees 60 degrees is represented with a[tex]\sqrt{3}[/tex]
The side length that sees 30 degrees is represented with 2a
Here the side length that sees angle measure 60 is given as [tex]\sqrt{6}[/tex]
so a[tex]\sqrt{3}[/tex] = [tex]\sqrt{6}[/tex] to find the value of a we divide [tex]\sqrt{6}[/tex] with [tex]\sqrt{3}[/tex]
[tex]\frac{\sqrt{6} }{\sqrt{3} }[/tex] = [tex]\sqrt{2}[/tex]
so y = [tex]\sqrt{2}[/tex] and x = 2[tex]\sqrt{2}[/tex]
for second question
the square value of hypotenuse is equal to sum of other two side length's square value
10^2 + 6^2 = x^2
100 + 36 = x^2
136 = x^2
[tex]\sqrt{136}[/tex] = x
Ten samples were taken from a plating bath used in an electronics manufacturing process, and the bath pH was determined. The sample pH value are 7.91, 7.85, 6.82, 8.01, 7.46, 6.95, 7.05, 7.35, 7.25, 7.42. Manufacturing engineering believes that pH has a median value of 7.0. Do the sample data indicate that this statement is correct
Answers
Answer:
There is no sufficient evidence to support the claim.
Step-by-step explanation:
Given the data:
7.91, 7.85, 6.82, 8.01, 7.46, 6.95, 7.05, 7.35, 7.25, 7.42
Sample size, n = 10
The sample mean, xbar = ΣX/ n = 74.07 / 10 = 7.407
The sample standard deviation, s = 0.41158 ( from calculator)
The hypothesis :
H0 : μ = 7
H0 : μ ≠ 7
The test statistic :
(xbar - μ) ÷ (s/√(n))
(7.047 - 7) ÷ (0.41158/√(10))
0.047 / 0.1301530
Test statistic = 0.361
Testing the hypothesis at α = 0.05
The Pvalue ;
df = n - 1 ; 10 - 1 = 9
Two tailed test
Pvalue(0.361, 9) = 0.7263
Since the Pvalue > α ; we fail to reject the Null and conclude that there isn't sufficient evidence to support the claim.
9. What is m JKM? A 28° C 90° B 58.5° D 117°
Answers
Step-by-step explanation:
her it Go i think it is helpful for u
(b) How much the selling price should be fixed for pulse bought for Rs.70 per kg. to earn a profit of Rs.6 after allowing a 5 % discount?
Answers
Answer:
Rs. 80
Step-by-step explanation:
Given that :
Purchase price = 70
Profit = 6
Discount = 5%
Let selling price = x
Selling price * (1 - discount) = (purchase price + profit)
x * (1 - 5%) = (70 + 6)
x * (1 - 0.05) = 76
x * 0.95 = 76
0.95x = 76
x = 76 / 0.95
x = 80
Hence, selling price = Rs. 80
Using the diagram, which of the following choices represent alternate exterior angles
Answers
Answer:
A
Step-by-step explanation:
The answer is choice A.
The two angles are alternate exterior angles of lines LG and KH cut by transversal JF.
A number increase by three is two more twice the number.find the number.
Answers
Answer:
1
Step-by-step explanation:
1 increased by 3=1+3=4
Twice of 1=1+1=2
4 is two more than 2.
Therefore, the number=1
How many mL of a 10% magnesium sulfate solution will contain 14 grams of magnesium sulfate?
Answers
Answer:
Upto 40 g or 160 mmols
Step-by-step explanation:
Can you plz mark me as brainliest?
Answer:
140 mL
Step-by-step explanation:
10% of 140 is 14
Two boats are travelling at 30 miles/hr, the first going north and the second going east. The second crosses the path of the first 10 minutes after the first one was there. At what rate is their distance increasing when the second has gone 10 miles beyond the crossing point
Answers
Answer:
their distance is increasing at the rate of 41.6 miles/hr
Step-by-step explanation:
Given the data in the question;
first we determine the distance travelled by the first boat in 10 min when the second boat was crossing its path;
⇒ ( 30/60 ) × 10 = 5 miles
so as illustrated in the diagram below;
y² = x² + ( x + 5 )²
2y[tex]\frac{dy}{dt}[/tex] = 2x[tex]\frac{dx}{dt}[/tex] + 2(x+5)[tex]\frac{dx}{dt}[/tex]
y[tex]\frac{dy}{dt}[/tex] = ( 2x + 5 ) ][tex]\frac{dx}{dt}[/tex]
[tex]\frac{dy}{dt}[/tex] = [( 2x + 5 )/y ][tex]\frac{dx}{dt}[/tex] ------ let this be equation 1
Now, given that, [tex]\frac{dx}{dt}[/tex] = 30 miles/hr, when x = 10
so
y = √( 10² + 15² ) = √325
so from equation 1
[tex]\frac{dy}{dt}[/tex] = [( 2x + 5 )/y ][tex]\frac{dx}{dt}[/tex]
we substitute
[tex]\frac{dy}{dt}[/tex] = [( 2(10) + 5 ) / √325 ]30
[tex]\frac{dy}{dt}[/tex] = [ 25 / √325 ] × 30
[tex]\frac{dy}{dt}[/tex] = 41.6 miles/hr
Therefore, their distance is increasing at the rate of 41.6 miles/hr
It is estimated that t months from now, the population of a certain town will be changing at the rate of 4+ 5t^2/3 people per month. If the current population is 10,000, what will the population be 8 months from now?
Answers
Answer:
240000
Step-by-step explanation:
Represent the exponential equation.
[tex]10000 (5 {t}^{ \frac{2}{3} } + 4) = [/tex]
Replace 8 with t
[tex]10000(5(8) {}^{ \frac{2}{3} } + 4)[/tex]
[tex]10000(5 \times 4 + 4) [/tex]
[tex]10000(24) = 240000[/tex]
The population of the town after 8 month will be 2,40,000.
What is exponential growth?
Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function.
Let P be the population of the town after 8 months
According to the given question
The current population of the town = 10,000.
Also, the population of the town is changing at the rate of [tex]4+5t^{\frac{2}{3} }[/tex].
Therefore, the population of the town after 8 month is given by the exponential function
[tex]P = 10000(4+5t^{\frac{2}{3} } )[/tex]
Substitute t =8 in the above equation
⇒[tex]P = 10000(4 + 5(8)^{\frac{2}{3} } )[/tex]
⇒[tex]P = 10000(4 + 5(2^{3}) ^{\frac{2}{3} } )[/tex]
⇒[tex]P = 10000(4+5(4))[/tex]
⇒[tex]P = 10000(24)[/tex]
⇒[tex]P = 240000[/tex]
Hence, the population of the town after 8 month will be 2,40,000.
Find out more information about exponential growth here:
https://brainly.com/question/11487261
#SPJ2
Pls if anyone knows the answer that will be greatly appreciated :)
Answers
Answers:
The areas from left to right are: 13 m^2, 49 m^2, 24 m^2, 14 m^2
The largest area occurs when the rectangle is a square
===========================================================
Explanation:
The area rectangle formula is base*height, or length*width, whichever you prefer.
From left to right, we have these areas:
1*13 = 13 m^27*7 = 49 m^212*2 = 24 m^25*9 = 14 m^2We get the largest area (49 m^2) when the figure is a square. This happens with any problem in which we have a fixed amount of fencing and we want to max out the area. So it's not particular to this specific problem only.
Why a square? Well an informal way to think of it would be to consider that as one dimension goes up, the other goes down, and vice versa. Think of it like a see-saw. As the examples show, if one dimension is particularly large, then its area wont be as big compared to when the dimensions are closer together. It's only when all dimensions are equal is when we max the area out entirely.
I am sry but he has don wrong calculation the answer is last one 9*5=45m and its the answer.
and the noticiable thing is the larger the shape is the area increases.
What is the maximum of f(x)= sin(x)?
-2π
-1
1
2π
Answers
Answer:
1
Step-by-step explanation:
the maximum of f(X)=sin(X) is 1
a train left town at 9:15am it arrived at 2:15 at an average speed of24km/h, how many km does it cover.
Answers
Answer:
168 km
Step-by-step explanation:
24*7 is equal to 168 km
Which expression is the best estimate of the product of 7/8and 8 1/10?
Answers
Answer:
7 7/80 or 7.0875
Step-by-step explanation:
product is the result of multiplication
7/8 * 81/10 = 567/80 = 7 7/80 or 7.0875
After completing the fraction division 5 / 5/3, Miko used the multiplication shown to check her work.
3 x 5/3 = 3/1 x 5/3 = 15/3 or 5
Answers
Answer:
its the same above
Step-by-step explanation:
I need help completing this problem ASAP
Answers
4/(√x - √(x - 2)) × (√x + √(x - 2))/(√x + √(x - 2))
= 4 (√x + √(x - 2)) / ((√x)² - (√(x - 2))²)
= 4 (√x + √(x - 2)) / (x - (x - 2))
= 4 (√x + √(x - 2)) / (x - x + 2)
= 4 (√x + √(x - 2)) / 2
= 2 (√x + √(x - 2))
The length of a rectangle is 7 inches
more inan its width. the area of
the rectangle is eqaul to 4 inches less
than 4 times the perimeter. Find the
length and width of the rectangle
Answers
Answer:
length = 20 inches
width = 13 inches
Step-by-step explanation:
l = length
w = width
area = l×w
perimeter = (2×l) + (2×w)
l = w + 7
l×w = 4×(2×l + 2×w) - 4
(w+7)×w = 4×(2×(w+7) + 2×w) - 4
w² + 7w = 4×(2w + 14 + 2w) - 4
w² + 7w = 8w + 56 + 8w - 4 = 16w + 52
w² - 9w - 52 = 0
the solution for a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
here we use now w instead of x.
and a=1
b=-9
c=-52
w = (9 ± sqrt(81 - -208))/2 = (9 ± sqrt(81+208))/2 =
= (9 ± sqrt(289))/2 = (9 ± 17)/2
w1 = (9+17)/2 = 26/2 = 13
w2 = (9-17)/2 = -8/2 = -4
and a negative length does not make any sense for a geometric shape.
so, only w1 = 13 applies.
l = w + 7 = 13 + 7 = 20
What percentage of area is above the mean on a normal curve?
Group of answer choices
34%
68%
97.35%
50%
Answers
Answer:
z=0
50%
Step-by-step explanation:
help with 27 please. thanks
Answers
Answer:
See Below.
Step-by-step explanation:
We are given the function:
[tex]\displaystyle y=\sqrt{\sin x}[/tex]
And we want to show that:
[tex]\displaystyle 4y^3\frac{d^2y}{dx^2}+y^4+1=0[/tex]
Find the first derivative of y using the chain rule:
[tex]\displaystyle \frac{dy}{dx} = \frac{1}{2\sqrt{\sin x}}\cdot \cos x = \frac{\cos x}{2\sqrt{\sin x}}[/tex]
And find the second derivative using the quotient and chain rules:
[tex]\displaystyle \begin{aligned} \frac{d^2y}{dx^2} &= \frac{1}{2}\left(\frac{(\cos x)'(\sqrt{\sin x})-(\cos x)(\sqrt{\sin x})'}{(\sqrt{\sin x})^2}\right) \\ \\ &=\frac{1}{2}\left(\frac{-\sin x\sqrt{\sin x} - \left(\cos x\right) \left (\dfrac{\cos x}{2\sqrt{\sin x}}\right)}{\sin x}\right) \\ \\ & = \frac{1}{2}\left(\frac{ -\sin x(2\sin x) -\cos x(\cos x) }{\sin x \left(2\sqrt{\sin x}\right) }\right) \\ \\ &= -\frac{1}{2} \left(\frac{2\sin^2 x + \cos^2 x}{2\sin^{{}^{3}\!/\! {}_{2}}x}\right)\end{aligned}[/tex]
Find y³:
[tex]\displaystyle y^3 = \left((\sin x)^{{}^{1}\!/\!{}_{2}}\right) ^3= \sin^{{}^{3}\! / \! {}_{2} }x[/tex]
And find y⁴:
[tex]\displaystyle y^4 = \left((\sin x)^{{}^{1}\!/\!{}_{2}}\right)^4 = \sin^2 x[/tex]
Substitute:
[tex]\displaystyle 4\left( \sin^{{}^{3}\! / \! {}_{2} }x\right)\left(-\frac{1}{2}\left(\frac{2\sin ^2x + \cos ^2 x}{2\sin^{{}^{3}\!/ \! {}_{2}}x}\right)\right)+\left(\sin ^2 x\right) + 1= 0[/tex]
Simplify:
[tex]-\left(2\sin^2 x + \cos^2 x\right) + \sin ^2 x + 1=0[/tex]
Distribute:
[tex]-2\sin ^2 x - \cos^2 x + \sin ^2 x + 1=0[/tex]
Simplify:
[tex]-\sin ^2 x - \cos^2 x + 1= 0[/tex]
Factor:
[tex]-(\sin ^2 x + \cos^2 x ) + 1=0[/tex]
Pythagorean Identity:
[tex]-(1)+1=0\stackrel{\checkmark}{=}0[/tex]
Q.E.D.
Help please!!!
I need this assignment done today
Answers
Answer:
x- 1
y-5
z-3
Step-by-step explanation:
all u have to do is calculate the distance, so for example y is 5 because - -4 -3 -2 -1 0 1 and that is a 5 number distance
Y-5
Z-3
Those should be the answers
I need help on this problem
Answers
9514 1404 393
Answer:
see attached
Step-by-step explanation:
(a) The graph is scaled by a factor of 2, and shifted up 1 unit. The scaling moves each point away from the x-axis by a factor of 2. The points on the x-axis stay there. The translation moves that scaled figure up 1 unit.
__
(b) The graph is reflected across the x-axis and shifted right 4 units. The point on the x-axis stays on the x-axis.
I need help this is confusing to me
Answers
Answer:i think it is b not really sure
Step-by-step explanation:
Find area of shaded region
Answers
Answer:
ayyyy i go to RSM too? Which location r u at? ANyways the answer is
Given the numbers 30 and 45, find the common factors of the two numbers.
Answers
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 45: 1, 3, 5, 9, 15, 45
The common factors between the two numbers are 1, 3, 5, 15.
Hope this helps!
write your answer as an integer or as a decimal rounded to the nearest tenth
Answers
Answer:
Step-by-step explanation:
CE and are the sides making up the sine of an angle.
CE is the side opposite the angle
DE is the side hypotenuse.
<D = 61 degrees
Sin(D) = opposite / hypotenuse
hypotenuse = 8
Sin(61) = 0.8746
CE = ?
sin(61) = CE / 8 multiply both sides by 8
8 sin(61) = CE
CE = 8 * 0.8746
CE = 6.9969
CE = 7.0
That 0 should be included in the answer, but I think it is safe to say that if you enter 7, you will get it right.
Answer:
7.0
Step-by-step explanation:
6w + 2(4w - 7) simplified
Answers
Answer:
14w -14
Step-by-step explanation:
6w + 2(4w - 7)
Distribute
6w+ 8w -14
Combine like terms
14w -14
Answer:
6w+(2×4w)-(2×7)
(6w+8w)-14=14w-14
square root of 12321 by prime factorization
Answers
12321-3x3x37x37
(3)^2×(37)^2
square root = 3×37=111
Hope it helps you..!!
A baseball stadium has a fixed cost of $12,000 per night game. In addition, there is a total cost of $2
each fan that attends the game that evening. Which of the following is the cost equation for the baseball
stadium?
C = 12,002x
C = 2x - 12,000
C= 12,000x + 2
C = 2x + 12,000
Answers
if f(x)=3x²-7 and f(x+n)=3x²+24x+41, what is the value of n?
Answers
Answer:
n=4
Step-by-step explanation:
f(x+n)=3(x+n)^2-7=3x^2+24x+41
3x^2+3n^2+6xn-7=3x^2+24x+41
Comparing and we will get, n=4
find the value of the trigonometric ratio
Answers
Answer:
15÷39
Step-by-step explanation:
I hope it will help you
cos x = adjacent÷ hypotenuse
cos x =15÷39
cos x = 5÷13
What is the solution set of the
equation?
(3x – 5)(2x – 10) = 0
Answers
Answer:
Step-by-step explanation:
3x - 5 = 0
3x = 5
x = 5/3
2x - 10 = 0
2x = 10
x = 10/2 = 5
