Answers
a. By the chain rule,
[tex]\dfrac{dy}{dx} = \dfrac{dy}{dt}\cdot\dfrac{dt}{dx} = \dfrac{dy}{dt}\cdot\dfrac{1}{\frac{dx}{dt}}[/tex]
Given that [tex]y=e^t+e^{-t}[/tex] and [tex]x=e^{-t}[/tex], we have the derivatives
[tex]\dfrac{dy}{dt} = e^t -e^{-t}[/tex]
[tex]\dfrac{dx}{dt} = -e^{-t}[/tex]
and so
[tex]\dfrac{dy}{dx} = \dfrac{e^t - e^{-t}}{-e^{-t}} = 1-e^{2t}[/tex]
Set this equal to zero and solve for t :
[tex]1-e^{2t} = 0[/tex]
[tex]1 = e^{2t}[/tex]
[tex]\ln(1) = \ln\left(e^{2t}\right)[/tex]
[tex]0 = 2t \ln(e)[/tex]
[tex]0 = 2t[/tex]
[tex]t=0[/tex]
This value of t corresponds to x = e⁰ = 1 and y = e⁰ - 1/e⁰ = 1 - 1 = 0. So the only point on the curve where the derivative dy/dx is zero is (1, 0).
b. Compute the second derivative. Since dy/dx is a function of t, we'll momentarily replace it with f(t). By the chain rule,
[tex]\dfrac{d^2}{dx^2} = \dfrac d{dx} \dfrac{dy}{dx} = \dfrac{df}{dx} = \dfrac{df}{dt}\cdot\dfrac{dt}{dx} = \dfrac{df}{dt}\cdot\dfrac{1}{\frac{dx}{dt}}[/tex]
We have
[tex]\dfrac{df}{dt} = -2e^{2t}[/tex]
and we've already committed dx/dt. So
[tex]\dfrac{d^2}{dx^2} = \dfrac{-2e^{2t}}{-e^{-t}} = 2e^{3t} [/tex]
Substitute the first and second derivative into the differential equation:
[tex]\left(\dfrac{d^2y}{dx^2}\right)^2 + \dfrac{dy}{dx} - 1 = 0[/tex]
[tex]\left(2e^{3t}\right)^2 + (1-e^{2t}) - 1 = 0[/tex]
[tex]4e^{6t} -e^{2t}= 0[/tex]
[tex]e^{2t} (4e^{4t} - 1) = 0[/tex]
[tex]4e^{4t} - 1 = 0[/tex]
[tex]4e^{4t} = 1[/tex]
[tex]e^{4t} = \dfrac14[/tex]
[tex]\ln\left(e^{4t}\right) = \ln\left( \dfrac14\right)[/tex]
[tex]4t \ln(e) = -\ln(4)[/tex]
[tex]4t = -\ln(4)[/tex]
[tex]t = -\dfrac{\ln(4)}4[/tex]
Related Questions
Solve the following system using the elimination method
−8y+6x=36
6x−y=15
Answers
Answer:
a
Step-by-step explanation:
Answer:
C. (2, -3)
Step-by-step explanation:
asap asnwerrrrrrrrrrrrrrrrrrrrrrrr
Answers
Hope this helps!
HELP HURRY !! is the graph a function ?
Answers
Answer:
Yes.
Step-by-step explanation:
If you use the vertical line test, you'll see that the line only touches the graph at one point.
use the correct trig function to find the value of x. SOH, CAH, TOA
Answers
Answer:
About 11.025
Step-by-step explanation:
[tex]\sin{58^\circ}=\dfrac{x}{13}\\\\x=13\cdot\sin{58^\circ} \approx11.025[/tex]
what 2.3892 decimal of the round to nearest hundredths
Answers
Answer:
2.38
Step-by-step explanation:
William wish1wish woman weeks whereby when wishful willful
As you may know, the 8 would be located in the hundreaths place.
2.3892
If the number right of 8 is five for less, it says the same. If it's 5 or more, then 8 would change. In this case, 9 is more than 5 meaning that 8 becomes 9. However, we aren't quite finished yet. 9 and everything left of that number must become 0.
Your answer would be 2.39.
Please hurry will mark brainliest if you answer correctly
Answers
What is the 5th term of the linear sequence? 1,7,13,19,25,...
Answers
Answer : 31
Explanation : Every numbers have a gap of 6 as you can see:
1 + 6 = 7
7 + 6 = 13
13 + 6 = 19
19 + 6 = 25
So,
25 + 6 = 31
Hope This Helps You!
it has been suggested that the percentage of drivers in the us that use a blinker when changing lanes is58% you plan to observe 50 cars during a lane change. What type of distribution could you use to determine the number of cars you will observe before finding a car the does not use a blinker during a lane change?
Answers
Since the number of trials is not fixed, an inverse binomial distribution is used.
For each car, there are only two possible outcomes, either they use a blinker during lane change, or they do not. The probability of a car using a blinker during lane change is independent of any other car, which means that the number of cars that do not use a blinker during a lane change is a binomial variable.
However, we are counting the number of trials until observing a car that does not use a blinker during a lane change, hence, it is an inverse binomial variable.A similar problem is given at https://brainly.com/question/25644451
plssssssssssss answer for brainiest
Answers
Answer:
E = 120
F = 60
G = 120
H = 60
Step-by-step explanation:
There are 30 pens in a box.
12 of the pens are black.
7 of the pens are green.
The rest of the pens are red.
One of the pens is chosen at random.
Find the probability that the pen is red.
Answers
Step-by-step explanation:
30-(12-7)= red pens
30-19
11
Probability=
11/30
pens that are lefted are red
what is the correct order of reasons to complete the proof??
Answers
Answer: Choice B
I, I, V, II
======================================================
Explanation:
When doing any proofs, we always repeat what is given to us. In this case, the given statements are
QT = RSQS = RTIt seems weirdly redundant to repeat these statements back to the teacher, but this is how all proofs start off. You state what is given and the reason is simply "given". So far the answer is narrowed down to choice B or choice C, due to the "statement I" listed twice like that.
From those statements, we build toward what we want to prove. Those two given statements tell us we have two pairs of congruent sides. If we could prove a third pair of sides are congruent, then we can use SSS which stands for "side side side". It turns out that we are more or less automatically given the third pair of sides in the form QR = QR, which is true by the reflexive property. This property says that any segment is congruent to itself. So we'll have statement V listed after statement I twice. The sequence so far is I, I, V. At this point, we can conclude that choice B must be the final answer. Sure enough, statement II involves SSS just like mentioned earlier.
12x-4=10x-10+30 find value of x
Answers
Answer:
12
Step-by-step explanation:
12x - 4 = 10x + 20
2x - 4 = 20
2x = 24
x = 12
What is the value for x if 2x+3=20 that makes the equation true
Answers
Answer:
x = 8.5
Step-by-step explanation:
2x+3=20
2x = 17...........subtract 3 from both sides
x = 8.5............divide both sides by 2
What is the y- intercept of a line that has a slope of 2 and passes through (-1,3)?
Answers
Answer:
y - intercept = 5
Step-by-step explanation:
Equation: y = mx + b
Input the given values
3 = -1(2) + b
3 = -2 + b
5 = b
-Chetan K
7. Fill in the columns in the following table. What quantity should
a profit-maximizing firm produce? Verify your answer with
marginal reasoning.
Answers
Answer:
The rule for a profit-maximizing perfectly competitive firm is to produce the level of output where Price= MR = MC, so the raspberry farmer will produce a quantity of 90, which is labeled as e in Figure 4 (a). Remember that the area of a rectangle is equal to its base multiplied by its height.
find a positive coterminal angle for 765 degrees
Answers
Answer:
45 degrees
Step-by-step explanation:
Subtract 360° 360 ° from 405° 405 ° . The resulting angle of 45° 45 ° is positive, less than 360° 360 ° , and coterminal with 765° 765 ° .
What is the answer to this question?
Answers
Answer:
The answer should be A.
Step-by-step explanation:
You make $85,000 per year and your company matches 50 cents for every dollar you deposit into your 401k plan, up to %5 of your salary. If you contribute $300
every month to your 401k, what will your company contribute each month?
Answers
Answer:
$150 each month
Step-by-step explanation:
divide 300 in half because they contribute 50¢ per dollar
Jace is deciding between two landscaping companies for his place of business. Company A charges $50 per hour and a $250 equipment fee. Company B charges $75 per hour and a $150 equipment fee. Let A represent the amount Company A would charge for tt hours of landscaping, and let B represent the amount Company B would charge for tt hours of landscaping. Graph each function and determine the number of hours, t, that would make the cost of each company the same.
give me the corrdent plan point i need 2 for example
(5,2) and (4,1) like that on the graph
Answer: ________
dollars per hour
dollars
hours per dollar
hours
my graph is the y axis is 1000 and the x axis is 20
Answers
Answer:
I don't know ✌️. ..............
Find the error(s) and solve the problem correctly. Be sure to explain and show your work. Where did the "wrong answer" come from? sin 52° Answer: sin 52 = .9866
Answers
The error of the evaluation comes from the fact that Sin 52° = 0.7880.
According to the question;
we are required to determine what's wrong with the evaluation..By evaluation;
Sin 52° when evaluated by means of a scientific calculator or tables gives; 0.7880.The error can be determined by evaluating; the inverse sine of 0.9866.
Sin-¹ .9866 = 80.6.Read more;
https://brainly.in/question/14817452
2 (n - 1) + 47 = 2 (3n - 1)
Answers
[tex]\\ \sf\longmapsto 2(n-1)+47=2(3n-1)[/tex]
[tex]\\ \sf\longmapsto 2n-2+47=6n-2[/tex]
[tex]\\ \sf\longmapsto 2n+47=6n[/tex]
[tex]\\ \sf\longmapsto 4n=47[/tex]
[tex]\\ \sf\longmapsto n\approx 12[/tex]
Answer:
n = 11.75
Step-by-step explanation:
2(n - 1) + 47 = 2(3n - 1) Distribute the multiplier
2n - 2 + 47 = 6n - 2 Combine terms
47 = 4n divide
n = 11.75
2 Which of the following would result in a
rotation 360° counterclockwise about the
origin?
® (x,y) - (-Y. -x)
© (x, y) - (-y, x)
(x, y) - (-x, -Y)
(x,y) - (x, y)
Answers
Answer:
(d) (x,y) ⇒ (x, y)
Step-by-step explanation:
Rotation 360° about any point brings the object back to its original location. The coordinates are unchanged.
(x,y) ⇒ (x, y)
I need word form and decimal
(I already shaded it in so that is all I need
Answers
The shading is correct for each of the three problems. Nice work.
==========================================
Problem 13
Word form = one tenth
Decimal form = 0.1
Explanation: The 1 is in the tenths place for the decimal value 0.1 to represent the fraction 1/10.
==========================================
Problem 14
Word form = eighty-five hundredths
Decimal form = 0.85
Explanation: The 8 is in the tenths place, and the 5 is in the hundredths place. Overall, 0.85 represents 85 out of 100, aka 85/100.
==========================================
Problem 15
Word form = thirty-nine thousandths
Decimal form = 0.039
Explanation: The second zero is in the tenths place. The 3 is in the hundredths place. The 9 is in the thousandths place. The decimal represents 39/1000 which you can think of it as 039/1000.
Eighty five hundredths, 0.85
Thirty nine thousandths, 0.039
Hope this helps!!!
Solve the equation for x:
4x2 + 15x = 4
Show steps please
Answers
First multiply the 4 and 2
8 + 15x = 4
Next, subtract the 8 on both sides
15x = -4
x=3.75
Original price of a bicycle $1,600.00 Discount 35.3% Tax 1%? Find the selling price of each item ???
Answers
Subtract the discount ($564.80) from the price then add the tax ($16).
-6 10/9 divided by 6/5
Answers
Answer:
-160/27
Step-by-step explanation:
I hope this helps if I’m wrong I apologize
In △ABC, m∠A = 75° and m∠C = 20°, What is m∠B?
Answers
Answer:
30 degree ok please make me brainliest
Answer:
∠ B = 85°
Step-by-step explanation:
The sum of the 3 angles in a triangle = 180° , that is
∠ A + ∠ B + ∠ C = 180° , substitute values
75° + ∠ B + 20° = 180°
∠ B + 95° = 180° ( subtract 95° from both sides )
∠ B = 85°
Write a related function for –6x + 8x – 5 = 3.
Answers
Answer:
Answer:
x=4
Step-by-step explanation:6x+8x=2x
5+3=8 (when you move a number to the other side of an equals sign, it changes its value)
2x=8
2x/2 and 8/2
x=4
The related function for –6x + 8x – 5 = 3 is;
x = 4
Algebraic ExpressionsWe are given the equation;
–6x + 8x – 5 = 3
Let us simplify the numbers that have the variable x first. Thus;
–6x + 8x – 5 = 3 gives;
8x - 6x - 5 = 3
>> 2x - 5 = 3
Use addition property of equality to add 5 to both sides;
2x - 5 + 5 = 3 + 5
>> 2x = 8
Use division property of equality to divide both sides by 2 to get;
x = 8/2
Thus;
x = 4
Read more about algebra expressions at; https://brainly.com/question/4344214
2x+3-3=-6
solve right please
Answers
Finding the value of x,
→ 2x + 3 - 3 = - 6
→ x + (3 - 3) = - 6/2
→ [ x = -3 ]
Thus, the value of x is -3.
If sin theta < 0 and cot theta < 0, which quadrant(s)
could the terminal side of theta lie?
Answers
Answer:
Step-by-step explanation:
sinθ < 0 means below the x axis
cotθ = cosθ/sinθ < 0
As sinθ is already < 0 means cosθ has to be > 0 meaning right of the y axis.
Right of y and below x is the IV quadrant.
The quadrant in which angle theta lie if sin theta < 0 and cot theta < 0 exists Quadrant IV (4).
What is meant by Trigonometric Identities?Trigonometric Identities are equality conditions that hold for all possible values of the variables in the equation and use trigonometry functions. There exists numerous unique trigonometric identities that relate a triangle's side length and angle.
here, we have,
Sine, cosine, and tangent are the fundamental three operations in trigonometry. The cotangent, secant, and cosecant functions are derived from these three basic functions. On these functions, all trigonometrical notions exists built.
Trigonometric identities are equality conditions in trigonometry that hold for all values of the variables that appear and are defined on both sides of the equivalence. These exists identities that, geometrically speaking, involve certain functions of one or more angles.
we have,
sinθ < 0 means below the x axis
cotθ = cosθ/sinθ < 0
As sinθ is already < 0 means cosθ has to be > 0 meaning right of the y axis.
Right of y and below x is the IV quadrant.
Hence, The quadrant in which angle theta lie if sin theta < 0 and cot theta < 0 exists Quadrant IV (4).
To learn more about trigonometric identities refer to:
brainly.com/question/7331447
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