1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Andrei [34K]
2 years ago
8

2. Calculate an expression for dy/dx and d2y/dx2 in terms of t if the parametric pair is given as tan(x) = e^at and e^y = 1 + e^

2at
Hint: Turn tan(x) = e^at and e^y = 1 + e^2at into expressions suited to your calculation.​

Mathematics
1 answer:
Ber [7]2 years ago
4 0

I assume a is a constant. If tan(x) = exp(at) (where exp(x) means eˣ), then differentiating both sides with respect to t gives

sec²(x) dx/dt = a exp(at)

Recall that

sec²(x) = 1 + tan²(x)

Then we have

(1 + tan²(x)) dx/dt = a exp(at)

(1 + exp(2at)) dx/dt = a exp(at)

dx/dt = a exp(at) / (1 + exp(2at))

If exp(y) = 1 + exp(2at), then differentiating with respect to t yields

exp(y) dy/dt = 2a exp(2at)

(1 + exp(2at)) dy/dt = 2a exp(2at)

dy/dt = 2a exp(2at) / (1 + exp(2at))

By the chain rule,

dy/dx = dy/dt • dt/dx = (dy/dt) / (dx/dt)

Then the first derivative is

dy/dx = (2a exp(2at) / (1 + exp(2at))) / (a exp(at) / (1 + exp(2at))

dy/dx = (2a exp(2at)) / (a exp(at))

dy/dx = 2 exp(at)

Since dy/dx is a function of t, if we differentiate dy/dx with respect to x, we have to use the chain rule again. Suppose we write

dy/dx = f(t)

By the chain rule, the derivative is

d²y/dx² = df/dx

d²y/dx² = df/dt • dt/dx

d²y/dx² = (df/dt) / (dx/dt)

d²y/dx² = 2a exp(at) / (a exp(at) / (1 + exp(2at)))

d²y/dx² = 2 (1 + exp(2at))

You might be interested in
Now just 2. Thank you
Levart [38]
The correct choice is B 
x is greater than equals to -4.25
8 0
3 years ago
Another one for you!
Eddi Din [679]
Is this from khan academy?
3 0
3 years ago
Read 2 more answers
What's the difference between 4.5 and -3.5​
suter [353]

Answer:

4.5 and -3.5 are different numbers for one then on is positive and one is negative

Step-by-step explanation:

4 0
3 years ago
Which of the following is the graph of y = cosine (2 (x + pi))?
bonufazy [111]
On a coordinate plane, a curve crosses the y- axis at (0,1). It has a minimum of negative 1 and a maximum of 1. It goes through 2 cycle at 2 pi.

4 0
3 years ago
Read 2 more answers
........................ ​
Oxana [17]

Answer:

what do u need help with?

Step-by-step explanation:

3 0
3 years ago
Other questions:
  • Find the measure of angle 2<br><br> Correct Answer is 2
    13·2 answers
  • Which expression is 3 times as large as the expression 610 – 97?
    6·1 answer
  • One x-intercept for a parabola is at the point (0.5,0). Use the factor method to find the other x-intercept for the parabola def
    14·1 answer
  • HELP ME I WILL LOVE U
    8·2 answers
  • 11. Which of the following points is on the
    9·1 answer
  • convert the following into percentage. 1)2.98. 2)7/36. 3)3is to 4/5. d)7.35.​
    9·1 answer
  • Does anyone know this i will give brainliest
    7·1 answer
  • Which is 1/3+1/4???
    7·2 answers
  • -3y^2(-4y^3+y-9) find the product.
    6·2 answers
  • A spinner is spun `40` times for a game.
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!