Question

find the formula for a function in the form y=a/1+be^-t with a y intercept of 5 and inflection point at t=1

Answer 1

The formula for the function in the given form with a y-intercept of 5 and an inflection point at t = 1 is:

`y = 5 - 5e^(-t)/(1 + 0.2e^(-t))`

Step-by-step explanation:

To find the formula for this function, we first identify the constants a and b from the given information. Since the function has a y-intercept of 5, we know that when t is equal to 0, y is equal to 5. This means that a is equal to 5. Next, we find the inflection point, which is where the second derivative of the function changes sign. The formula for the second derivative of this function is:

`y'' = (a - 10be^(-t))/(1 + be^(-t))^3`

To find the inflection point, we set y'' equal to 0 and solve for t:

`0 = (a - 10be^(-t))/(1 + be^(-t))^3`

Simplifying this expression, we get:

`a - 10be^(-t) = 0`

Since we already know that a is equal to 5, we can substitute this value into the equation and solve for b:

`5 - 10be^(-t) = 0`

Dividing both sides by -10, we get:

`b = e^t/(-10)`

Now that we have found the value of b, we can substitute it into the formula for y and simplify:

`y = a/(1 + be^-t) = 5/(1 - e^(-t)/2) = 5(2 - e^(-t))/(2 - e^(-t) + e^(-2t))`

To find the value of y when t is equal to 1, we substitute this value into the formula:

`y = 5(2 - e^(-1))/(2 - e^(-1) + e^(-2)) = 3.4698.`.. (rounded to four decimal places)

Answer 2

y = (5e + 5sqrt(3) + 5) / (1 + (e + sqrt(3)) * e^(-t))

Step-by-step explanation:

To find the formula for a function in the form y = a/(1 + be^(-t)) with a y-intercept of 5 and an inflection point at t = 1, we can use the following steps:

Step 1: Find the value of a

Since the y-intercept of the function is 5, we know that the point (0, 5) lies on the graph of the function. So, we substitute t = 0 and y = 5 into the equation and solve for a:

5 = a / (1 + be^(0))

5 = a / (1 + b1)

5 = a / (1 + b)

a = 5*(1 + b)

Step 2: Find the value of b

To find the value of b, we use the fact that the function has an inflection point at t = 1. The inflection point is where the concavity of the function changes from upward to downward or vice versa. It is also the point where the second derivative of the function is zero or undefined.

The first derivative of the function is:

y' = -abe^(-t) / (1 + b*e^(-t))^2

The second derivative of the function is:

y'' = abe^(-t)(be^(-t) - 2) / (1 + b*e^(-t))^3

Setting t = 1 and y'' = 0, we get:

0 = abe^(-1)(be^(-1) - 2) / (1 + b*e^(-1))^3

Simplifying and using the value of a from Step 1, we get:

0 = 5be^(-1)(be^(-1) - 2) / (1 + b*e^(-1))^3

Multiplying both sides by (1 + be^(-1))^3 and simplifying, we get:

0 = 5b^2e^(-2) - 10b*e^(-1) + 1

Solving for b using the quadratic formula, we get:

b = (10e^(-1) ± sqrt(100e^(-2) - 451e^(-2))) / (25*e^(-2))

b = (2e + sqrt(4e^2 - e^2)) / (2e^(-1))

b = e + sqrt(3)

Step 3: Write the function in the form y = a/(1 + be^(-t))

Using the values of a and b from Steps 1 and 2, we get:

a = 5*(1 + b) = 5*(1 + e + sqrt(3)) = 5e + 5sqrt(3) + 5

b = e + sqrt(3)

So, the function in the form y = a/(1 + be^(-t)) with a y-intercept of 5 and an inflection point at t = 1 is:

y = (5e + 5sqrt(3) + 5) / (1 + (e + sqrt(3)) * e^(-t))