A person's temperature is f (t) degrees Fahrenheit t days after the start of a ten-days sickness and f(t)=98.6+1. 2t-0.12t^2, 0 is less than or equal to t, and t is less than or equal to 10. find the rate of change of the temperature when the person has been sick for 8 days.
To find the rate of change of the temperature when the person has been sick for 8 days, we need to find the derivative of the temperature function f(t) with respect to t, and then substitute t = 8 into the derivative.
Step 1: Find the derivative of f(t)
f(t) = 98.6 + 1.2t - 0.12t^2
Differentiating each term separately:
f'(t) = d/dt (98.6) + d/dt (1.2t) - d/dt (0.12t^2)
Since the derivative of a constant is 0, the first term becomes:
f'(t) = 0 + d/dt (1.2t) - d/dt (0.12t^2)
= 1.2 - d/dt(0.12t^2)
Using the power rule for differentiation, we can find the derivative of the second term:
d/dt (0.12t^2) = 2 * 0.12t * 1
= 0.24t
Putting it all together:
f'(t) = 1.2 - 0.24t
Step 2: Substitute t = 8 into f'(t)
f'(8) = 1.2 - 0.24 * 8
= 1.2 - 1.92
= -0.72
Therefore, the rate of change of the temperature when the person has been sick for 8 days is -0.72 degrees Fahrenheit per day.
To find the rate of change of temperature when the person has been sick for 8 days, we need to calculate the derivative of the temperature function f(t).
The given temperature function is f(t) = 98.6 + 1.2t - 0.12t^2.
To find the derivative of f(t) with respect to t, we can use the power rule and the constant rule of differentiation.
Differentiating the first term, 98.6, with respect to t gives us 0 because it is a constant.
Differentiating the second term, 1.2t, with respect to t gives us 1.2 because the derivative of t is 1.
Differentiating the third term, -0.12t^2, with respect to t gives us -0.24t because the derivative of t^2 is 2t, and multiplying by the coefficient -0.12.
Now, let's simplify the expression:
f'(t) = 1.2 - 0.24t
Now that we have the derivative of f(t), we can find the rate of change of temperature when the person has been sick for 8 days by substituting t = 8 into f'(t):
f'(8) = 1.2 - 0.24(8)
= 1.2 - 1.92
= -0.72
Therefore, when the person has been sick for 8 days, the rate of change of the temperature is -0.72 degrees Fahrenheit per day.
df/dt = 1.2 - 0.24t
now just plug in t=8