1. A car travels 20 mi/hr for 2 hours and then 30 mi/hr for 3 hours.

a) For 0<t<2 let c(t) be the distance the car has traveled in miles from time 0 to time t, where t is measured in hours. Find c(t).

b) Extend your function c(t) from part (a) to the time period 2< =t< =5. You will have a piecewise continuous function.

c) For the function c(t) in the previous problem, find c'(1) and c'(3).

a) To find the function c(t) for 0 < t < 2, we need to determine the distance the car has traveled during that time period.

The car travels at a constant speed of 20 mi/hr for 2 hours. So, during the first 2 hours, the car travels a distance of 20 mi/hr * 2 hr = 40 miles.

Therefore, the function c(t) for 0 < t < 2 is given by c(t) = 40.

b) To extend the function c(t) from 0 < t < 2 to 2 <= t <= 5, we need to consider the second part of the car's journey.

The car travels at a constant speed of 30 mi/hr for 3 hours. So, during this time period, the car travels a distance of 30 mi/hr * 3 hr = 90 miles.

Therefore, the function c(t) for 2 <= t <= 5 can be defined as c(t) = 90 + (t - 2) * 30.

c) To find c'(1), we need to find the derivative of the function c(t) at t = 1.

Taking the derivative of c(t) = 40 with respect to t, we get c'(t) = 0, as the function is constant.

Therefore, c'(1) = 0.

To find c'(3), we need to find the derivative of c(t) = 90 + (t - 2) * 30 at t = 3.

Taking the derivative, we get c'(t) = 30.

Therefore, c'(3) = 30.