Josephood7

For what positive values of [math] does the function [math] satisfy the differential equation [math]?


For what negative values of [math] does the function [math] satisfy the differential equation [math]?

Newton's law of cooling says that the rate of cooling of an object is proportional to the difference between the temperature of the object and that of its surroundings (provided the difference is not too large). If [math] represents the temperature of a (warm) object at time [math], [math] represents the ambient (cool) temperature, and [math] is a negative constant of proportionality, which equation(s) accurately characterize Newton's law?

 A. [math]
 B. [math]
 C. [math]
 D. [math]
 E. All of the above
 F. None of the above

The function [math] satisfies the differential equation [math] List the (distinct) constant solutions [math] to the differential equation in ascending order. (If there are fewer than four solutions, leave the latter blanks empty)
   

For what values of [math] (in interval notation) is [math] increasing? Use the strings "plus_infinity" or "minus_infinity" if appropriate, and if there is only one interval, leave the second one blank. Finally, list your intervals so that the first interval is to the left of the second (on the real line).
Interval 1: [math] , [math]
Interval 2:[math] , [math]