Differentiation

Question

The number of crimes, C, in a particular country between 2000 and 2010 was given by the function C=11.9+ 0.1t + ln(t+1), where t is the number of years passed since 2000. Show that C is an increasing function of t for 0≤ t ≤ 10.

 

Thank You! :)

Answer 1

To show that a function is increasing, you use the first derivative test. When the derivative has a positive value, it's increasing, negative value, decreasing. So your function has the derivative, C'(t) = 0.1 + 1/(t+1). 

The critical points for this is when it equals 0 or is undefined.

 

It is undefined when t+1 = 0 or t = -1. remember the denominator cannot be 0.

 

The other critical point is when .1 + 1/(t+1) = 0. Solving this you get t = -11.

 

These critical values split the number line into sections where you can test if the derivative fumction is positive or negative. You're concerned with the interval between 0 and 10 which does not contain either of my critical points so it's either always positive or always negative. plug in 1 and let's find out the sign of the derivative there. C'(1) = .1 + 1/2.....obviously positive. Therfore yourderivative is always positive on this interval, therfore your function is always increasing on this interval.

 

Hope this makes sense to you.