exponential order c is defined as Which functions are of exp

exponential order \'c\' is defined as:

Which functions are of exponential order c? Please find c if they are of exponential order.

Solution

In the first case

y(t) = 6t^2 + 5t + 1

6t^2 + 5t + 1 <= M*e^(ct)

since the growth of e^(x) is greater than x*2 => growth of 6t^2 + 5t+ 1 can be bound with the function e^(x) * some constant

Hence the value of c = 1 for first case (in order to bound the condition)

In the second case

sin(4t)

Sin function varies from -1 to 1

Hence -1 < sin(4t) < 1

Hence the function is not exponential it can be easily satisfied by putting the value of M=2

no exponential order, value of c will be 0 in this case

In the third case

5e^(2t) < M*e^(ct)

The given function can be satisfied for value of M greater than 5 and putting as c=2

Hence value of c in this case is 2

In the fourth case

cos(t*e^(3t))

Hence -1 < cos(t*e^(3t)) < 1

Hence the function is not exponential it can be easily satisfied by putting the value of M=2

no exponential order, value of c will be 0 in this case

In the fifth case

cos(t)*e^(3t)

since -1 < cos(3t)< 1

Hence value of M can be greater than 1 and c must be 3 in order to satisfy the inequality

Exponential order function, value of c=3