help with exponents

[jtowich]

I'm going through an old college textbook and have a problem I can't figure out. I have the answer, just not how it is derived. Scary thinking that I must have known how at one time.

The problem is given the following equation (^ indicates an exponent, K is a constant):

v(t) = K(e^(-at) - e^(-bt))

The largest value of K needs to be determined in terms of 'a' and 'b' such that v(t) is limited to the range from -1 to 1.

The answer given (in the back of the book) is:

|K| <= 1/|((a/b)^(-a/(a-b)) - (a/b)^(b/(a-b)))| the vertical lines indicating absolute value

any idea how to approach this?

 

[daon2]

I'm going through an old college textbook and have a problem I can't figure out. I have the answer, just not how it is derived. Scary thinking that I must have known how at one time.

The problem is given the following equation (^ indicates an exponent, K is a constant):

v(t) = K(e^(-at) - e^(-bt))

The largest value of K needs to be determined in terms of 'a' and 'b' such that v(t) is limited to the range from -1 to 1.

The answer given (in the back of the book) is:

|K| <= 1/|((a/b)^(-a/(a-b)) - (a/b)^(b/(a-b)))| the vertical lines indicating absolute value

any idea how to approach this?

I find that hard to believe. If a=-1, and b=0 then there is no constant K that will stop v(t) =Ke^t-K, from shooting off to infinity (unless K=0).

 

[jtowich]

Both exponents are negative, so the equation can be rewritten as

v(t) = K(1/e^(at) - 1/e^(bt))

so as long as 'a' and 'b' are not negative (which I guess is assumed), as 'a' or 'b' is increased the result will only approach 0, not infinity.