Brief Calculus.

Question

The annual earnings of a certain company f (t) =⎷10t^2+t+229 are thousand dollars t years after its opening in January 2005.

a) Find the formula for the rate of change.

b) Find and Interpret f (10)

c) Find and Interpret f' (10)

This is the symbol of the square root⎷.

Answer 1

The rate of change is the derivative with respect to time, so, in this case that's f '(t). To take the derivative, of a function with a radical like this, I find it easier to write it as a fractional exponent. The fractional exponent that goes with square root is 1/2 so:

f(t) = (10t² + t + 229)^1/2

To take the derivative, use the Power Rule, then the Chain Rule:

f '(t) = 1/2(10t² + t + 229)^-1/2(20t + 1) which can also be written as:

f(10) = (10(10)² + (10) + 229)^1/2 = √1239 = 35.199 meaning 10 years after the company's opening (so in Jan 2015), the company's annual earnings were $35,199 (remember that earnings are in thousands of dollars)

f '(10) = [20(10) + 1]/[2√(10(10)² + (10) + 229)] = 201/(2√1239) = 2.855 meaning the rate of change of earnings in Jan 2015 is $2855 per year