To determine
To calculate: The rate of change of the length of Alaskan rockfish at age 12.
Answer
The instantaneous rate of change of the length of Alaskan rockfish at age 12 is 1.718 inches per year.
Explanation
Given:
The length (L) of Alaskan rockfish at age A is L=0.0155A3−0.372A2+3.95A+1.21 where L is measured in inches and A in years.
Derivative rules:
(1) Derivative of Constant Function: ddt(c)=0
(2) Constant Multiple Rule: ddx[c⋅f(x)]=c⋅ddxf(x)
(3) Power Rule: ddx(xn)=nxn−1
(4) Difference Rule: ddx[f(x)−g(x)]=ddx(f(x))−ddx(g(x))
(5) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))
Calculation:
Obtain the first derivative of L with respect to A.
dLdA=ddA(0.0155A3−0.372A2+3.95A+1.21)
Apply the derivative rules (4) and (5) to get,
dLdA=ddA(0.0155A3)−ddA(0.372A2)+ddA(3.95A)+ddA(1.21)
Apply the derivative rules (1) and (2) to get
dLdA=ddA(0.0155A3)−ddA(0.372A2)+ddA(3.95A)+0=ddA(0.0155A3)−ddA(0.372A2)+ddA(3.95A)=0.0155ddA(A3)−0.372ddA(A2)+3.95ddA(A)
Apply the power rule (3) and simplify the terms,
dLdA=0.0155(3A3−1)−0.372(2A2−1)+3.95(1A1−1)+0=0.0155(3A2)−0.372(2A)+3.95(1)=0.0465A2−0.744A+3.95
Here, the derivative of L with respect to A (dLdA) means the instantaneous rate of change of the length of Alaskan rockfish.
Obtain the values of dLdA when A=12 .
dLdA|A=12=0.0465(12)2−0.744(12)+3.95=0.0465(144)−0.744(12)+3.95=6.696−8.928+3.95=1.718
Therefore, the instantaneous rate of change of the length of Alaskan rockfish at age 12 is 1.718 inches per year.