The Arctic Sea ice extent, the area of sea covered by ice,...

The Arctic Sea ice extent, the area of sea covered by ice, grows over the winter months, typically from November to March. Let $F(t)$ be the Arctic Sea ice extent, in million of square kilometers, as a function of time, $t,$ in days since November 1,2014 . Then $F^{\prime}(t)=0.073$ on January $1,2015 .^{11}$ (a) Give the units of the $0.073,$ and interpret the number in practical terms. (b) Estimate $\Delta F$ between January 1 and January $6,$ $2015 .$ Explain what this tells us about Arctic Sea ice.

The Arctic Sea ice extent, the area of sea covered by ice, grows over the winter months, typically from November to March. Let $F(t)$ be the Arctic Sea ice extent, in million of square kilometers, as a function of time, $t,$ in days since November 1,2014 . Then $F^{\prime}(t)=0.073$ on January $1,2015 .^{11}$
(a) Give the units of the $0.073,$ and interpret the number in practical terms.
(b) Estimate $\Delta F$ between January 1 and January $6,$ $2015 .$ Explain what this tells us about Arctic Sea ice.

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Key Concepts

Linear Approximation (Differentials)

Linear approximation is a method to estimate small changes in a function by using its derivative. When the derivative is known, the change in the function over a short time interval can be approximated by multiplying the derivative by the length of the time interval. This method is practical for estimating changes when the function does not vary dramatically over the interval.

Units Analysis for Derivatives

Understanding the units of a derivative involves dividing the units of the function by the units of the independent variable. Here, since the function F represents sea ice extent in million square kilometers and time is measured in days, the derivative has units of million square kilometers per day. This unit quantifies how much the sea ice extent changes per day.

Derivative as Rate of Change

The derivative of a function at a point gives the instantaneous rate at which the function is changing with respect to a variable. In this context, it tells us how quickly the sea ice extent is increasing or decreasing at a specific moment in time.

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