Daily oil imports to the United States from Mexico can be...

Daily oil imports to the United States from Mexico can be approximated by $I(t)=-0.015 t^{2}+0.1 t+1.4$ million barrels $\quad(0 \leq t \leq 8)$ where $t$ is time in years since the start of $2000 .^{58}$ Find the derivative function $\frac{d I}{d t} .$ At what rate were oil imports changing at the start of $2007(t=7) ?$ HINT [See Example 4.]

Daily oil imports to the United States from Mexico can be approximated by $I(t)=-0.015 t^{2}+0.1 t+1.4$ million barrels $\quad(0 \leq t \leq 8)$
where $t$ is time in years since the start of $2000 .^{58}$ Find the derivative function $\frac{d I}{d t} .$ At what rate were oil imports changing at the start of $2007(t=7) ?$ HINT [See Example 4.]

Key Concepts

Derivative

The derivative of a function represents the rate at which the function is changing at any given point. It is found by applying differentiation techniques and gives the slope of the tangent line to the function's graph at that point. In many practical problems, like changes over time, the derivative quantifies how one quantity responds instantaneously to a change in another.

Power Rule

The power rule is a fundamental technique in calculus used to differentiate polynomial functions. It states that for any term of the form t^n, the derivative is n*t^(n-1). This rule is particularly useful when differentiating functions composed of variables raised to a power, making the process of finding derivatives straightforward.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a specific point is given by the value of its derivative at that point. This concept is essential in understanding how a quantity changes at an exact moment in time, rather than over an interval. In applications, it enables the approximation of dynamic behavior in various contexts, such as speed, growth, or change in imports over time.

Evaluation of the Derivative

Evaluating the derivative at a particular point involves substituting the specific value of the independent variable into the derivative function. This process yields the exact rate of change of the original function at that point, providing insights into the behavior of the function at that specific moment.

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