Let Q(t)=t^2 . As in the previous exercise, find a formula...

Let $Q(t)=t^{2} .$ As in the previous exercise, find a formula for the average rate of change of $Q$ over the interval $[1, t]$ and use it to estimate the instantaneous rate of change at $t=1 .$ Repeat for the interval $[2, t]$ and estimate the rate of change at $t=2$.

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Key Concepts

Average Rate of Change

This concept refers to the change in the value of a function divided by the change in the independent variable over a specific interval. It is calculated using the difference quotient, essentially finding the slope of the secant line connecting two points on the function's graph. This measure provides an overall sense of how the function is changing between two points.

Instantaneous Rate of Change

This is the rate at which the function is changing at a single specific point and is found by taking the limit of the average rate of change as the interval shrinks to zero. It is mathematically defined as the derivative at that point, representing the slope of the tangent line to the graph.

Derivative

The derivative is a fundamental concept in calculus that provides a precise measure of the instantaneous rate of change of a function with respect to its variable. It is obtained as the limit of the difference quotient and is used to analyze and predict the behavior of functions locally, such as determining slopes of tangent lines at specific points.

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