Let f(t)=1 / t for t ≠0 a. Find the average rate of change...

Let $f(t)=1 / t$ for $t \neq 0$ a. Find the average rate of change of $f$ with respect to $t$ over the intervals (i) from $t=2$ to $t=3,$ and (ii) from $t=2$ to $t=T$ b. Make a table of values of the average rate of change of $f$ with respect to $t$ over the interval $[2, T],$ for some values of $T$ approaching $2,$ say $T=2.1,2.01,2.001,2.0001,2.00001$ and 2.000001 c. What does your table indicate is the rate of change of $f$ with respect to $t$ at $t=2 ?$

Let $f(t)=1 / t$ for $t \neq 0$
a. Find the average rate of change of $f$ with respect to $t$ over the intervals (i) from $t=2$ to $t=3,$ and (ii) from $t=2$ to $t=T$
b. Make a table of values of the average rate of change of $f$ with respect to $t$ over the interval $[2, T],$ for some values of $T$ approaching $2,$ say $T=2.1,2.01,2.001,2.0001,2.00001$ and 2.000001
c. What does your table indicate is the rate of change of $f$ with respect to $t$ at $t=2 ?$

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Key Concepts

Limits

Limits are used to rigorously define the behavior of functions as the input approaches a particular value. In the context of derivatives, limits allow us to transition from the average rate of change to the instantaneous rate of change.

Instantaneous Rate of Change

This is the limit of the average rate of change as the interval shrinks to zero. It represents the derivative of the function at a specific point and indicates how the function changes at that very point.

Difference Quotient

The difference quotient is the expression (f(t + h) - f(t)) / h and is used to calculate the average rate of change of a function over a small interval. It serves as the foundational step for defining the derivative.

Average Rate of Change

This concept measures the change in a function's output relative to the change in its input over a specific interval. It is computed using the formula (f(b) - f(a)) / (b - a), reflecting how the function behaves on average between the two points.

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