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 $t=2 ?$ d. Calculate the limit as $T$ approaches 2 of the average rate of change of $f$ with respect to $t$ over the interval from 2 to $T$. You will have to do some algebra before you can substitute $T=2$

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 $t=2 ?$
d. Calculate the limit as $T$ approaches 2 of the average rate of change of $f$ with respect to $t$ over the interval from 2 to $T$. You will have to do some algebra before you can substitute $T=2$

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Key Concepts

Instantaneous Rate of Change (Derivative)

The instantaneous rate of change of a function at a specific point is defined as the limit of the average rate of change (difference quotient) as the interval between the two points approaches zero. This limit, when it exists, is the derivative of the function at that point, representing the slope of the tangent line to the function's graph.

Limits

The notion of a limit is used to analyze the behavior of a function as the input approaches a particular value. In the context of the average rate of change, limits allow us to investigate what happens as the interval between two points shrinks, especially as one endpoint approaches a given value.

Average Rate of Change

The average rate of change of a function between two points is the ratio of the change in the function's value to the change in the independent variable over that interval. It is essentially the slope of the secant line that connects the two points on the function's graph and is computed using the difference quotient.

Difference Quotient

The difference quotient is an expression that represents the average rate of change of a function over an interval. It is given by the formula (f(b) - f(a)) / (b - a) for a function evaluated at two points a and b. This concept is fundamental in understanding how functions change over intervals.

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