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Suppose $d(t) = 40t$ represents miles traveled after $t$ hours.

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Average velocity is $\underline{\hskip 1cm}$

Instantaneous velocity at $t = t_0$ is $\underline{\hskip 1cm}$
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Suppose $d(t) = t^2$ represents miles traveled after $t$ hours.

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$\matrix{
t_0 & \hbox{change in time} & \hbox{change in distance} & \hbox{average 
velocity} \cr
& \hbox{btwn } t = 0 \hbox{ and } t = t_0 
& \hbox{btwn } t = 0 \hbox{ and } t = t_0 
& \hbox{btwn } t = 0 \hbox{ and } t = t_0 \cr
~ & ~ & ~ & \cr
2 & 2 - 0 & 2^2 - 0^2 & { 2^2 - 0^2 \over 2 - 0} = 2
~ & ~ & ~ & \cr
1 & 1 - 0 & 1^2 - 0^2 & { 1^2 - 0^2 \over 1 - 0} = 1
~ & ~ & ~ & \cr
.5 & .5 - 0 & (.5)^2 - 0^2 & { (.5)^2 - 0^2 \over .5 - 0} = .5
~ & ~ & ~ & \cr
.1 & .1 - 0 & (.1)^2 - 0^2 & { (.1)^2 - 0^2 \over .1 - 0} = .1
~ & ~ & ~ & \cr
.01 & .01 - 0 & (.01)^2 - 0^2 & { (.01)^2 - 0^2 \over .01 - 0} = .01
~ & ~ & ~ & 
}$
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Instantaneous velocity at $t = 0$ is $\underline{\hskip 1cm}$

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Suppose $d(t) = t^2$ represents miles traveled after $t$ hours.

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$\matrix{
t_0 & \hbox{change in time} & \hbox{change in distance} & \hbox{average 
velocity} \cr
& \hbox{btwn } t = 2 \hbox{ and } t = t_0 
& \hbox{btwn } t = 2 \hbox{ and } t = t_0 
& \hbox{btwn } t = 2 \hbox{ and } t = t_0 \cr
~ & ~ & ~ & \cr
4 & 4 - 2 & 4^2 - 2^2 & { 4^2 - 2^2 \over 4 - 2} = 6
~ & ~ & ~ & \cr
3 & 3 - 2 & 3^2 - 2^2 & { 3^2 - 2^2 \over 3 - 2} = 5
~ & ~ & ~ & \cr
2.5 & 2.5 - 2 & (2.5)^2 - 2^2 & { (2.5)^2 - 2^2 \over 2.5 - 2} = 4.5
~ & ~ & ~ & \cr
2.1 & 2.1 - 2 & (2.1)^2 - 2^2 & { (2.1)^2 - 2^2 \over 2.1 - 2} = 4.1
~ & ~ & ~ & \cr
1.9 & 1.9 - 2 & (1.9)^2 - 2^2 & { (1.9)^2 - 2^2 \over 1.9 - 2} = 3.9
~ & ~ & ~ & \cr
1.5 & 1.5 - 2 & (1.5)^2 - 2^2 & { (1.5)^2 - 2^2 \over 1.5 - 2} = 3.5
~ & ~ & ~ & \cr
1 & 1 - 2 & 1^2 - 2^2 & { 1^2 - 2^2 \over 1 - 2} = 3
~ & ~ & ~ & \cr
}$
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Instantaneous velocity at $t = 2$ is $\underline{\hskip 1cm}$
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SLOPE OF SECANT LINE = AVERAGE VELOCITY

SLOPE OF TANGENT LINE = INSTANTANEOUS VELOCITY


in general, SLOPE = RATE OF CHANGE

SLOPE OF SECANT LINE = AVERAGE RATE OF CHANGE

SLOPE OF TANGENT LINE = INSTANTANEOUS RATE OF CHANGE


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