Find the slope of the curve at the indicated point.
-Assume that a watermelon dropped from a tall building falls $y = 16 \mathrm { t } ^ { 2 } \mathrm { ft }$ in $\mathrm { t }$ sec. Find the watermelon's average speed during the first $4 \sec$ of fall and the speed at the instant $t = 4 \mathrm { sec }$.
A) $128 \mathrm { ft } / \mathrm { sec } ; 65 \mathrm { ft } / \mathrm { sec }$
B) $32 \mathrm { ft } / \mathrm { sec } ; 64 \mathrm { ft } / \mathrm { sec }$
C) $64 \mathrm { ft } / \mathrm { sec } ; 128 \mathrm { ft } / \mathrm { sec }$
D) $65 \mathrm { ft } / \mathrm { sec } ; 130 \mathrm { ft } / \mathrm { sec }$

Question

Quizplus · Accepted Answer

The average speed during the first 4 seconds is calculated by the total distance divided by time, which is $y(4) = 16(4)^2 = 256 \, \text{ft}$ divided by 4 seconds, giving $64 \, \text{ft/sec}$. The speed at the instant $t = 4 \, \text{sec}$ is found by differentiating $y = 16t^2$ to get $y' = 32t$, and then substituting $t = 4$ to get $32(4) = 128 \, \text{ft/sec}$.

Find the Slope of the Curve at the Indicated Point y=16t2fty = 16 \mathrm { t } ^ { 2 } \mathrm { ft }y=16t2ft

Find the Slope of the Curve at the Indicated Point $y = 16 \mathrm { t } ^ { 2 } \mathrm { ft }$