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Basic Differentiation Rules and Rates of Change
121
05. Velocity Verify that the average velocity over the time interval
[t_(0)-Delta t,t_(0)+Delta t] is the same as the instantaneous velocity at
t=t_(0) for the position function

Basic Differentiation Rules and Rates of Change $\newline$ $121$ $\newline$ $05$ . Velocity Verify that the average velocity over the time interval $\left[t_{0}-\Delta t, t_{0}+\Delta t\right]$ is the same as the instantaneous velocity at $t=t_{0}$ for the position function

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Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. Basic Differentiation Rules and Rates of Change

\newline

121

\newline

05

. Velocity Verify that the average velocity over the time interval

\left[t_{0}-\Delta t, t_{0}+\Delta t\right]

is the same as the instantaneous velocity at

t=t_{0}

for the position function

Define Average Velocity Formula: To find the average velocity, we need to use the formula: Average Velocity = $\frac{\text{Position at final time} - \text{Position at initial time}}{\text{Final time} - \text{Initial time}}$ .
Calculate Average Velocity Interval: Let's denote the position function as $s(t)$ . The average velocity over the interval $[t_{0}-\Delta t, t_{0}+\Delta t]$ is $\frac{s(t_{0}+\Delta t) - s(t_{0}-\Delta t)}{(t_{0}+\Delta t) - (t_{0}-\Delta t)}$ .
Simplify Denominator: Simplify the denominator: $(t_{0}+\Delta t) - (t_{0}-\Delta t) = 2\Delta t$ .
Plug in Simplified Denominator: Now, plug in the simplified denominator into the average velocity formula: Average Velocity = $\frac{s(t_{0}+\Delta t) - s(t_{0}-\Delta t)}{2\Delta t}$ .
Find Instantaneous Velocity: To find the instantaneous velocity at $t=t_{0}$ , we need to take the derivative of the position function $s(t)$ with respect to time $t$ and then evaluate it at $t=t_{0}$ .
Apply Mean Value Theorem: Let's denote the derivative of $s(t)$ as $s'(t)$ . The instantaneous velocity at $t=t_{0}$ is $s'(t_{0})$ .
Approach Instantaneous Velocity: If the position function $s(t)$ is differentiable, then by the Mean Value Theorem, there exists some point $c$ in $(t_{0}-\Delta t, t_{0}+\Delta t)$ such that $s'(c)$ is equal to the average velocity over that interval.
Equivalence of Average and Instantaneous Velocity: Since we are looking at the interval shrinking to a single point $t_{0}$ , as $\Delta t$ approaches $0$ , the point $c$ approaches $t_{0}$ , and thus the average velocity approaches the instantaneous velocity at $t=t_{0}$ .
Equivalence of Average and Instantaneous Velocity: Since we are looking at the interval shrinking to a single point $t_{0}$ , as $\Delta t$ approaches $0$ , the point $c$ approaches $t_{0}$ , and thus the average velocity approaches the instantaneous velocity at $t=t_{0}$ .Therefore, the average velocity over the interval $[t_{0}-\Delta t, t_{0}+\Delta t]$ is the same as the instantaneous velocity at $t=t_{0}$ for the position function $s(t)$ .

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