Position, Velocity, and Acceleration Using Integrals - AP Calculus AB
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What is the derivative of position with respect to time?
What is the derivative of position with respect to time?
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Velocity. Rate of change of position is velocity by definition.
Velocity. Rate of change of position is velocity by definition.
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If $a(t) = 0$, what can be said about $v(t)$?
If $a(t) = 0$, what can be said about $v(t)$?
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$v(t)$ is a constant function. Zero acceleration implies constant velocity function.
$v(t)$ is a constant function. Zero acceleration implies constant velocity function.
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What does the area under a velocity-time graph represent?
What does the area under a velocity-time graph represent?
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Displacement. Area under curve equals change in position.
Displacement. Area under curve equals change in position.
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Given $s(t) = 4t^2 - 2t + 1$, find $v(t)$ and $a(t)$.
Given $s(t) = 4t^2 - 2t + 1$, find $v(t)$ and $a(t)$.
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$v(t) = 8t - 2$, $a(t) = 8$. Differentiate position once for $v(t)$, twice for $a(t)$.
$v(t) = 8t - 2$, $a(t) = 8$. Differentiate position once for $v(t)$, twice for $a(t)$.
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What is the initial condition used for when integrating velocity to find position?
What is the initial condition used for when integrating velocity to find position?
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To find the constant of integration. Determines the arbitrary constant $C$ from integration.
To find the constant of integration. Determines the arbitrary constant $C$ from integration.
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How do you find average velocity over an interval $[a, b]$?
How do you find average velocity over an interval $[a, b]$?
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Average velocity = $\frac{s(b) - s(a)}{b - a}$. Displacement divided by time elapsed.
Average velocity = $\frac{s(b) - s(a)}{b - a}$. Displacement divided by time elapsed.
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If velocity is constant, what is acceleration?
If velocity is constant, what is acceleration?
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Zero. Derivative of constant velocity is zero.
Zero. Derivative of constant velocity is zero.
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Given $v(t) = 6t - 3t^2$, find $s(t)$ if $s(0) = 5$.
Given $v(t) = 6t - 3t^2$, find $s(t)$ if $s(0) = 5$.
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$s(t) = 3t^2 - t^3 + 5$. Integrate velocity and use initial condition.
$s(t) = 3t^2 - t^3 + 5$. Integrate velocity and use initial condition.
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If $s(t) = t^2 - 3t + 2$, what is $v(t)$?
If $s(t) = t^2 - 3t + 2$, what is $v(t)$?
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$v(t) = 2t - 3$. Differentiate position function using power rule.
$v(t) = 2t - 3$. Differentiate position function using power rule.
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Given $v(t) = 5t - 4$, find the integral to determine position.
Given $v(t) = 5t - 4$, find the integral to determine position.
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Integrate $v(t)$ to get $s(t) = \frac{5t^2}{2} - 4t + C$. Apply antiderivative rules for polynomial functions.
Integrate $v(t)$ to get $s(t) = \frac{5t^2}{2} - 4t + C$. Apply antiderivative rules for polynomial functions.
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If $v(t) = t^3 - 6t^2 + 9t$, find $a(t)$.
If $v(t) = t^3 - 6t^2 + 9t$, find $a(t)$.
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$a(t) = 3t^2 - 12t + 9$. Take the derivative of velocity using power rule.
$a(t) = 3t^2 - 12t + 9$. Take the derivative of velocity using power rule.
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Given $v(t) = 4t - 7$, find the acceleration $a(t)$.
Given $v(t) = 4t - 7$, find the acceleration $a(t)$.
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$a(t) = 4$. Take the derivative of velocity function.
$a(t) = 4$. Take the derivative of velocity function.
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If $a(t) = 3$, find $v(t)$ given $v(0) = 5$.
If $a(t) = 3$, find $v(t)$ given $v(0) = 5$.
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$v(t) = 3t + 5$. Integrate constant acceleration and apply initial condition.
$v(t) = 3t + 5$. Integrate constant acceleration and apply initial condition.
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Given $s(t) = 2t^3 - 5t^2 + 3t + 1$, find $v(t)$.
Given $s(t) = 2t^3 - 5t^2 + 3t + 1$, find $v(t)$.
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$v(t) = 6t^2 - 10t + 3$. Take the derivative of position function.
$v(t) = 6t^2 - 10t + 3$. Take the derivative of position function.
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How do you find total distance traveled from a velocity function $v(t)$?
How do you find total distance traveled from a velocity function $v(t)$?
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Integrate $|v(t)|$ over the interval. Absolute value accounts for direction changes.
Integrate $|v(t)|$ over the interval. Absolute value accounts for direction changes.
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What is the integral of zero acceleration over time?
What is the integral of zero acceleration over time?
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Constant velocity. Integrating zero gives constant velocity.
Constant velocity. Integrating zero gives constant velocity.
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If $s(t) = 4t - \text{ln}(t)$, what is $v(t)$?
If $s(t) = 4t - \text{ln}(t)$, what is $v(t)$?
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$v(t) = 4 - \frac{1}{t}$. Differentiate using power and logarithmic rules.
$v(t) = 4 - \frac{1}{t}$. Differentiate using power and logarithmic rules.
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If $v(t) = 0$ for all $t$, what can be said about $s(t)$?
If $v(t) = 0$ for all $t$, what can be said about $s(t)$?
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$s(t)$ is constant. Zero velocity means no change in position.
$s(t)$ is constant. Zero velocity means no change in position.
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What is the implication of $a(t) = 0$ over an interval?
What is the implication of $a(t) = 0$ over an interval?
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Velocity is constant. Zero acceleration means no change in velocity.
Velocity is constant. Zero acceleration means no change in velocity.
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If $v(t) = 3t$, find $s(t)$ given $s(0) = 0$.
If $v(t) = 3t$, find $s(t)$ given $s(0) = 0$.
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$s(t) = \frac{3t^2}{2}$. Integrate velocity and apply zero initial position.
$s(t) = \frac{3t^2}{2}$. Integrate velocity and apply zero initial position.
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What condition must be checked when integrating to determine total distance traveled?
What condition must be checked when integrating to determine total distance traveled?
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Check for sign changes in $v(t)$. Must split integral where velocity changes sign.
Check for sign changes in $v(t)$. Must split integral where velocity changes sign.
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If $v(t) = 2t + 1$, what is the integral to find total distance?
If $v(t) = 2t + 1$, what is the integral to find total distance?
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Integrate $|2t + 1|$ over the interval. Use absolute value when velocity changes sign.
Integrate $|2t + 1|$ over the interval. Use absolute value when velocity changes sign.
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How do you determine if an object is speeding up or slowing down?
How do you determine if an object is speeding up or slowing down?
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Check if $v(t)$ and $a(t)$ have the same sign. Same sign means speeding up, opposite means slowing down.
Check if $v(t)$ and $a(t)$ have the same sign. Same sign means speeding up, opposite means slowing down.
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What is the physical interpretation of $v(t) = 0$?
What is the physical interpretation of $v(t) = 0$?
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Object is momentarily at rest. Zero velocity means no instantaneous motion.
Object is momentarily at rest. Zero velocity means no instantaneous motion.
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What is the formula for displacement from time $t_1$ to $t_2$?
What is the formula for displacement from time $t_1$ to $t_2$?
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$s(t_2) - s(t_1) = \text{integral of } v(t) \text{ from } t_1 \text{ to } t_2$. Net change in position equals integral of velocity.
$s(t_2) - s(t_1) = \text{integral of } v(t) \text{ from } t_1 \text{ to } t_2$. Net change in position equals integral of velocity.
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If $a(t) = 8 - 2t$, find $v(t)$ given $v(0) = 0$.
If $a(t) = 8 - 2t$, find $v(t)$ given $v(0) = 0$.
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$v(t) = 8t - t^2$. Integrate acceleration and apply zero initial velocity.
$v(t) = 8t - t^2$. Integrate acceleration and apply zero initial velocity.
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For $v(t) = 2t^3 + 3$, find the acceleration $a(t)$.
For $v(t) = 2t^3 + 3$, find the acceleration $a(t)$.
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$a(t) = 6t^2$. Differentiate velocity using power rule.
$a(t) = 6t^2$. Differentiate velocity using power rule.
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What is the algebraic expression for displacement?
What is the algebraic expression for displacement?
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$\text{integral of } v(t) \text{ from } t_1 \text{ to } t_2$. Definite integral of velocity over time interval.
$\text{integral of } v(t) \text{ from } t_1 \text{ to } t_2$. Definite integral of velocity over time interval.
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What is the significance of the constant of integration when finding $s(t)$?
What is the significance of the constant of integration when finding $s(t)$?
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It represents initial position. The constant $C$ represents position at $t = 0$.
It represents initial position. The constant $C$ represents position at $t = 0$.
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If $s(t) = \text{ln}(t)$, what is $v(t)$?
If $s(t) = \text{ln}(t)$, what is $v(t)$?
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$v(t) = \frac{1}{t}$. Differentiate logarithmic function.
$v(t) = \frac{1}{t}$. Differentiate logarithmic function.
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