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Movement Functions with Integrals: AP Calculus AB Rundown

This document provides a concise overview of movement functions and their relationship to integrals, as relevant to AP Calculus AB.

Position, Velocity, and Acceleration

These three concepts are fundamental when dealing with movement. They are related through differentiation and integration.

• Position: Denoted by $ s(t) $ or $ x(t) $ . It represents the location of an object at a given time $ t $ .

• Velocity: Denoted by $ v(t) $ . It represents the rate of change of position with respect to time. Velocity is the derivative of position:

$$ v(t) = \frac{ds}{dt} = s’(t) $$

• Acceleration: Denoted by $ a(t) $ . It represents the rate of change of velocity with respect to time. Acceleration is the derivative of velocity (and the second derivative of position):

$$ a(t) = \frac{dv}{dt} = v’(t) = s’’(t) $$

Integration and Movement

Integration allows us to move “backwards” in the position-velocity-acceleration hierarchy. Specifically:

• Finding Velocity from Acceleration: Given the acceleration function $ a(t) $ and the initial velocity $ v(0) $ , we can find the velocity function $ v(t) $ by integrating $ a(t) $ :

$$ v(t) = v(0) + \int_{0}^{t} a(\tau) , d\tau $$
Note that we are using $ \tau $ as the variable of integration here to avoid confusion with the upper limit, $ t $ . Understanding Dummy Variables

• Finding Position from Velocity: Given the velocity function $ v(t) $ and the initial position $ s(0) $ , we can find the position function $ s(t) $ by integrating $ v(t) $ :

$$ s(t) = s(0) + \int_{0}^{t} v(\tau) , d\tau $$

Displacement vs. Total Distance

This is a critical distinction.

• Displacement: The change in position of an object over a given time interval. It’s calculated by integrating the velocity function over that interval:

$$ \text{Displacement} = \int_{a}^{b} v(t) , dt = s(b) - s(a) $$
Displacement can be positive, negative, or zero. A positive displacement means the object moved to the right (or up, depending on the context), a negative displacement means it moved to the left (or down), and a zero displacement means it ended up at the same position where it started.

• Total Distance: The total length of the path traveled by an object over a given time interval. It’s calculated by integrating the absolute value of the velocity function over that interval:

$$ \text{Total Distance} = \int_{a}^{b} |v(t)| , dt $$
The absolute value ensures that we’re always adding up positive distances, regardless of the direction of movement. To evaluate this integral, you’ll need to determine where $ v(t) $ is positive and negative on the interval $ [a, b] $ . Then, split the integral into intervals where $ v(t) $ has a constant sign, and negate $ v(t) $ where it’s negative:

$$ \int_{a}^{b} |v(t)| , dt = \int_{a}^{c} v(t) , dt - \int_{c}^{d} v(t) , dt + \int_{d}^{b} v(t) , dt $$
Where $ v(t) \ge 0 $ on $ [a, c] $ , $ v(t) \le 0 $ on $ [c, d] $ , and $ v(t) \ge 0 $ on $ [d, b] $ . This means $ v(c) = v(d) = 0 $ .

Average Velocity and Average Speed

• Average Velocity: The displacement divided by the time interval:

$$ \text{Average Velocity} = \frac{s(b) - s(a)}{b - a} = \frac{1}{b - a} \int_{a}^{b} v(t) , dt $$
This is also the average value of the velocity function over the interval $ [a, b] $ . Average of a function

• Average Speed: The total distance traveled divided by the time interval:

$$ \text{Average Speed} = \frac{\text{Total Distance}}{b - a} = \frac{1}{b - a} \int_{a}^{b} |v(t)| , dt $$

Key Concepts and Cautions

• Initial Conditions: Don’t forget to use initial conditions ( $ s(0) $ and $ v(0) $ ) when finding position and velocity functions. These are crucial for determining the constant of integration.

• Units: Pay attention to the units of measurement (e.g., meters, seconds, meters/second, meters/second²).

• Direction: Velocity is a vector quantity (it has both magnitude and direction). Speed is a scalar quantity (it only has magnitude). The sign of the velocity indicates direction.

• When $ v(t) = 0 $ : When the velocity is zero, the object is momentarily at rest. This is often a turning point, where the object changes direction. Finding these points is essential for calculating total distance. Finding Roots

Understanding Dummy Variables

In the equations: $$ v(t) = v(0) + \int_{0}^{t} a(\tau) , d\tau $$ $$ s(t) = s(0) + \int_{0}^{t} v(\tau) , d\tau $$
The variable $ \tau $ is a “dummy variable”. It’s just a placeholder variable used within the integral. The important thing is that the limits of integration are in terms of time ( $ 0 $ to $ t $ ). The variable of integration inside the integral ( $ a(\tau) $ or $ v(\tau) $ ) doesn’t affect the final result, as long as it’s consistent with the differential ( $ d\tau $ ). We use a different variable than $ t $ to avoid confusion between the upper limit of integration and the variable in the function we are defining.

Average of a function

The Average Value Theorem states that if a function $ f(x) $ is continuous on the closed interval $ [a, b] $ , then there exists a number $ c $ in the interval $ (a, b) $ such that:

$$ f(c) = \frac{1}{b-a} \int_a^b f(x) , dx $$
In the context of velocity, the average velocity over the interval $ [a,b] $ is the average value of the velocity function $ v(t) $ over that interval.

To find the roots (or zeros) of a function, such