Position, Velocity, Acceleration, and Increasing/Decreasing Functions in Calculus

Position, Velocity, and Acceleration

Definitions and Relationships

In calculus, the concepts of position, velocity, and acceleration are fundamental for describing the motion of objects. These quantities are related through differentiation:

• Position function s(t): Describes the location of an object at time t.

• Velocity function v(t): The rate of change of position with respect to time. It is the first derivative of the position function.

• Acceleration function a(t): The rate of change of velocity with respect to time. It is the second derivative of the position function.

The relationships can be written as:

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

The speed of an object is the absolute value of its velocity: $|v(t)|$.

Example: Vertical Motion

• A potato is launched vertically upward with an initial velocity of 100 ft/s from a platform 85 ft above the ground. The position function is $s(t) = -16t^2 + 100t + 85$.

• To find when the potato hits the ground, set $s(t) = 0$ and solve for $t$:

$0 = -16t^2 + 100t + 85$

$t = \frac{-100 \pm \sqrt{(100)^2 - 4(-16)(85)}}{-32}$

• To find the velocity at impact, substitute the value of $t$ into $v(t) = s'(t) = -32t + 100$.

Example: General Position Function

• Given $s(t) = 4t^5 - 5t^2 + t - 6$, find velocity and acceleration at $t = 2$.

• Velocity: $v(t) = s'(t) = 20t^4 - 10t + 1$

• At $t = 2$: $v(2) = 20(2^4) - 10(2) + 1 = 320 - 20 + 1 = 301$

• Acceleration: $a(t) = s''(t) = 80t^3 - 10$

• At $t = 2$: $a(2) = 80(8) - 10 = 640 - 10 = 630$

Increasing and Decreasing Functions

Definitions

A function's behavior (increasing or decreasing) on an interval is determined by the sign of its derivative:

• Increasing function: On an interval $(a, b)$, $f(x)$ is increasing if for any $x_1 < x_2$ in $(a, b)$, $f(x_1) < f(x_2)$. Equivalently, $f'(x) > 0$ on $(a, b)$.

• Decreasing function: On an interval $(a, b)$, $f(x)$ is decreasing if for any $x_1 < x_2$ in $(a, b)$, $f(x_1) > f(x_2)$. Equivalently, $f'(x) < 0$ on $(a, b)$.

Graphical Interpretation

• For an increasing function, the slope of the tangent line at any point is positive.

• For a decreasing function, the slope of the tangent line at any point is negative.

Mathematically, the slope between two points $x_1$ and $x_2$ is:

$\text{slope} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• If $f(x_2) > f(x_1)$, the slope is positive (increasing).

• If $f(x_2) < f(x_1)$, the slope is negative (decreasing).

Example: Piecewise Behavior

• Suppose $f(x)$ behaves as follows:

• For $a < x < b$, $f$ is increasing, $f'(x) > 0$

• For $b < x < c$, $f$ is decreasing, $f'(x) < 0$

• For $c < x < d$, $f$ is increasing, $f'(x) > 0$

Polynomial Model: Application to Biology

Modeling Growth with Polynomials

Polynomials are often used to model real-world phenomena, such as biological growth. For example, the femur length $L$ (in mm) of a fetus as a function of age $t$ (in weeks) can be modeled by:

$L = -37.50 + 3.71t - 6.33 \times 10^{-4} t^3$

• To find the rate of growth at a specific time, compute the derivative $\frac{dL}{dt}$ and evaluate at the desired $t$ values (e.g., $t = 10, 20, 30$ weeks).

Example: Rate of Growth

• Differentiate $L$ with respect to $t$:

$\frac{dL}{dt} = 3.71 - 3 \times 6.33 \times 10^{-4} t^2$

• Evaluate at $t = 10, 20, 30$ to analyze how the rate of growth changes as time increases.

Summary Table: Derivatives and Function Behavior

Interval	Sign of $f'(x)$	Function Behavior
$a < x < b$	$f'(x) > 0$	Increasing
$b < x < c$	$f'(x) < 0$	Decreasing
$c < x < d$	$f'(x) > 0$	Increasing

Key Takeaways

• The sign of the derivative determines whether a function is increasing or decreasing on an interval.

• Position, velocity, and acceleration are related through differentiation and are essential for modeling motion.

• Polynomial models and their derivatives can be used to analyze rates of change in real-world contexts, such as biological growth.