Calculus Study Notes: Position, Velocity, Acceleration, and Increasing/Decreasing Functions

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.

Formulas:

• Velocity: $v(t) = s'(t)$

• Speed: $|v(t)|$

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

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 $(a, b)$, if for every $x_1, x_2$ in $(a, b)$ with $x_1 < x_2$, $f(x_1) < f(x_2)$, then $f$ is increasing.

• Decreasing function: On an interval $(a, b)$, if for every $x_1, x_2$ in $(a, b)$ with $x_1 < x_2$, $f(x_1) > f(x_2)$, then $f$ is decreasing.

The derivative $f'(x)$ provides a test for increasing or decreasing behavior:

• If $f'(x) > 0$ on $(a, b)$, then $f$ is increasing on $(a, b)$.

• If $f'(x) < 0$ on $(a, b)$, then $f$ is decreasing on $(a, b)$.

Graphical Interpretation

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

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

Example intervals:

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

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

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

Examples and Applications

Projectile Motion Example

Example: A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot building. 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 the velocity function:

• $v(t) = s'(t) = -32t + 100$

• At $t = 7.0081$, $v(7.0081) = -32(7.0081) + 100$

Velocity and Acceleration Example

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

• Velocity: $v(t) = s'(t) = 24t^5 - 10t + 1$

• At $t = 2$, $v(2) = 24(2^5) - 10(2) + 1 = 768 - 20 + 1 = 749$

• Acceleration: $a(t) = s''(t) = 120t^4 - 10$

• At $t = 2$, $a(2) = 120(2^4) - 10 = 1920 - 10 = 1910$

A Polynomial Model: Application in Biology

Modeling Fetal Growth

Polynomial models are used in biology to estimate growth rates. For example, the femur length $L$ (in mm) of a fetus as a function of age $t$ (in weeks) can be modeled as:

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

To find the rate of growth, compute the derivative $\frac{dL}{dt}$:

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

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

Summary Table: Increasing/Decreasing Functions and Derivatives

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

Key Takeaways:

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

• Position, velocity, and acceleration are related through differentiation.

• Polynomial models and their derivatives are useful for analyzing rates of change in real-world applications.