chem 150

Chain Rule Usage: The derivative of the function arcsin(3x) involves applying the chain rule.
- The basic derivative formula is:
  \frac{d}{dx}(\text{arcsin}(x)) = \frac{1}{\sqrt{1 - x^2}}
- Under the chain rule, if the input is not just x but rather 3x, we need to apply the rule as follows:
- The derivative of arcsin(3x) becomes:
  \frac{1}{\sqrt{1 - (3x)^2}} \cdot \frac{d}{dx}(3x)
- This results in:
  \frac{3}{\sqrt{1 - 9x^2}}
- Implication: Multiply the outer function's derivative by the derivative of the inside function (3).

The focus shifts to sections 3.4 and 3.5, focusing on:
- Rates of change.
- Applications, particularly concerning motion problems, such as projectile motion.
- Importance of understanding first and second derivatives to analyze function behavior.
Concept of Higher Order Derivatives:
- Higher order derivatives, such as the fourth and subsequent derivatives, can reveal patterns in functions;
- Example: Derivatives of sine function exhibit a cyclical pattern every four derivatives.

Prime Notation:
- For first, second, and third derivatives use:
  - f'(x), f''(x), f'''(x) respectively.
- Beyond the third derivative, we use:
  - f^(n)(x) for the n-th derivative.
Leibniz Notation:
- Can also be expressed as:
  - \frac{d^n}{dx^n} for the n-th derivative.
Example Function:
- Generally, for f(x) = x^2 + 3ln(x):
  - f' = 2x + \frac{3}{x}
  - f'' = 2 - \frac{3}{x^2}
  - f''' = \frac{6}{x^3}

When modeling a projectile influenced only by gravity:
- Acceleration: Constant at:
- -a = -9.8 \, \text{m/s}^2 or -32 \, \text{ft/s}^2
Height Function:
- The general model for height, given constant acceleration, is a quadratic polynomial:
  - h(t) = pt^2 + qt + r where:
  - p varies based on gravitational acceleration, and is negative if the motion is downward.
  - q = initial velocity (v₀).
  - r = initial height (h₀).
Deriving velocity and acceleration from the height function:
- Velocity can be derived as:
- v(t) = h'(t) = 2pt + q
- Acceleration is derived as:
- a(t) = v'(t) = h''(t) = 2p

To derive initial velocity and height using a specific height function:
- Example: for height function h(t) = 24 + 40t - 16t², we analyze:
  - At t=0, h(0) = 24 (initial height).
  - Finding Initial Velocity: Use the first derivative, yielding 40 ft/s.

To determine when a projectile strikes the ground, set the height equation to zero:
- Solve quadratic equations to find time.
Velocity upon impact can be calculated using the derivative of the height function and plugging in the time found_.

The maximum height is found when the velocity function equals zero, indicating that the directed motion has ceased to ascend and is about to descend.
This can also be done using the vertex formula for quadratics, identifying the peak of a parabola.

Consider objects falling purely under gravitational influence:
- The outcome involves recognizing constant acceleration leading to behaviors that can be predicted from the polynomial nature of the governing equations.
Example Scenario with a Falling Piano:
- Using similar quadratic characteristics, determine the ground impact behavior and the effects of constant acceleration on height and velocity profiles.

The derivative behavior elucidates broader mathematical implications:
- For any polynomial higher than its degree, derivatives continuously lead to zero.
- Connections to continuous motion and acceleration provide substantive insight throughout calculus.

Increasing/decreasing intervals are established through the first derivative:
- If f'(x) > 0, f increases;
- If f'(x) < 0, f decreases.
In contrast, concavity is determined using second derivatives:
- If f''(x) > 0, function is concave up;
- If f''(x) < 0, function is concave down.

Visual assessment includes observing tangent slopes and concavity to identify increasing/decreasing behavior alongside their concavity.
A negative slope in f indicates a downward motion, leading to corresponding downward movements in derivatives reflecting these rates of change.