Calculus - Applications of Differentiation (Tangents, Normals, Optimization, Kinematics)

The Gradient as a Derivative: The derivative $\frac{dy}{dx}$ represents the instantaneous rate of change of a function. Geometrically, this is the gradient of the tangent to the curve at any given point $(x, y)$. If the graph is viewed as a path, the tangent line indicates the direction of travel at that specific moment.

Tangents and Normals: A tangent is a straight line that just touches a curve at a point, sharing the same gradient as the curve at that point. A normal is a line perpendicular to the tangent at the point of contact. On a graph, the normal and tangent intersect at a $90^{\circ}$ angle, and their gradients are negative reciprocals of each other.

Increasing and Decreasing Functions: A function is strictly increasing on an interval if $f'(x) > 0$, shown visually as the graph rising from left to right. Conversely, a function is strictly decreasing if $f'(x) < 0$, shown as the graph falling from left to right. Stationary points occur where $f'(x) = 0$, where the graph momentarily 'flattens out'.

Local Extrema and Stationary Points: Local maximums and minimums occur where the derivative is zero or undefined. A local maximum appears as a 'peak' on a graph where the gradient changes from positive to negative. A local minimum appears as a 'valley' where the gradient changes from negative to positive. The Second Derivative Test can confirm this: $f''(x) < 0$ indicates a maximum (concave down), while $f''(x) > 0$ indicates a minimum (concave up).

Concavity and Points of Inflection: Concavity describes the 'bend' of a curve. A curve is concave up (like a cup) if $f''(x) > 0$ and concave down (like a cap) if $f''(x) < 0$. A point of inflection is a point where the concavity changes (e.g., from cup-shaped to cap-shaped). At these points, $f''(x) = 0$ and there is a sign change in the second derivative.

Optimization: This involves finding the absolute maximum or minimum value of a function within a specific context, such as maximizing the area of a shape or minimizing the cost of production. Visually, this requires identifying the highest or lowest point on the function's graph within the feasible domain, often by checking stationary points and interval endpoints.

Kinematics of a Particle: Differentiation is used to relate displacement $s(t)$, velocity $v(t)$, and acceleration $a(t)$. Velocity is the rate of change of displacement $v(t) = s'(t)$, and acceleration is the rate of change of velocity $a(t) = v'(t) = s''(t)$. On a displacement-time graph, the gradient represents velocity; on a velocity-time graph, the gradient represents acceleration.