Calculus Applications Classrooms

54.1 Rates of Change

One of the most powerful applications of differentiation is to find the rate at which a quantity changes with respect to another. If a quantity $y$ is a function of time $t$, i.e., $y = f(t)$, then $\frac{dy}{dt}$ represents the instantaneous rate of change of $y$ with respect to $t$.

This concept is widely used in physics (velocity, acceleration), economics (marginal cost, marginal revenue), and biology (population growth rates).

54.2 Velocity & Acceleration

In kinematics, if the displacement of an object is given by a function $s(t)$, where $t$ is time:

• The velocity ($v(t)$) is the first derivative of displacement with respect to time: $$ v(t) = \frac{ds}{dt} $$
• The acceleration ($a(t)$) is the first derivative of velocity with respect to time (or the second derivative of displacement): $$ a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2} $$

These relationships allow us to analyze the motion of objects by calculating their instantaneous speed and how their speed changes over time.

54.3 Turning Points (Stationary Points)

A turning point (also known as a stationary point or critical point) of a curve is a point where the gradient of the curve is zero. At these points, the tangent line is horizontal.

Turning points can be:

• Local Maxima: The function reaches a peak in a local region. The gradient changes from positive to zero to negative.
• Local Minima: The function reaches a trough in a local region. The gradient changes from negative to zero to positive.
• Stationary Points of Inflexion: The gradient is zero, but the curve does not change direction (e.g., $y=x^3$ at $x=0$).

To find turning points, set the first derivative $\frac{dy}{dx}$ to zero and solve for $x$.

54.4 Maxima & Minima Problems (Optimization)

Differentiation is extensively used to solve optimization problems, where the goal is to find the maximum or minimum value of a quantity (e.g., maximizing profit, minimizing cost, finding the largest area, or smallest volume).

The general steps involve:

1. Formulate the quantity to be optimized as a function of one variable.
2. Differentiate the function and set the derivative equal to zero to find the stationary points.
3. Use the first or second derivative test to determine if these points are maxima or minima.
4. Verify the answer with respect to the problem's context (e.g., check endpoints or physical constraints).

54.5 Points of Inflexion

A point of inflexion (or inflection point) is a point on a curve where the concavity changes. This means the curve changes from being "concave up" (like a cup) to "concave down" (like a frown), or vice-versa.

At a point of inflexion, the second derivative $\frac{d^2y}{dx^2}$ is typically zero or undefined. However, a second derivative of zero is a necessary but not sufficient condition; you must also check that the sign of the second derivative changes across the point.

Points of inflexion indicate where the rate of change of the gradient itself changes (e.g., where acceleration is zero).

54.6 Tangents & Normals

Differentiation can be used to find the equations of tangent and normal lines to a curve at a specific point $(x_1, y_1)$:

• The tangent line touches the curve at that single point and has a gradient equal to the derivative of the curve at that point: $$ m_{\text{tangent}} = \frac{dy}{dx} \Big|_{(x_1, y_1)} $$ Equation of tangent: $y - y_1 = m_{\text{tangent}}(x - x_1)$
• The normal line is perpendicular to the tangent line at the point of contact. Its gradient is the negative reciprocal of the tangent's gradient: $$ m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}} $$ Equation of normal: $y - y_1 = m_{\text{normal}}(x - x_1)$

54.7 Small Changes (Approximations)

For a small change $\delta x$ in $x$, the corresponding small change $\delta y$ in $y$ (where $y = f(x)$) can be approximated using the derivative:

This approximation is based on the idea that for a very small interval, the curve can be approximated by its tangent line. This is particularly useful for estimating errors or changes in a quantity when the change in an independent variable is small.