Distance Distance is the magnitude of the total movement from the start point or a fixed point. For example, a particle moves \( 5 \) units forwards from \( O \), and moves \( 3 \) backwards. Its distance travelled is \( 5 + 3= 8 \) units. Displacement The displacement of a moving position […]

# Differentiation

# Maxima and Minima with Trigonometric Functions

Periodic motions can be modelled by a trigonometric equation. By differentiating these functions we are then able to solve problems relating to maxima (maximums) and minima (minimums). Remember that the following steps are used when solving a maximum or minimum problem. Step 1: Find $f'(x)$ to obtain the gratest function. Step 2: Solve for $x$ [...]

# Optimisation

There are many cases for which we need to identify the maximum or minimum value for a situation. The solution is often referred to as the optimum solution and the process is called optimisation. There are several ways of finding optimum solutions including: graphing the function accurately and search for the maximum or minimum from [...]

# Rates of Change

Assume $x(t)$ is a displacement, then $x'(t)$ or $\dfrac{dx}{dt}$ is the instantaneous rate of change in displacement with respect to time, which is velocity. Example where quantities vary with time, or with respect to some other value. temperature changes height of the surface of water in a pond speed of a car $\dfrac{dy}{dx}$ gives the [...]

# Velocity and Acceleration

If a particle $P$ moves in a straight line and its position is given by the displacement function $x(t)$, then: the velocity of $P$ at time $t$ is given by $v(t) = x'(t)$ the acceleration of $P$ at time $t$ is given by $a(t)=v'(t)=x^{\prime \prime}(t)$ $x(0)$, $v(0)$ and $a(0)$ give the position, velocity and acceleration [...]

# Motion Kinematics

Displacement Suppose an object $P$ moves along a straight line so that its position $s$ from an origin $O$ is given as some function of time $t$. We write $x=x(t)$ where $t \ge 0$. $x(t)$ is a displacement function and for any value of $t$ it gives the displacement from the origin. On the horizontal [...]

# Inflection Points (Points of Inflection)

Horizontal (stationary) point of inflection (inflection point) If $x \lt a$, then $f'(x) \gt 0$ and $f^{\prime \prime}(x) \le 0 \rightarrow$ concave down. If $x = a$, then $f'(x) = 0$ and $f^{\prime \prime}(x) = 0 \rightarrow$ horizontal point inflection. If $x \gt a$, then $f'(x) \gt 0$ and $f^{\prime \prime}(x) \ge 0 \rightarrow$ concave [...]

# Turning Points and Nature

A turning point of a function is a point where $f'(x)=0$. A maximum turning point is a turning point where the curve is concave up (from increasing to decreasing ) and $f'(x)=0$ at the point. $$ \begin{array}{|c|c|c|} \hline f'(x) \gt 0 & f'(x) = 0 & f'(x) \lt 0 \\ \hline & \text{maximum} & \\ [...]

# Increasing Functions and Decreasing Functions

Increasing and Decreasing We can determine intervals where a curve is increasing or decreasing by considering $f'(x)$ on the interval in question. $f'(x) \gt 0$: $f(x)$ is increasing $f'(x) \lt 0$: $f(x)$ is decreasing Monotone (Monotonic) Increasing or Decreasing Many functions are either increasing or decreasing for all $x \in \mathbb{R}$. These functions are called [...]

# Finding the Normal Equation

A normal to a curve is a straight line passing through the point where the tangent touches the curve and is perpendicular (at right angles) to the tangent at that point. The gradient of the tangent to a curve is $m$, then the gradient of the normal is $\displaystyle -\dfrac{1}{m}$, as the product of the [...]