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The stationary points can be found by solving the derivatives to zero. Some cubic equations have one root if the stationary points are in the same sign, such as two stationary points are either above the x-axis or below the x-axis. In these cases, the cubic graph cuts the x-axis only once, so the cubic […]
Distance Distance is the magnitude of the total movement from the start point or a fixed point. Displacement The displacement of a moving position relative to a fixed point. Displacement gives both the distance and direction that a particle is from a fixed point. For example, a particle moves \( 5 \) units forwards from […]
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 $f^{\prime}(x)$ to obtain the gratest function. Step 2: Solve for $x$ […]
Example 1 Find the maximum and minimum values of $y=x^2$, for $1 \le x \le 2$. From the graph, its maximum value is $4$, when $x=2$, and its minimum value is $1$, when $x=1$. Example 2 Find the maximum and minimum values of $y=x^2$, for $-2 \le x \le -1$. From the graph, its maximum […]
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 […]
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 […]
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 […]
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 $f^{\prime}(x)=0$ at the point. $$ \begin{array}{|c|c|c|} \hline f^{\prime}(x) \gt 0 & f'(x) = 0 & f'(x) \lt 0 \\ \hline & \text{maximum} & \\ […]
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 […]
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 […]