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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 […]
Consider a curve $y=f(x)$. A tangent to a curve is a straight line which touches the curve at a given point and represents the gradient of the curve at that point.If $A$ is the point with $x$-coordinate $a$, then the gradient of the tangent line to the curve at this point is $f'(a)$. The equation […]