7. The level h in centimetres of water varies with time t in hours after 5 a.m. (t = 0) according to the equation h = 16 t 2 − 180t + 600, 0 ≤ t ≤ 12. Find the minimum value of the level of the water and when it occurs, to the nearest minute.
8. A cannon fi res a shell along the path given by the formula h = x(40 – x), where h is the height of the shell above the ground in metres and x is its horizontal distance in metres from O(0, 0). The shell lands at B.
h(m) Height
O (a) Find:
(i) how far from O it lands, (ii) its maximum height above the ground.
(b) Find the slope of the tangent at (i) O and (ii) B.
Curve sketching The most important points in sketching a curve are the local maxima and minima.
Steps for sketching a curve 1. Find all local maxima and local minima (turning points). 2. Find where the curve crosses: (i) the y-axis (ii) the x-axis, if possible
3. Find all other points in the domain. 4. Plot a smooth curve on graph paper.
Types of curve (a) Quadratic curves
(b) Cubic curves (c) Exponential curves
(a) Quadratic curves: y = a x 2 + bx + c 1. Local maxima and minima: y = ax 2 + bx + c
(i)
___ dx = 2ax + b = 0
dy
x = − b ___
2a
Quadratic functions always have one and only one stationary point at x = − b ___
(ii)
___ dx 2 = 2a
d 2 y
If a > 0 the slope is increasing Therefore, the shape is
If a < 0 the slope is decreasing Therefore, the shape is
318 always.
and x = − b ___
2a
and x = − b ___
gives a local minimum. always. 2a gives a local maximum. 2a .
