CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find the approimate value of 8. 0. Eample. (page 15): The radius of a circle is measured with a possible error %. Find the possible percentage error in calculating the area of the circle.. Rates of Change (page 155) d If is a function of, then is the change of changes d with respect to. Eample, if V represents the volume of an 1
object and r represents the radius, V is the function in terms dv of r, then represents the rate of change of the volume dr with respect to radius. Eample.6 (page 156): The radius, r cm of a sphere at t seconds is given b r 4 t t a) Find the initial radius of the sphere. b) Find the rate of change of r at t...1.1 Constant Rate of Change (page 158) If dr dr c, where c is a constant value, c is known as a dt dt constant rate of change. It means that for ever value of t, dr is alwas the same. Hence for the constant rate of dt change, dr dt changing changing of of r t Eample.9 (page 158): The radius r cm of a circle increases at a constant rate of 0.5 cm s -1. If the initial radius is.5 cm, find the radius of the circle after 10 seconds.
.. Related Rates (page 158) The problem involving the rate of change of several related quantities is called the related rate of change problem. In general, this problem can be solved b using the differentiation of composite functions (chain rule). Eample.14 (page 16): The length of each side of a cube being heated is increasing at a constant rate of 10 cm/s. a) State the total area of the surface, A in terms of, the da length of its side, and find. d b) If the initial length of the sides is 10 cm, after the cube has been heated for 50 seconds calculate the length of the sides. c) Calculate the rate of increase in the surface area at that time.. Motion Along a Line (Omitted).4 Gradient of Curve at a Point (page 176) The gradient of curve at a point can be defined as the gradient of tangent line at the point of the curve.
Eample. (page 176): Find the gradient of the curve 8 5 at (, 19).4.1 Equation of Tangent to a Curve The tangent to the curve f ( at an point P is the straight line PA which touches the curve at point P. To find the equation of the tangent to the curve at P, we need to find the gradient of the tangent to curve at that point. Eample. (page 177): Find the equation of tangent of the parabola 5 at. Eample.4 (page 177): Find the equation of tangent to the ellipse 4 48 at (, ). Eample.5 (page 178): The parametric equation of a curve is given b t 1, t. Show that the point (5, -8) is a point on the curve and find the equation of the tangent to the curve at that point. 4
.4. Equation of Normal to a Curve The normal line equation to the curve f ( at an point P can be defined as a straight line PB that is perpendicular to the tangent PA. If the gradient of the tangent to the curve at P is m, then the gradient of the normal at P is 1. m (gradient of the tangent) (gradient of the normal)= -1 Eample.9 (page 18): Find the equation of the normal to the curve 5 at (, 9). Eercise at home: (Tutorial 5) Quiz A (page 15) : 1, 4b) Quiz B (page 167) : 4, 5 Quiz D (page 184) : 5b), 11 Eercise (page 1) : 1, 9, 16, 5
.5 Maimum and Minimum (page 186) For an two values of, such as 1 and where 1, (i) f ( is increasing if f ) f ( ) ( 1 (ii) f ( is decreasing if f ) f ( ) ( 1 (iii) f ( is constant if f ) f ( ) ( 1 Increasing function: positive gradient everwhere (page 186) Decreasing function: negative gradient everwhere (page 187) Assume that f ( is a continuous function and differentiable on the open interval a b. (page 186) (i) If f '( 0, then f ( is increasing on the interval. (ii) If f '( 0, then f ( is decreasing on the interval. 6
f '( is increasing, f is concave (page 188 & 19) f '( is decreasing, f is conve (i) If f ''( 0, then f '( is increasing and f ( is concave. (ii) If f ''( 0, then f '( is decreasing and f ( is conve. Definition.1 (Critical Points) (page 188) A point ( c, f ( c)) of the function f ( is a critical point if f '( c) 0 or if f '( c) does not eist. First Derivative Test (page 189) Given that f (. 1. If f '( 0, or, if f '( does not eist, is a critical point.. A critical point is a maimum point if f '( changes sign from positive to negative as is increasing through the critical point. The curve is conve. 7
. A critical point is a minimum point if f '( changes sign from negative to positive as is increasing through the critical point. The curve is concave. Eample. (page 189): Find the maimum and minimum points (if an) of the functions below. (a) f ( 4 1 (b) f ( 9 15 5 Definition. (Absolute Maimum / Minimum Values) (page 19) Absolute maimum value the function has the largest value in the domain. Similarl, the absolute minimum value is the smaller than all other values of f( in the domain. 8
Definition. (Local Maimum / Minimum Values) (Page 19) A local or relative maimum/minimum value of a function occurs when the function has the larger/smaller value in its neighborhood. Maimum and minimum values are called etreme values of the function. Second Derivative Test (page 19) Assuming that f ( has a critical point at 0. 1. If f ''( 0, the graph is conve and f ( has a maimum at 0.. If f ''( 0, the graph is concave and f ( has a minimum at 0.. If f ''( 0, or does not eist, the second derivative test fails. We have to use the first derivative to determine the propert of the etreme point at 0. Eample: Solve Eample. (b) (Page 189) Eample.6 (page 196): Determine the maimum and minimum points of f ( 1 4 9
Definition.4 (Point of Inflection) (page 197) A point which separates a conve and a concave sections of a continuous function is called a point o inflection. There are three possibilities when f ''( 0) 0 at the critical point 0. The are (page 198) (a) maimum point (b) minimum point (c) point of inflection (A point of inflection need not be a critical point. Onl the condition of f ( = 0 is a necessit). Theorem.1 (Point of Inflection and Sign f '( )) ' 0 If f ''( 0) 0 or if f '( ) does not eist, and if the value of ' 0 f ''( changes sign when passing through 0, then the point, f ( )) on the curve is a point of inflection. ( 0 0 Eample.7 (page 198): Determine maimum, minimum and the point of inflection (if an) of f ( ( 1). 10
.6 Curve Sketching (omitted) (page 1).7 L Hospital s Rule (page 7) Definition.8 (L Hospital s Rule) (page 7) Assume that f ( and g ( are differentiable functions in the interval ( a, b) containing c ecept (possibl) at the point c itself. If f ( g( has an improper form 0 0 or at c and if g '( 0 for 0, then provided lim c lim c f ( f '( lim, g( c g'( f '( f '( eists or lim g'( c g'( Note (page 9): If the first derivatives still give an undetermined form then we appl L Hospital s once again. The repeated use of L Hospital s rule is permitted. Eample.5 (page 8): (a) lim 6 9 11
(d) lim 0 sin Eample.55 (page 9): (c) lim e Eercise at home (Tutorial 6): (page 08), Quiz E: no. (a), (a), (b) (page 1), Quiz G: no. 1,, 7, 1, 15 (page 40 & 4), Eercise : no. 56, 6 1