1. Techniques of Differentiation (For Upload)

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: TECHNIQUES OF DIFFERENTIATION Page | 1 H2 MATHEMATICS TOPIC TECHNIQUES OF DIFFERENTIATION 2022/JC1 1 Limits 1.1 Limit of a Function Consider the behaviour of a function  2f2 xxx  for values of x close to 2. x f x x f x 1.0 2.000000 3.0 8.000000 1.5 2.750000 2.5 5.750000 1.8 3.440000 2.2 4.640000 1.9 3.710000 2.1 4.310000 1.95 3.852500 2.05 4.152500 1.99 3.970100 2.01 4.030100 1.995 3.985025 2.005 4.015025 1.999 3.997001 2.001 4.003001 x values approaching 2 from the left x values approaching 2 from the right The above table gives the corresponding values of f x for values of x close to 2 but not equal to 2. Content Outline:  Differentiation of simple functions defined implicitly.  Finding the approximate value of a derivative at a given point using a graphing calculator.

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: TECHNIQUES OF DIFFERENTIATION Page | 2 From the graph and table of values, we observe that as x approaches 2, f x gets closer and closer to 4. We express this in words as “the limit of the function  2f2 xxx  as x approaches 2 is 4”. Mathematically, we write it as  2 lim f 4 x x   . In general, given a function f , if f x approaches a fixed number L as x approaches a , then L is the limit of f x as x approaches a . We write it as lim f xa xL   __________________________________________________________________________________  Remark: To find the limit of a function f , f x need not be defined at xa . __________________________________________________________________________________ For example, consider the graph of  sinf xx x for π3π 3x  as shown below. We observe that even though f0 does not exist, it is clear that the limit of the function  sinf xx x as x approaches 0 is 1. If f is defined at xa and  lim f f xa xa   , we say f is continuous at xa . If f is continuous at every point on an interval  ,ab , f is a continuous function on the interval  ,ab . Intuitively, continuous functions are the functions whose graphs can be drawn without lifting the pen off the paper. In the above example, f is not continuous at 0x (hence it is excluded graphically with an empty circle). In H2 Maths syllabus, the func tions are all assumed to be c ontinuous on the interval under consideration. y O x 3 3 1 Note that when 0x ,  f x is not defined.

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: TECHNIQUES OF DIFFERENTIATION Page | 3 1.2 Properties of Limits (  Self-Reading) If lim f xa x  and lim g xa x  both exist, then Properties Examples (a)  lim f lim f xa xa kx k x    , where k is a constant    33 44 3 lim 5 5 lim 54 320 xx x x    