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Intutive [dea of Continuity: A function is continuous in ils domain if its graph is a curve without breaks or jumps throughout its domain, The graph of y = sinx is continuous in the domain R. > A function y = f(x) is said to be continuous at a point ‘a’ if limy sat f() = limysa- f C= f(a) RHL = LHL = f(a) > A function y = f(x) is said to be discontinuous at x = a@ if limyoar f(x) # limysa- f x) > A function y = f(x) will have removable discontinuity 1f [ Limyagtf 0) = limysg-f CD # f(a) | ‘ a _(lifx Illustration: Let f(x) = is ifx>0 In this case LHL # RHL Hence Discontinuous Function yf) (@,2) »(0,1) X45 +x Y lifx+0 Illustration II: f(x) = {) as =0 LHL = RHL # f(0) Fence it has removable discontinuity. Scanned with CamScanner CONTINUPEY OF A FUNCTION AL A POINT ' r . 6 “'a’ be int in its domain, Then ‘f" is Let '// be a real function on a subset of the real numbers and let ‘a! be a point in its f continuous at ‘a’ if Hiya) = f(a) More elaborately, if the lef hand limit, right hand limit and the value of the function al x = a exists are equal to cach other, Lamy ogt fQ) = Umyag-f Oe) = f(a) then’ f" is said to be continuous at xX = a. CONTINUITY IN AN INTERVAL: (1) fis said to be continuous in an open interval (a,b) if it is continuous at every point in this interval. (II) f is said to be continuous in the closed interval [@, b] if * f is continuous in (a, b) © limyoar fQX) = f(a) o limon fOr) = f(b) GEOMETRICAL MEANING OF CONTINUITY QQ) Function f will be continuous x = a if there is no break in the graph of the function at the point (a, f(a)). (U) [nan interval, function is said to be continuous if there is no break in the graph of the function in the entire internal. DISCONTINUIT. ‘The function ‘f” will be discontinuous at x = a in any of the following cases. © limyoatfOx) = f(x) and limy spf (x) = f(x) exists but not equal. © lima f(x) = f(x) and limy.y- fC) = F(x) exists and are equal but not equal to f(a). * f(a) is not defined, CONTINUITY OF SOME COMMON FUNCTIONS Function Interval in which f is continuous 1. The constant function f(x) = ¢ R 2. The identity function f(x) = x R 3. The polynomial function R F(X) = gx" + aQxT) bret ayia + ay 4. |x -al (-«, «) 5.x", nis a positive integer (—«, x) — {0} 6. a p(x)and q(x aay POXand q(x) R-(x:9q(x) = 0) Polynomials in x 7, sinx ,cosx R 8. tanx,secx R-{(2n+ 1)3}]nez 9. colx,cosec x R-(nj,néZ 10, @ R I. log x (0,02) 12. The inverse functions In their respective domains Scanned with CamScanner IMPLICIT FUNCTIONS: A function of the form y = f(x), I'the variables are connected by are the form f(x,y) = 0. I y can’t be expressible directly in terms of x, then itis called an implicit function, *= tann-+by (I) loge? + y?) = 2 tan7! = Exponential Function: The exponential function with positive base b > 1 in the function y = f(x) = b* its domain is R (sets of all real numbers) and the range is the set of all positive real numbers. * — Exponential function with base 10 is called the common exponential function. * — Exponential function with base ‘e! is called natural exponential function, LOGARITHMIC FUNCTIONS: Let b > 1 be areal number. Then we say logarithm of ’a’ to a base b is x if b* = a + Logarithm of ‘a’ to the base ‘b'is denoted by logya * Ifthe base is 10, we say it is a common logarithm + = Ifthe base is ‘e’, we say it is a natural logarithm « — logx denotes log,x «* = The domain of logx is Rt * Range is set of all positive real numbers The properties of logarithmic functions + logy (xy) = logyx + logyy * logy (2) = logyx - logy © log,x” =n log,x * logyx = mn (c > 1), itis called base change formula a 1 + logyx = logyb * log,b = 1and log,1 =0 LOGARITHMIC DIFFRENTIATION Let y = [f(x)]9™ bea function * Taking logarithm of both sides © logy = log [f(x)]9 = g(x) log fC} * Differentiating with respect to x © EP a 9.5 ES) + log f)-E(9@) 6 Sa y[2.5 fe) + og f@)-5 (9) DIFFERENTIATION OF PARAMETRIC FUNCTIONS * x and y are two functions ofa single variable. « — Insuch a case x and y are called parametric functions « x= @(t),y='V(t), t is called parameter. dy dx and then we differentiate it with respect to ‘x’. * To find = of parametric functions, we first obtain the relationship between x and y by eliminating 't’ * — [tis not always convenient to eliminate the parameter, _ - dy — dy/dt _ dy. dt sfore, = = = — x — can be used. e Therelore, Ty = aryar ~ at ax Scanned with CamScanner (Cts parameter) dy dy de 2at dy dt “xe Qa Example: x = acos 0, y = bcos 0 (2 in parameter) ax dy —=-asin 0,— = -bsin0 FT asin 7) bsin dy ay dO —bsind _b dx dg ra -asingd a DIFFENTIATION OR INVERSE TRIGONOMETRIC FUNCTIONS © By the use of chain rule, we can lind the derivative of inverse trigonometric function ¢ By using trigometrical substitutions as following . . 2lanx o sin2x = 2sinx.cosx = ; ittanty + - .y -tan2x o cos 2x = cos* x — sin¢x = 2cos*x ~— 1 = 1- 2sin’x = ren annex o 1+cos2x = 2cos?x o 1—cos2x = 2sin*x 2tanx o tan2x = 3 1-tan*x 3tanx-tan3x o tan3x =———_ 1-3tan?x o sin3x = 3sinx —4sin’x @ cos3x = dcos*x ~ 3 cosx Some substitutions useful in finding derivatives © @ +x? +x =atand or acosd © a -x? +x =asin@ or x =acosd © x*-a? >x=asin@ or x=acosecd . or ~>x=acos20 fam {a x at—x? a2 +x? 2 sso x= s20 a OF aay? OX" = ACO DIFFERENTIATION OF A FUNCTION WITH RESPECT TO ANPTHER FUNCTION © Letw = f(x) and v = g(x) be two functions of 'x’, du _ du | dx =—x de dx dv Scanned with CamScanner 
