Class name Subject Book title Chapter number Content Created at
Class XI Mathematics Mathematics 11 +(y−4) +(z+5) 2 = (x+2) +(y−1) +(z −4) 2 or 10 + 6y – 18z – 29 = 0. Example 9 The centroid of a triangleABC is at the point (1, 1, 1). If the coord...
Class XI Mathematics Mathematics 11 from the YZ, ZX and XY-planes. (i) Any point on x-axis is of the form (x, 0, 0) fi (ii) Any point on y-axis is of the form (0, y, 0) (iii) Any point...
Class XI Mathematics Mathematics 12 Chapter 12 LIMITSANDDERIV ATIVES v With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature –...
Class XI Mathematics Mathematics 12 1 1 Table 13.2 gives the average velocity (v), t = t 1 2.5 30.625 seconds and t = 2 seconds. 3 44.1 4 78.4 Table12.2 t 0 1 1.5 1.8 1.9 1.95 1.99 v ...
Class XI Mathematics Mathematics 12 = 2 is between 19.551m/s and 19.649 m/s. Technically, we say that the instantaneous velocity at t = 2 is between 19.551 m/s and 19.649 m/s. As is w...
Class XI Mathematics Mathematics 12 tends to zero is to be thought of as the value f (x) should assume at x = 0. In general as x → a, f (x) → l, then l is called limit of the function...
Class XI Mathematics Mathematics 12 3x. Let us try to find the limit of this function at x = 2. The following Table 12.6 is now self-explanatory. Table12.6 x 1.9 1.95 1.99 1.999 2.001...
Class XI Mathematics Mathematics 12 x + x2 shown in the Fig 12.5, it is clear that as x approaches 1, the graph approaches (1, 2). Here, again we observe that the lim f (x) = f (1) x→...
Class XI Mathematics Mathematics 12 MATHEMATICS Illustration 8 Consider the functionf( )= x 2 for x > 0. We want to know lx→0f (x). Here, observe that the domain of the function is gi...
Class XI Mathematics Mathematics 12 exist. Graph of this function is given in the Fig12.6. Here, we remark that the value of the function at x = 0 is well definedandis,indeed,equalto0...
Class XI Mathematics Mathematics 12 (iii) Limit of product of two functions is product of the limits of the functions, i.e., lim lim lim x→a [f(x) . g(x)] =x→a f(x). x→a g(x). (iv) Li...
Class XI Mathematics Mathematics 12 for each step in the above!) g( ) A function f is said to be a rational function, if f(xh( ) , where g(x) andh(x) are polynomials such that h(x) ≠ ...
Class XI Mathematics Mathematics 12 x 10 =1+ −1 + −1 +...+ −1 10 x→−1   ( ) ( ) ( ) . =1−1+1...+1=1 Example 2 Find the limits:  x +1  x −4x + 4x  (i) lim   (ii) lim  2  x→1...
Class XI Mathematics Mathematics 12 = x→2 (x−2 )(3 ) x −5x +6 x 2 ( )2 4 = lim = = = −4 . x→2( x −3) 2−3 −1 Reprint 2026-27 232 MATHEMATICS (v) First, we rewrite the function as a rat...
Class XI Mathematics Mathematics 12 x→a x−a x→a = an – + a a n–2+. . . + a–2(a) +a n–l = an–1+ a n – +...+an–1+ an–1(n terms) n−1 = na Example 3 Evaluate: x −1 1+ x −1 (i) lim 10 (ii)...
Class XI Mathematics Mathematics 12 the following important inequalityrelatingtrigonometricfunctions. sinx π cosx < <1 for 0< x < (*) x 2 Reprint 2026-27 LIMITS AND DERIVATIVES 235 Pr...
Class XI Mathematics Mathematics 12 trigonometric identity 1 – cos x   .in   Then 2sin2  sin  1−cosx       x→0 x = x→0 = x→0 x .si  x   sin      = x→0 x .x→0si...
Class XI Mathematics Mathematics 12  22 1. limx +3 2. lim  −  3. limπr2 x→3 x→π  7  r→1 10 5 5 4. lim4x +3 5. lim x + x +1 6. lim (x +1)−1 x→4 x −2 x→ −1 x −1 →0 x 2 4 7. lim 3x...
Class XI Mathematics Mathematics 12 of a and b? Reprint 2026-27 LIMITS AND DERIVATIVES 239 29. Let a 1 a2, . . .nbe fixed real numbers and define a function f( ) (x −a 1)(x− a 2) (.x−...
Class XI Mathematics Mathematics 12 of f at a is defined by f(a +h ) f a( ) h→0 h provided this limit exists. Derivative of f (x) at a is denoted by f Observethatf ′(a)quantifiesthech...
Class XI Mathematics Mathematics 12 = sin x. Then f(0 + h)− f ( ) f ′(0) =h→0 h sin(0+ h)−sin ( ) sin h = lim = lim =1 h→0 h h→0 h Example 8 Find the derivative of f(x) = 3 at x = 0 a...
Class XI Mathematics Mathematics 12 f(x +h ) f x( ) h→0 h wherever the limit exists is defined to be the derivative of f at x and is denoted by f(x). This definition of derivative is ...
Class XI Mathematics Mathematics 12 v Now, let us tackle derivatives of some standard functions. It is easy to see that the derivative of the function f(x) = x is the constant f(x + )...
Class XI Mathematics Mathematics 12 nand hence (x + h) – x = h(nx n – +... + h –). Thus df (x) (x +h )n− xn = lim dx h→0 h h (x n−1+....+ h−1) = lim h→0 h = lim (x n−1+...+ h−1) = n−....
Class XI Mathematics Mathematics 12 equals ( )( ) 1 + 2(1) + 3(1) + .. . + 50(1) = 1 + 2 + 3 + . . . + 50 = = 1275. x+1 Example 15 Find the derivative of f(x) = x Solution Clearly thi...