Limits and Derivatives | Class 11 Mathematics Notes

Derivatives

Derivatives measure the rate at which a function changes at any point and are central to calculus. The derivative of a function f at a point a is defined as the limit of the difference quotient: f'(a) = lim h→0 [f(a + h) - f(a)] / h, provided this limit exists. This definition captures the instantaneous rate of change of the function at a. Geometrically, the derivative at a point corresponds to the slope of the tangent to the curve y = f(x) at x = a. The section explains that the derivative can be interpreted as the velocity of a moving object at an instant when the position is given by a function of time. Examples include finding the derivative of f(x) = x² at a point. The section also discusses the concept of differentiability and the relationship between differentiability and continuity. If a function is differentiable at a point, it is continuous there, but the converse is not always true.

📊 Diagram: Table on page 25 (2×4) illustrating derivative calculation examples.

🧪 Activity: Computing derivatives from first principles for various functions.

🔗 Connection: Prepares for rules of differentiation to simplify derivative calculations.

Table on page 25 (2×4)

| of various rules, limits are subjected to. The following illustrates this. Example 7 Find the derivative of sin x at x = 0. Solution Let f(x) = sin x. Then f (0+h)− f ( 0) f ′(0) = lim h→0 h sin (0+h)−sin ( 0) sinh = lim = lim =1 h→0 h h→0 h Example 8 Find the derivative of f(x) = 3 at x = 0 and at x = 3. Solution Since the derivative measures the change in function, intuitively it is clear that the derivative of the constant function must be zero at every point. This is indeed, supported by the following computation. f (0+h)− f ( 0) 3−3 0 f ' ( 0)= lim =lim =lim =0. h→0 h h→0 h h→0h f (3+h)− f ( 3) 3−3 Similarly f ' ( 3) = lim =lim =0. h→0 h h→0 h We now present a geomet- ric interpretation of derivative of a function at a point. Let y = f(x) be a function and let P = (a, f(a)) and Q = (a + h, f(a + h) be two points close to each other on the graph of this function. The Fig 12.11 is now self explanatory. | | | |

Table on page 2 (2×4)

| = 2 1 Time interval (t −t ) 2 1 (19.6−0)m = =9.8m /s. (2−0)s | t | s | |

| | 0 1 1.5 | 0 4.9 11.025 | | | 1.8 15.876 Similarly, the average velocity between t = 1 1.9 17.689 and t = 2 is 1.95 18.63225 (19.6 – 4.9)m 2 19.6 = 14.7 m/s (2 −1)s 2.05 20.59225 Likewise we compute the average velocitiy 2.1 21.609 between t = t and t = 2 for various t . The following 2.2 23.716 1 1 Table 13.2 gives the average velocity (v), t = t 2.5 30.625 1 seconds and t = 2 seconds. 3 44.1 4 78.4 Table 12.2 t 0 1 1.5 1.8 1.9 1.95 1.99 1 v 9.8 14.7 17.15 18.62 19.11 19.355 19.551 From Table 12.2, we observe that the average velocity is gradually increasing. As we make the time intervals ending at t = 2 smaller, we see that we get a better idea of the velocity at t = 2. Hoping that nothing really dramatic happens between 1.99 seconds and 2 seconds, we conclude that the average velocity at t = 2 seconds is just above 19.551m/s. This conclusion is somewhat strengthened by the following set of computation. Compute the average velocities for various time intervals starting at t = 2 seconds. As before the average velocity v between t = 2 seconds and t = t seconds is 2 Distance travelled between 2 seconds and t seconds = 2 t −2 2 | 1.8 1.9 1.95 2 2.05 2.1 2.2 2.5 3 4 | 15.876 17.689 18.63225 19.6 20.59225 21.609 23.716 30.625 44.1 78.4 | |