Limits and Derivatives | Class 11 Mathematics Notes

Average Velocity and Instantaneous Velocity

This section elaborates on the concepts of average velocity and instantaneous velocity using the example of a freely falling body. The distance s travelled by the body in time t seconds is given by s = 4.9t² (formula_1). The average velocity between two time points t1 and t2 is defined as the distance travelled divided by the time interval (formula_5 and formula_6). By calculating average velocities over intervals that end at t = 2 seconds and gradually reducing the length of these intervals, we observe that the average velocity approaches a certain value. Tables 12.2 and 12.3 (table_2, table_4) list average velocities for various intervals approaching t = 2 from the left and right, respectively. The values approach approximately 19.6 m/s, suggesting that the instantaneous velocity at t = 2 seconds is about 19.6 m/s. This is further supported by the graphical representation of the function s = 4.9t² and the slope of the tangent at t = 2 (figure_7). Thus, instantaneous velocity is conceptualized as the limit of average velocities over smaller and smaller intervals, leading to the need for the mathematical concept of limits.

📊 Diagram: The adjoining Table 13.1 gives the distance travelled in metres at various intervals; Table 13.2 gives the average velocity (v), t = t1; The following Table 12.3 gives the average velocity v in metres per second; etc. From the Fig 12.1 it is safe to conclude that this latter sequence approaches the

🧪 Activity: Activity involves calculating average velocities over decreasing intervals to approximate instantaneous velocity.

🔗 Connection: Introduces the need for the concept of limits to rigorously define instantaneous velocity.

Table on page 2 (1×8)

| t 1 | 0 | 1 | 1.5 | 1.8 | 1.9 | 1.95 | 1.99 |

Table on page 3 (1×8)

| t 2 | 4 | 3 | 2.5 | 2.2 | 2.1 | 2.05 | 2.01 |

Table on page 6 (2×10)

| f(x) 14.9 14.95 14.99 14.995 15.001 15.01 15.1 From the Table 12.4, we deduce that value of f(x) at x = 5 should be greater than 14.995 and less than 15.001 assuming nothing dramatic happens between x = 4.995 and 5.001. It is reasonable to assume that the value of the f(x) at x = 5 as dictated by the numbers to the left of 5 is 15, i.e., lim f ( x)=15. x→5– Similarly, when x approaches 5 from the right, f(x) should be taking value 15, i.e., lim f ( x)=15. x→5+ Hence, it is likely that the left hand limit of f(x) and the right hand limit of f(x) are both equal to 15. Thus, lim f ( x)= lim f ( x)=lim f ( x)=15. x→5− x→5+ x→5 This conclusion about the limit being equal to 15 is somewhat strengthened by seeing the graph of this function which is given in Fig 2.16, Chapter 2. In this figure, we note that as x approaches 5 from either right or left, the graph of the function f(x) = x +10 approaches the point (5, 15). We observe that the value of the function at x = 5 also happens to be equal to 15. Illustration 2 Consider the function f(x) = x3. Let us try to find the limit of this function at x = 1. Proceeding as in the previous case, we tabulate the value of f(x) at x near 1. This is given in the Table 12.5. Table 12.5 x 0.9 0.99 0.999 1.001 1.01 1.1 f(x) 0.729 0.970299 0.997002999 1.003003001 1.030301 1.331 | f(x) | 14.9 | 14.95 | 14.99 | 14.995 | 15.001 | 15.01 | 15.1 | |

Table on page 6 (1×7)

Table on page 7 (1×8)

Table on page 8 (1×7)

Table on page 9 (1×5)

| x | π −0.1 2 | π −0.01 2 | π +0.01 2 | π +0.1 2 |

Table on page 9 (1×7)

Table on page 10 (1×5)

Table on page 10 (1×7)

Table on page 11 (1×7)

Table on page 39 (3×2)

| | sinx lim =1 x→0 x 1−cosx lim =0 x→0 x ® |

| The derivative of a function f at a is defined by f (a+h)− f ( a) f′( a) =lim h→0 h ® Derivative of a function f at any point x is defined by df ( x) f (x+h)− f ( x) f′( x)= =lim dx h→0 h ® For functions u and v the following holds: (u±v)′=u′±v′ (uv)′=u ′v +uv′ u′ u ′v −uv′   = provided all are defined. v v2 ® Following are some of the standard derivatives. d (xn)=nxn−1 dx d (sin x) =cosx dx d (cos x)=−sinx dx Historical Note In the history of mathematics two names are prominent to share the credit for inventing calculus, Issac Newton (1642 – 1727) and G.W. Leibnitz (1646 – 1717). | The derivative of a function f at a is defined by f (a+h)− f ( a) f′( a) =lim h→0 h ® Derivative of a function f at any point x is defined by df ( x) f (x+h)− f ( x) f′( x)= =lim dx h→0 h ® For functions u and v the following holds: (u±v)′=u′±v′ (uv)′=u ′v +uv′ u′ u ′v −uv′   = provided all are defined. v v2 ® Following are some of the standard derivatives. d (xn)=nxn−1 dx d (sin x) =cosx dx d (cos x)=−sinx dx | | | Historical Note In the history of mathematics two names are prominent to share the credit for inventing calculus, Issac Newton (1642 – 1727) and G.W. Leibnitz (1646 – 1717). | | | Both of them independently invented calculus around the seventeenth century. After the advent of calculus many mathematicians contributed for further development of calculus. The rigorous concept is mainly attributed to the great |