Limits and Derivatives

Introduction

This chapter introduces the fundamental concepts of calculus, focusing on limits and derivatives. Calculus is a branch of mathematics that deals with continuous change and motion. It provides tools to analyze functions that describe varying quantities, such as velocity, growth, and decay. The study begins by motivating the need for limits through the problem of finding instantaneous velocity, which cannot be directly measured but can be approached by considering average velocities over smaller and smaller intervals. This leads to the formal definition of limits, a foundational concept that describes the behavior of a function as its input approaches a particular value. Once limits are understood, the chapter proceeds to define derivatives, which measure the rate of change of a function at any point. Derivatives are crucial in understanding how quantities change instantaneously and have wide applications in physics, engineering, and other sciences. The chapter also introduces basic differentiation rules and standard derivatives of common functions, enabling efficient computation without always resorting to the limit definition.

• Calculus studies continuous change and motion.
• Limits describe the behavior of functions near a point.
• Instantaneous velocity motivates the concept of limits.
• Derivatives measure the rate of change of functions.
• Basic rules simplify the process of differentiation.
• Standard derivatives serve as building blocks for complex problems.
• 📌 Calculus: Branch of mathematics dealing with continuous change.
• 📌 Limit: The value a function approaches as the input approaches a point.
• 📌 Derivative: The instantaneous rate of change of a function.

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. **Table on page 2 (1×8)** | t 1 | 0 | 1 | 1.5 | 1.8 | 1.9 | 1.95 | 1.99 | | --- | --- | --- | --- | --- | --- | --- | --- | | v | 9.8 | 14.7 | 17.15 | 18.62 | 19.11 | 19.355 | 19.551 | **Table on page 3 (1×8)** | t 2 | 4 | 3 | 2.5 | 2.2 | 2.1 | 2.05 | 2.01 | | --- | --- | --- | --- | --- | --- | --- | --- | | v | 29.4 | 24.5 | 22.05 | 20.58 | 20.09 | 19.845 | 19.649 | **Table on page 6 (2×10)** | | x | 4.9 | 4.95 | 4.99 | 4.995 | 5.001 | 5.01 | 5.1 | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | | | | | | | | | | 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)** | 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 | **Table on page 7 (1×8)** | x | 1.9 | 1.95 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 | | --- | --- | --- | --- | --- | --- | --- | --- | | f(x) | 5.7 | 5.85 | 5.97 | 5.997 | 6.003 | 6.03 | 6.3 | **Table on page 8 (1×7)** | x | 0.9 | 0.99 | 0.999 | 1.01 | 1.1 | 1.2 | | --- | --- | --- | --- | --- | --- | --- | | f(x) | 1.71 | 1.9701 | 1.997001 | 2.0301 | 2.31 | 2.64 | **Table on page 9 (1×5)** | x | π −0.1 2 | π −0.01 2 | π +0.01 2 | π +0.1 2 | | --- | --- | --- | --- | --- | | f(x) | 0.9950 | 0.9999 | 0.9999 | 0.9950 | **Table on page 9 (1×7)** | x | – 0.1 | – 0.01 | – 0.001 | 0.001 | 0.01 | 0.1 | | --- | --- | --- | --- | --- | --- | --- | | f(x) | 0.9850 | 0.98995 | 0.9989995 | 1.0009995 | 1.00995 | 1.0950 | **Table on page 10 (1×5)** | x | 1 | 0.1 | 0.01 | 10–n | | --- | --- | --- | --- | --- | | f(x) | 1 | 100 | 10000 | 102n | **Table on page 10 (1×7)** | x | – 0.1 | – 0.01 | – 0.001 | 0.001 | 0.01 | 0.1 | | --- | --- | --- | --- | --- | --- | --- | | f(x) | – 2.1 | – 2.01 | – 2.001 | 2.001 | 2.01 | 2.1 | **Table on page 11 (1×7)** | x | 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 | | --- | --- | --- | --- | --- | --- | --- | | f(x) | 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 | **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 |

• Distance travelled by a falling body is s = 4.9t².
• Average velocity = (change in distance) / (change in time).
• Average velocity over intervals approaching t = 2 seconds approaches about 19.6 m/s.
• Instantaneous velocity is the limit of average velocities as interval length approaches zero.
• Tables show average velocities for intervals from left and right of t = 2.
• Graphical slope of tangent at t = 2 corresponds to instantaneous velocity.
• 📌 Average velocity: Total distance divided by total time over an interval.
• 📌 Instantaneous velocity: Velocity at a specific instant, defined as the limit of average velocities.