Continuity and Differentiability

Definition of Continuity

  • A function f(x) is continuous at a point x = a if three conditions are satisfied:
    1. f(a) is defined
    2. The limit of f(x) as x approaches a exists
    3. The limit of f(x) as x approaches a is equal to f(a)

Definition of Differentiability

  • A function f(x) is differentiable at a point x = a if the limit of the difference quotient of f(x) as x approaches a exists.

Determining Continuity at a Point

  • To determine the continuity of a function at a point, we need to check the following:
    1. Function is defined at that point
    2. Left-hand limit is equal to the right-hand limit at that point
    3. Value of the function is equal to the limit at that point

Determining Differentiability at a Point

  • To determine the differentiability of a function at a point, we need to check the following:
    1. The function is continuous at that point
    2. The derivative of the function exists at that point

Types of Discontinuity

  1. Removable Discontinuity:
  • A removable discontinuity occurs when a point in the function can be made continuous by changing or redefining the function at that point.
  1. Jump Discontinuity:
  • A jump discontinuity occurs when there is an abrupt change in the function’s value at a specific point.
  1. Infinite Discontinuity:
  • An infinite discontinuity occurs when the limit of the function as it approaches a certain point is either positive or negative infinity.
  1. Non-Removable Discontinuity:
  • A non-removable discontinuity occurs when the function cannot be made continuous at a specific point, even by redefining the function at that point.

Differentiation

  • Differentiation is the process of finding the derivative of a function.

Derivative of a Function

  • The derivative of a function f(x) is denoted as f’(x) or dy/dx and represents the rate of change of the function with respect to x.

Rules of Differentiation

  1. Constant Rule: the derivative of a constant is 0.
  1. Power Rule: the derivative of x raised to the power n is n * x^(n-1).
  1. Sum/Difference Rule: the derivative of the sum or difference of two functions is the sum or difference of their derivatives.

Common Differentiation Techniques

  1. Chain Rule: Used to find the derivative of a composite function.
    • If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).
  1. Product Rule: Used to find the derivative of the product of two functions.
    • If y = u(x) * v(x), then dy/dx = u’(x) * v(x) + u(x) * v’(x).
  1. Quotient Rule: Used to find the derivative of the quotient of two functions.
    • If y = u(x) / v(x), then dy/dx = (u’(x) * v(x) - u(x) * v’(x)) / (v(x))^2.

Important Trigonometric Derivatives

  1. sin x → cos x
  1. cos x → -sin x
  1. tan x → sec^2 x

Important Exponential and Logarithmic Derivatives

  1. e^x → e^x
  1. log a (x) → 1 / (x * ln a)

Important Rules for Derivatives

  1. Derivative of a constant times a function: d/dx (k * f(x)) = k * f’(x)
  1. Derivative of the sum or difference of two functions: d/dx (f(x) ± g(x)) = f’(x) ± g’(x)

Example – Finding Continuity and Differentiability of a Function

Let’s consider the function f(x) = 2x - 1.

  1. Continuity:
    • The function f(x) is defined for all real values of x.
    • The left-hand limit as x approaches a is equal to the right-hand limit as x approaches a.
    • The value of f(x) is equal to the limit as x approaches a.
  1. Differentiability:
    • The function f(x) is a linear function and, therefore, it is continuous.
    • The derivative of f(x) is equal to 2 for all values of x.
    • Hence, the function f(x) is differentiable for all real values of x.

Summary

  • Continuity refers to the smoothness of a function, while differentiability refers to the existence of the derivative of a function.
  • There are different types of discontinuities, such as removable, jump, infinite, and non-removable.
  • Differentiation is the process of finding the derivative of a function.
  • Derivatives follow certain rules and techniques, including the power rule, chain rule, product rule, and quotient rule.
  • Important trigonometric and exponential/logarithmic derivatives are essential to know.
  • Checking for continuity and differentiability requires evaluating the function, its limits, and derivative at a specific point.
  1. Differentiation Rules:
  • Chain Rule: Used to find the derivative of a composite function.
    • If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).
  • Product Rule: Used to find the derivative of the product of two functions.
    • If y = u(x) * v(x), then dy/dx = u’(x) * v(x) + u(x) * v’(x).
  • Quotient Rule: Used to find the derivative of the quotient of two functions.
    • If y = u(x) / v(x), then dy/dx = (u’(x) * v(x) - u(x) * v’(x)) / (v(x))^2.
  • Power Rule: Used to find the derivative of x raised to the power n.
    • If y = x^n, then dy/dx = n * x^(n-1).
  • Sum/Difference Rule: Used to find the derivative of the sum or difference of two functions.
    • If y = f(x) ± g(x), then dy/dx = f’(x) ± g’(x).
  1. Derivatives of Trigonometric Functions:
  • sin x → cos x
  • cos x → -sin x
  • tan x → sec^2 x
  • csc x → -csc x cot x
  • sec x → sec x tan x
  • cot x → -csc^2 x
  1. Derivatives of Exponential and Logarithmic Functions:
  • e^x → e^x
  • ln x → 1/x
  • log_a (x) → 1/ (x * ln a)
  1. Derivatives of Inverse Trigonometric Functions:
  • d/dx (arcsin x) = 1 / √(1 - x^2)
  • d/dx (arccos x) = -1 / √(1 - x^2)
  • d/dx (arctan x) = 1 / (1 + x^2)
  • d/dx (arccosec x) = -1 / (x * √(x^2 - 1))
  • d/dx (arcsec x) = 1 / (x * √(x^2 - 1))
  • d/dx (arccot x) = -1 / (1 + x^2)
  1. Implicit Differentiation:
  • Implicit differentiation is a technique used to find the derivative of an implicitly defined function. Steps to perform implicit differentiation:
  1. Differentiate both sides of the equation with respect to x.
  1. Treat y as a function of x and use the chain rule when differentiating terms involving y.
  1. Solve for the derivative dy/dx.
  1. Example – Finding the Derivative Using Implicit Differentiation:
  • Let’s consider the equation x^2 + y^2 = 4.
  • Differentiating both sides with respect to x, we get 2x + 2y * (dy/dx) = 0.
  • Solving for dy/dx, we get dy/dx = -x/y.
  1. Higher Order Derivatives:
  • The derivative of a function can also be differentiated to obtain higher order derivatives.
  • The second derivative represents the rate of change of the first derivative.
  • The notation for the second derivative is d^2y/dx^2 or f’’(x).
  • Higher order derivatives can be obtained by differentiating the function multiple times.
  1. Example – Finding Higher Order Derivatives:
  • Let’s consider the function f(x) = x^4 - 3x^3 + 2x^2.
  • The first derivative is f’(x) = 4x^3 - 9x^2 + 4x.
  • The second derivative is f’’(x) = 12x^2 - 18x + 4.
  • The third derivative is f’’’(x) = 24x - 18.
  1. Rate of Change and Tangent Line:
  • The derivative of a function gives the rate of change of the function with respect to x.
  • The tangent line to a function at a given point represents the instantaneous rate of change of the function at that point.
  • The slope of the tangent line is equal to the derivative of the function at that point.
  • The equation of the tangent line can be found using the point-slope form: y - y1 = m(x - x1), where m is the slope of the tangent line and (x1, y1) is a point on the line.
  1. Example – Finding the Equation of a Tangent Line:
  • Let’s consider the function f(x) = x^2 - 3x + 2.
  • The derivative of the function is f’(x) = 2x - 3.
  • To find the equation of the tangent line at x = 2, we need the slope and a point on the line.
  • At x = 2, the slope is f’(2) = 2(2) - 3 = 1.
  • The point (2, f(2)) on the line is (2, 0).
  • Using the point-slope form, the equation of the tangent line is y - 0 = 1(x - 2), which simplifies to y = x - 2.
  1. Problem Solving - Finding the Derivative of a Function:
  • Let’s consider the function f(x) = 3x^4 + 2x^3 - 5x^2 + 3.
  • To find the derivative of f(x), we can differentiate each term separately.
  • The derivative of 3x^4 is 12x^3.
  • The derivative of 2x^3 is 6x^2.
  • The derivative of -5x^2 is -10x.
  • The derivative of 3 is 0.
  • Therefore, the derivative of f(x) is f’(x) = 12x^3 + 6x^2 - 10x.
  1. Problem Solving - Using the Chain Rule:
  • Let’s consider the function f(x) = (3x^2 + 5)^4.
  • To find the derivative of f(x), we can use the chain rule.
  • Let u = 3x^2 + 5. Then, f(x) = u^4.
  • The derivative of u^4 with respect to u is 4u^3.
  • The derivative of u with respect to x is 6x.
  • Therefore, the chain rule gives us the derivative of f(x) as f’(x) = 4(3x^2 + 5)^3 * 6x.
  1. Problem Solving - Finding the Equation of a Tangent Line:
  • Let’s consider the function f(x) = x^2 - 4x + 3.
  • To find the equation of the tangent line at x = 2, we need the slope and a point on the line.
  • The slope is given by the derivative of f(x), which is f’(x) = 2x - 4.
  • At x = 2, the slope is f’(2) = 2(2) - 4 = 0.
  • The point (2, f(2)) on the line is (2, 3).
  • Using the point-slope form, the equation of the tangent line is y - 3 = 0(x - 2), which simplifies to y = 3.
  1. Problem Solving - Higher Order Derivatives:
  • Let’s consider the function f(x) = x^3 - 2x^2 + 4x - 1.
  • The first derivative of f(x) is f’(x) = 3x^2 - 4x + 4.
  • The second derivative of f(x) is f’’(x) = 6x - 4.
  • The third derivative of f(x) is f’’’(x) = 6.
  1. Problem Solving - Implicit Differentiation:
  • Let’s consider the equation x^2 + y^2 = 9.
  • Differentiating both sides with respect to x, we get 2x + 2y * (dy/dx) = 0.
  • Solving for dy/dx, we get dy/dx = -x/y.
  1. Problem Solving - Derivatives of Inverse Trigonometric Functions:
  • Let’s consider the function y = sin^(-1)(2x).
  • Applying implicit differentiation, we get dy/dx = 1 / √(1 - (2x)^2).
  • Simplifying further, we get dy/dx = 1 / √(1 - 4x^2).
  1. Problem Solving - Finding Continuity at a Point:
  • Let’s consider the function f(x) = (x^2 - 3x + 2) / (x - 2).
  • To determine the continuity at x = 2, we need to check the following conditions:
    1. The function is defined at x = 2.
    2. The left-hand limit as x approaches 2 is equal to the right-hand limit.
    3. The value of f(x) is equal to the limit as x approaches 2.
  • Evaluating the function, limits, and values, we find that f(x) is continuous at x = 2.
  1. Problem Solving - Determining Differentiability at a Point:
  • Let’s consider the function g(x) = 2x^2 - 4x + 1.
  • To determine the differentiability at x = 2, we need to check the following conditions:
    1. The function is continuous at x = 2.
    2. The derivative of g(x) exists at x = 2.
  • Evaluating the function and its derivative at x = 2, we find that g(x) is differentiable at x = 2.
  1. Problem Solving - Differentiation Using the Product Rule:
  • Let’s consider the function h(x) = x^2 * sin(x).
  • To find the derivative of h(x), we can use the product rule.
  • Let u = x^2 and v = sin(x).
  • The derivative of u with respect to x is u’ = 2x.
  • The derivative of v with respect to x is v’ = cos(x).
  • Applying the product rule, we get h’(x) = u’v + uv'.
  • Therefore, the derivative of h(x) is h’(x) = 2x * sin(x) + x^2 * cos(x).
  1. Problem Solving - Differentiation Using the Quotient Rule:
  • Let’s consider the function k(x) = (2x^2 + x + 1) / (x^2 - 3x + 2).
  • To find the derivative of k(x), we can use the quotient rule.
  • Let u = 2x^2 + x + 1 and v = x^2 - 3x + 2.
  • The derivative of u with respect to x is u’ = 4x + 1.
  • The derivative of v with respect to x is v’ = 2x - 3.
  • Applying the quotient rule, we get k’(x) = (u’v - uv’) / v^2.
  • Therefore, the derivative of k(x) is k’(x) = [(4x + 1)(x^2 - 3x + 2) - (2x - 3)(2x^2 + x + 1)] / (x^2 - 3x + 2)^2.