Continuity and Differentiability - Geometrically defining differentiability

• In this chapter, we will explore the concept of continuity and differentiability in calculus
• We will start by understanding geometrically defining differentiability
• This concept helps us understand the behavior of a function and its derivatives geometrically

Definition of Continuity

• A function is said to be continuous at a point if the left-hand limit, right-hand limit, and the value of the function at that point are equal
• Mathematically, for a function f(x), it is continuous at a point ‘a’ if:
  • lim(x -> a-) f(x) = lim(x -> a+) f(x) = f(a)

Example:

Consider the function f(x) = x^2. Is this function continuous at x = 1?

• Evaluate the left-hand limit: lim(x -> 1-) f(x) = lim(x -> 1-) x^2 = 1^2 = 1
• Evaluate the right-hand limit: lim(x -> 1+) f(x) = lim(x -> 1+) x^2 = 1^2 = 1
• Evaluate the value of the function at x = 1: f(1) = 1^2 = 1 Since lim(x -> 1-) f(x), lim(x -> 1+) f(x), and f(1) are all equal to 1, the function f(x) = x^2 is continuous at x = 1.

Geometric Interpretation of Differentiability

• The derivative of a function at a particular point represents the slope of the tangent line to the graph of the function at that point
• A function is said to be differentiable at a point if the tangent line exists and is unique at that point

Example:

Consider the function f(x) = x^3. Is this function differentiable at x = 0?

• Let’s find the derivative of f(x) using the limit definition: f’(x) = lim(h -> 0) [f(x + h) - f(x)] / h
• Substitute x = 0 in the derivative equation: f’(0) = lim(h -> 0) [f(0 + h) - f(0)] / h
• Simplify the equation: f’(0) = lim(h -> 0) [h^3 - 0^3] / h = lim(h -> 0) (h^3) / h = lim(h -> 0) h^2 = 0 Since f’(0) = 0, the function f(x) = x^3 is differentiable at x = 0.

Conditions for Geometric Differentiability

1. The function must be continuous at the point where we are checking for differentiability

2. The function should have a well-defined tangent line at that point

• If the function satisfies both these conditions, it is said to be geometrically differentiable at that point

Example:

Consider the function f(x) = |x|. Is this function differentiable at x = 0?

• The function f(x) = |x| is continuous at x = 0 because lim(x -> 0-) f(x) = lim(x -> 0+) f(x) = f(0) = 0
• However, the function does not have a well-defined tangent line at x = 0 as the slope of the tangent line changes abruptly Thus, the function f(x) = |x| is not differentiable at x = 0.

Differentiability Implies Continuity

• If a function is differentiable at a point, it must be continuous at that point
• The converse is not always true
• Thus, differentiability is a stronger condition than continuity

Example:

Consider the function f(x) = x^2. Is this function differentiable at x = 1?

• We have already shown earlier that f(x) = x^2 is continuous at x = 1
• Now, let’s find the derivative of f(x): f’(x) = 2x
• Substitute x = 1 in the derivative equation: f’(1) = 2(1) = 2 Since f’(1) = 2, the function f(x) = x^2 is differentiable at x = 1.

11. Differentiability and Differentiation Rules

• Differentiability is an important concept in calculus that allows us to find the derivative of a function at a certain point
• The derivative is a measure of how the function changes with respect to its input variable
• Differentiation rules are helpful in finding derivatives quickly and efficiently

12. Differentiation Rules: Constant Rule

• The derivative of a constant is always zero
• Mathematically, if f(x) = c, where c is a constant, then f’(x) = 0 Example: Let f(x) = 5. The derivative of f(x) is f’(x) = 0.

13. Differentiation Rules: Power Rule

• The derivative of a power function f(x) = x^n, where n is a constant, is given by f’(x) = nx^(n-1) Example: Let f(x) = x^3. The derivative of f(x) is f’(x) = 3x^(3-1) = 3*x^2.

14. Differentiation Rules: Sum and Difference Rule

• The derivative of the sum of two functions is the sum of their derivatives
• The derivative of the difference of two functions is the difference of their derivatives
• Mathematically, (f ± g)’(x) = f’(x) ± g’(x) Example: Let f(x) = x^2 and g(x) = 2x. The derivative of f(x) + g(x) is (f + g)’(x) = f’(x) + g’(x) = 2x + 2.

15. Differentiation Rules: Product Rule

• The derivative of the product of two functions is given by the product of the first function’s derivative with the second function, plus the product of the second function’s derivative with the first function
• Mathematically, (f * g)’(x) = f’(x) * g(x) + f(x) * g’(x) Example: Let f(x) = x^2 and g(x) = 2x. The derivative of f(x) * g(x) is (f * g)’(x) = f’(x) * g(x) + f(x) * g’(x) = 2x * 2x + x^2 * 2 = 4x^2 + 2x^2 = 6x^2.

16. Differentiation Rules: Quotient Rule

• The derivative of the quotient of two functions is given by the difference of the product of the first function’s derivative with the second function, minus the product of the second function’s derivative with the first function, all divided by the square of the second function
• Mathematically, (f / g)’(x) = (f’(x) * g(x) - f(x) * g’(x)) / (g(x))^2 Example: Let f(x) = x^2 and g(x) = 2x. The derivative of f(x) / g(x) is (f / g)’(x) = (f’(x) * g(x) - f(x) * g’(x)) / (g(x))^2 = (2x * 2x - x^2 * 2) / (2x)^2 = (4x^2 - 2x^2) / 4x^2 = 2x^2 / 4x^2 = 1/2.

17. Differentiation Rules: Chain Rule

• The chain rule allows us to find the derivative of a function composed with another function
• Mathematically, if y = f(g(x)), then dy/dx = f’(g(x)) * g’(x) Example: Let f(x) = x^3 and g(x) = 2x. The derivative of f(g(x)) is d/dx [f(g(x))] = f’(g(x)) * g’(x) = 3(g(x))^2 * 2 = 3(2x)^2 * 2 = 12x^2.

18. Differentiability in terms of the Derivative

• A function is differentiable at a point if and only if the derivative exists at that point
• If a function is differentiable at a point, its derivative gives the slope of the tangent line at that point
• A function can be differentiable on an interval if its derivative is defined for every point in that interval

19. Examples of Differentiable Functions

• Polynomial functions are differentiable everywhere
• Exponential and logarithmic functions are differentiable on their domains
• Trigonometric functions are differentiable on their domains
• Piecewise functions can be differentiable on certain intervals

20. Differentiability and Continuity

• Differentiability implies continuity, but continuity does not imply differentiability
• A function can be continuous at a point without being differentiable at that point
• Discontinuities in a function can prevent differentiability

Continuity and Differentiability - Geometrically defining differentiability

Differentiability and Continuity

• Differentiability implies continuity, but continuity does not imply differentiability
• A function can be continuous at a point without being differentiable at that point
• Discontinuities in a function can prevent differentiability

Example:

Consider the function f(x) = |x|. Is this function differentiable at x = 0?

• The function f(x) = |x| is continuous at x = 0 because lim(x -> 0-) f(x) = lim(x -> 0+) f(x) = f(0) = 0
• However, the function does not have a well-defined tangent line at x = 0 as the slope of the tangent line changes abruptly Thus, the function f(x) = |x| is not differentiable at x = 0.

Continuity at a Point

• A function is said to be continuous at a point if the left-hand limit, right-hand limit, and the value of the function at that point are equal
• Mathematically, for a function f(x), it is continuous at a point ‘a’ if:
  • lim(x -> a-) f(x) = lim(x -> a+) f(x) = f(a)

Example:

Consider the function f(x) = x^2. Is this function continuous at x = 1?

• Evaluate the left-hand limit: lim(x -> 1-) f(x) = lim(x -> 1-) x^2 = 1^2 = 1
• Evaluate the right-hand limit: lim(x -> 1+) f(x) = lim(x -> 1+) x^2 = 1^2 = 1
• Evaluate the value of the function at x = 1: f(1) = 1^2 = 1 Since lim(x -> 1-) f(x), lim(x -> 1+) f(x), and f(1) are all equal to 1, the function f(x) = x^2 is continuous at x = 1.

Geometric Interpretation of Differentiability

• The derivative of a function at a particular point represents the slope of the tangent line to the graph of the function at that point
• A function is said to be differentiable at a point if the tangent line exists and is unique at that point

Example:

Consider the function f(x) = x^3. Is this function differentiable at x = 0?

• Let’s find the derivative of f(x) using the limit definition: f’(x) = lim(h -> 0) [f(x + h) - f(x)] / h
• Substitute x = 0 in the derivative equation: f’(0) = lim(h -> 0) [f(0 + h) - f(0)] / h
• Simplify the equation: f’(0) = lim(h -> 0) [h^3 - 0^3] / h = lim(h -> 0) (h^3) / h = lim(h -> 0) h^2 = 0 Since f’(0) = 0, the function f(x) = x^3 is differentiable at x = 0.

Conditions for Geometric Differentiability

1. The function must be continuous at the point where we are checking for differentiability

2. The function should have a well-defined tangent line at that point

• If the function satisfies both these conditions, it is said to be geometrically differentiable at that point

Example:

Consider the function f(x) = |x|. Is this function differentiable at x = 0?

• The function f(x) = |x| is continuous at x = 0 because lim(x -> 0-) f(x) = lim(x -> 0+) f(x) = f(0) = 0
• However, the function does not have a well-defined tangent line at x = 0 as the slope of the tangent line changes abruptly Thus, the function f(x) = |x| is not differentiable at x = 0.

Differentiability Implies Continuity

• If a function is differentiable at a point, it must be continuous at that point
• The converse is not always true
• Thus, differentiability is a stronger condition than continuity

Example:

Consider the function f(x) = x^2. Is this function differentiable at x = 1?

• We have already shown earlier that f(x) = x^2 is continuous at x = 1
• Now, let’s find the derivative of f(x): f’(x) = 2x
• Substitute x = 1 in the derivative equation: f’(1) = 2(1) = 2 Since f’(1) = 2, the function f(x) = x^2 is differentiable at x = 1.

Differentiation Rules: Constant Rule

• The derivative of a constant is always zero
• Mathematically, if f(x) = c, where c is a constant, then f’(x) = 0 Example: Let f(x) = 5. The derivative of f(x) is f’(x) = 0.

Differentiation Rules: Power Rule

• The derivative of a power function f(x) = x^n, where n is a constant, is given by f’(x) = nx^(n-1) Example: Let f(x) = x^3. The derivative of f(x) is f’(x) = 3x^(3-1) = 3*x^2.

Differentiation Rules: Sum and Difference Rule

• The derivative of the sum of two functions is the sum of their derivatives
• The derivative of the difference of two functions is the difference of their derivatives
• Mathematically, (f ± g)’(x) = f’(x) ± g’(x) Example: Let f(x) = x^2 and g(x) = 2x. The derivative of f(x) + g(x) is (f + g)’(x) = f’(x) + g’(x) = 2x + 2.

Differentiation Rules: Product Rule

• The derivative of the product of two functions is given by the product of the first function’s derivative with the second function, plus the product of the second function’s derivative with the first function
• Mathematically, (f * g)’(x) = f’(x) * g(x) + f(x) * g’(x) Example: Let f(x) = x^2 and g(x) = 2x. The derivative of f(x) * g(x) is (f * g)’(x) = f’(x) * g(x) + f(x) * g’(x) = 2x * 2x + x^2 * 2 = 4x^2 + 2x^2 = 6x^2.