Continuity and Differentiability

Continuity of a Function

Continuity is a fundamental concept in calculus that describes whether a function has any breaks, jumps, or holes at a given point or interval.

• Definition: A function f(x) is continuous at x = a if:

  • 1. f(a) is defined.

  • 2. \( \lim_{x \to a} f(x) \) exists.

  • 3. \( \lim_{x \to a} f(x) = f(a) \)

• Example: If a graph has a jump or hole at x = 1, the function is not continuous at that point.

Differentiability of a Function

A function is differentiable at a point if its derivative exists at that point. Differentiability implies continuity, but not vice versa.

• Definition: f(x) is differentiable at x = a if \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \) exists.

• Key Points Where Differentiability Fails:

  • Sharp corners or cusps

  • Vertical tangents

  • Discontinuities

• Example: The absolute value function f(x) = |x| is not differentiable at x = 0.

Derivatives and Tangent Lines

Finding Where the Slope of the Tangent is Zero

The slope of the tangent line to a function f(x) at a point is given by f'(x). The tangent is horizontal where f'(x) = 0.

• Example: For f(x) = x^2 - 6x - 9:

  • Find f'(x) = 2x - 6

  • Set 2x - 6 = 0 to find x = 3

Finding Where the Slope of the Tangent is a Given Value

• Example: For f(x) = 3x^2 + 6x^2 + 15x + 3, find x where f'(x) = 15:

  • Compute f'(x) = 6x + 12x + 15

  • Solve 18x + 15 = 15 to get x = 0

Basic Differentiation Rules

Power Rule, Product Rule, and Chain Rule

• Power Rule: \( \frac{d}{dx} x^n = n x^{n-1} \)

• Product Rule: \( \frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \)

• Chain Rule: \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)

Examples

• Find the derivative: f(x) = (6 - x^2)(x^2 - 3x + 4)

  • Apply the product rule:

  • \( f'(x) = (6 - x^2) \cdot (2x - 3) + (x^2 - 3x + 4) \cdot (-2x) \)

• Find the derivative: f(x) = \frac{7 - 2x}{5x - x^2}

  • Apply the quotient rule:

  • \( f'(x) = \frac{(5x - x^2)(-2) - (7 - 2x)(5 - 2x)}{(5x - x^2)^2} \)

Trigonometric and Inverse Trigonometric Derivatives

Common Derivatives

• \( \frac{d}{dx} \sin x = \cos x \)

• \( \frac{d}{dx} \cos x = -\sin x \)

• \( \frac{d}{dx} \tan x = \sec^2 x \)

• \( \frac{d}{dx} \sec x = \sec x \tan x \)

• \( \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \)

• \( \frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}} \)

• \( \frac{d}{dx} \arctan x = \frac{1}{1 + x^2} \)

Examples

• Find the derivative: f(x) = 2 \sin \theta \cos \theta

  • Use the product rule and trigonometric identities:

  • \( \frac{d}{d\theta} [2 \sin \theta \cos \theta] = 2 \cos^2 \theta - 2 \sin^2 \theta \)

  • Or, using the double angle identity: \( 2 \sin \theta \cos \theta = \sin 2\theta \), so derivative is \( 2 \cos 2\theta \)

Implicit Differentiation

Implicit Differentiation Technique

Used when a function is not given explicitly as y = f(x), but rather as a relation involving both x and y.

• Example: For 4x^2 + xy = 28 - 3y^2:

  • Differentiate both sides with respect to x, treating y as a function of x:

  • \( 8x + y + x \frac{dy}{dx} = -6y \frac{dy}{dx} \)

  • Solve for \( \frac{dy}{dx} \)

Applications: Velocity, Acceleration, and Related Rates

Velocity and Acceleration

• Velocity: The derivative of the position function s(t) with respect to time t:

• \( v(t) = s'(t) \)

• Acceleration: The derivative of the velocity function:

• \( a(t) = v'(t) = s''(t) \)

• Example: If s(t) = 2t^3 - 21t^2 + 60t, then:

  • \( v(t) = 6t^2 - 42t + 60 \)

  • \( a(t) = 12t - 42 \)

Related Rates

• Used to find the rate at which one quantity changes with respect to another, often involving geometric formulas.

• Example: The volume of a sphere is increasing at a rate of 5 cm3/sec. Find the rate of change of its surface area when the volume is \( \frac{500\pi}{3} \) cm3.

  • Volume: \( V = \frac{4}{3} \pi r^3 \)

  • Surface area: \( S = 4\pi r^2 \)

  • Differentiate both with respect to time and use the chain rule to relate \( \frac{dV}{dt} \) and \( \frac{dS}{dt} \).

Logarithmic Differentiation

Derivatives Involving Logarithms

• \( \frac{d}{dx} \ln x = \frac{1}{x} \)

• \( \frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)} \)

• Example: \( \frac{d}{dx} (\ln(x^2 + 2)) = \frac{2x}{x^2 + 2} \)

Summary Table: Common Derivative Rules

Additional info:

• Some questions require interpreting graphs for continuity and differentiability, which is a standard Calculus I skill.

• Implicit differentiation and related rates are key applications in introductory calculus.

• All formulas and rules provided are foundational for first-semester calculus students.