Calculus Functions and Derivatives Quiz

Functions and their Domains

• f(x) = log x, g(x) = 1/x
• Domain of f(x): x > 0
• Domain of g(x): x ≠ 0
• (f°g)(x) = f(g(x)) = log(1/x) = -log(x)

Composite Functions

• (f°g)(-1) is undefined as x = -1 is outside the domain of g(x)

Mean Value Theorem

• For a differentiable function f(x), there exists a c such that f'(c) = (f(b) - f(a))/(b - a) in [a, b]
• |sin a - sin b| < |a - b| (proven using Mean Value Theorem)

Differentiable Function

• f(0) = 5, -1 < f'(x) < 3
• -5 ≤ f(10) (proven by using Mean Value Theorem)

Motion of a Car

• Car passes a camera at A with speed 50 km/h
• One hour later, the car passes another camera.

Equation of a Circle

• x² + y² = 25
• Find acceleration at a specific point

Implicit Differentiation

• Differentiating x² - y² = 1 implicitly shows that y" = ...

Tangent to a Curve

• Find points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent is parallel to the x-axis

Derivatives

• Find dy/dx for multiple functions:
  • y² = cos(sin√(x² + 1))
  • x²/ (x + y) = x - y
  • y = cos⁻¹(2x)
  • y = x³ ln(2x)
  • y = x^x
  • y = (x² + 9)²(x - 3)⁴/(x² + 2)

Test your understanding of functions, their domains, and the Mean Value Theorem in this comprehensive calculus quiz. Explore concepts such as composite functions, implicit differentiation, and the motion of a car, alongside equations of circles and derivatives. Perfect for students looking to reinforce their calculus knowledge.