Quiz Calculus 101: derivatives, integrals, limits and history
Test your calculus basics: derivatives, integrals, limits, the chain rule and the Newton-Leibniz feud. Twelve questions with clear explanations.
12 questions~6 minen
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What is the derivative of x² with respect to x?
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Q1. What is the derivative of x² with respect to x?
- x
- 2x
- x²/2
- 2
By the power rule d/dx(xⁿ) = n·xⁿ⁻¹, so d/dx(x²) = 2x. This rule, formalised in the 17th century, is the workhorse of differential calculus.Q2. Who are jointly credited with inventing calculus in the 17th century?
- Euler and Gauss
- Newton and Leibniz
- Archimedes and Euclid
- Cauchy and Riemann
Isaac Newton (around 1666) and Gottfried Wilhelm Leibniz (published 1684) developed calculus independently. The bitter priority dispute lasted decades, but Leibniz's dy/dx notation is the one we still use today.Q3. What is the integral of 1/x with respect to x?
- x
- ln|x| + C
- 1/x² + C
- -1/x² + C
∫(1/x) dx = ln|x| + C. The natural logarithm is precisely defined as the antiderivative of 1/x with ln(1) = 0, a property exploited since Napier and Euler.Q4. What is the limit of (sin x)/x as x approaches 0?
- 0
- 1
- Undefined
- ∞
lim_{x→0} sin(x)/x = 1 — a classical squeeze-theorem result. It underpins the derivative of sin(x), which equals cos(x).Q5. What is the derivative of e^x?
- x·e^(x-1)
- e^x
- x·e^x
- 1
The exponential function e^x is the unique function (up to scaling) that equals its own derivative. This property makes e ≈ 2.71828 the natural base for growth and decay models.Q6. What does the Fundamental Theorem of Calculus link?
- Addition and multiplication
- Derivatives and integrals
- Limits and series
- Vectors and matrices
It states that integration and differentiation are inverse operations: if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a). This bridge was the central insight of Newton and Leibniz.Q7. What is the derivative of sin(x)?
- cos(x)
- -cos(x)
- -sin(x)
- tan(x)
d/dx(sin x) = cos(x). Differentiating again gives −sin(x), then −cos(x), then sin(x): the cycle has period 4.Q8. What is the chain rule for d/dx(f(g(x)))?
- f'(x)·g'(x)
- f'(g(x))·g'(x)
- f(g'(x))
- f'(g(x)) + g'(x)
The chain rule states d/dx(f(g(x))) = f'(g(x))·g'(x). It is the cornerstone for differentiating composite functions and is used in every backpropagation step of neural networks.Q9. ∫₀¹ x dx = ?
- 0
- 1/2
- 1
- 2
∫ x dx = x²/2 + C, so evaluated from 0 to 1 it gives 1/2 − 0 = 1/2. Geometrically this is the area of the triangle under y = x from x = 0 to x = 1.Q10. What is a limit, intuitively?
- A maximum value
- The value a function approaches
- An integral with bounds
- A type of derivative
A limit is the value a function approaches as the input approaches some point. Cauchy and Weierstrass formalised the ε-δ definition in the 19th century, giving calculus its rigorous foundation.Q11. What is the derivative of a constant?
- The constant itself
- 1
- 0
- Undefined
The derivative of any constant c is 0, since a constant function has zero slope. This is why integration adds the constant of integration '+ C'.Q12. Which notation dy/dx for derivatives is due to whom?
- Newton
- Leibniz
- Euler
- Lagrange
Leibniz introduced dy/dx around 1675 to express derivatives as ratios of infinitesimals. Newton preferred dot notation (ẏ), still used in physics, but Leibniz's notation dominates modern math.
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