Calculus Quiz

Calculus studies change and accumulation. The quiz covers limits, derivatives, basic differentiation rules, integrals and simple applications.

What this quiz covers

Study tip: Think of a derivative as a rate of change and an integral as a total; that idea ties most questions together.

Questions, answers and explanations

1. Calculus is mainly the study of:

  1. Shapes
  2. Change
  3. Numbers only
  4. Probability

Answer: Change

Calculus studies rates of change and accumulation.

2. The derivative of a constant is:

  1. 1
  2. 0
  3. The constant itself
  4. Undefined

Answer: 0

The derivative of any constant is always zero.

3. What is the derivative of x²?

  1. x
  2. 2x
  3. x²
  4. 2

Answer: 2x

Using the power rule, d/dx(x²) = 2x.

4. The symbol ∫ represents:

  1. Derivative
  2. Integration
  3. Multiplication
  4. Division

Answer: Integration

The symbol ∫ is called the integral sign.

5. The derivative measures:

  1. Area
  2. Volume
  3. Rate of change
  4. Length

Answer: Rate of change

A derivative tells how quickly a quantity changes.

6. What is the derivative of x³?

  1. 3x²
  2. x²
  3. 3x
  4. x³

Answer: 3x²

Using the power rule: d/dx(x³) = 3x².

7. The integral of 0 is:

  1. 0
  2. 1
  3. C
  4. A constant (C)

Answer: A constant (C)

∫0 dx = C, where C is the constant of integration.

8. Which rule is used to differentiate xⁿ?

  1. Chain Rule
  2. Product Rule
  3. Power Rule
  4. Quotient Rule

Answer: Power Rule

The Power Rule states: d/dx(xⁿ) = n·xⁿ⁻¹.

9. What is the derivative of 5x?

  1. 0
  2. 1
  3. 5
  4. 5x

Answer: 5

The derivative of ax is simply a.

10. Integration is mainly used to find:

  1. Slope
  2. Area under a curve
  3. Angles
  4. Perimeter

Answer: Area under a curve

A definite integral gives the area under a curve.

11. What is the derivative of x⁴?

  1. 4x³
  2. 3x⁴
  3. x³
  4. 4x

Answer: 4x³

Using the power rule: d/dx(x⁴) = 4x³.

12. The derivative of x is:

  1. 0
  2. 1
  3. x
  4. 2

Answer: 1

The derivative of x with respect to x is 1.

13. Which branch of calculus deals with derivatives?

  1. Integral Calculus
  2. Differential Calculus
  3. Analytical Geometry
  4. Statistics

Answer: Differential Calculus

Differential Calculus is the study of derivatives and rates of change.

14. The integral of a function is also called its:

  1. Derivative
  2. Antiderivative
  3. Slope
  4. Limit

Answer: Antiderivative

An indefinite integral is also known as an antiderivative.

15. What is d/dx(7)?

  1. 7
  2. 1
  3. 0
  4. Undefined

Answer: 0

The derivative of any constant is always zero.

16. What is the derivative of x⁵?

  1. 5x⁴
  2. 4x⁵
  3. 5x
  4. x⁴

Answer: 5x⁴

Using the power rule: d/dx(x⁵) = 5x⁴.

17. A definite integral is used to calculate:

  1. Area under a curve
  2. Slope of a line
  3. Equation of a circle
  4. Perimeter

Answer: Area under a curve

A definite integral gives the area under a curve between two limits.

18. What is the derivative of 3x²?

  1. 3x
  2. 6x
  3. 9x
  4. 6

Answer: 6x

d/dx(3x²) = 3 × 2x = 6x.

19. The constant added after an indefinite integral is called:

  1. Limit
  2. Coefficient
  3. Constant of Integration
  4. Derivative

Answer: Constant of Integration

Every indefinite integral includes the constant of integration (C).

20. Which notation represents the derivative of y with respect to x?

  1. ∫y dx
  2. dy/dx
  3. x/y
  4. Δx

Answer: dy/dx

dy/dx is the standard notation for the derivative of y with respect to x.

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