Q1: 1. What is the definition of continuity of a function at a point?

• A) The function is differentiable at that point
• B) The function has a finite limit at that point
• C) The function has a limit at that point that equals the function value
• D) The function has a jump discontinuity at that point

Q2: 2. A real-life problem involving continuity is

• A) Finding the maximum height a ball reaches when thrown in the air
• B) Determining the average speed of a moving car
• C) Calculating the area under a curve
• D) Identifying the critical points of a function

Q3: 3. Which of the following functions is continuous for all real numbers?

• A) f(x) = |x|
• B) f(x) = 1/x
• C) f(x) = x^2
• D) f(x) = sqrt(x)

Q4: 4. Differentiability of a function at a point implies

• A) The function is continuous at that point
• B) The function has a finite limit at that point
• C) The function has a tangent line at that point
• D) The function has a removable discontinuity at that point

Q5: 5. A real-life problem involving differentiability is

• A) Calculating the slope of a curve at a given point
• B) Approximating the area under a curve using rectangles
• C) Solving a system of linear equations
• D) Finding the domain of a function

Q6: 6. Which of the following functions is not differentiable at x = 0?

• A) f(x) = |x|
• B) f(x) = x^2
• C) f(x) = sqrt(x)
• D) f(x) = 1/x

Q7: 7. The function f(x) = 3x^2 - 2x is differentiable

• A) Everywhere
• B) Only at x = 0
• C) Only at x = 1
• D) Nowhere

Q8: 8. A real-life problem involving continuity and differentiability is

• A) Modeling population growth of a species
• B) Solving a system of linear equations
• C) Calculating the volume of a sphere
• D) Approximating the area under a curve using trapezoids

Q9: 9. Which of the following functions is continuous but not differentiable at x = 0?

• A) f(x) = |x|
• B) f(x) = x^3
• C) f(x) = sqrt(x)
• D) f(x) = sin(x)

Q10: 10. The function f(x) = |x - 2| is continuous at x = 2

• A) True
• B) False
• C) Not enough information given
• D) Undefined

Q11: 11. A real-life example of a function that is continuous but not differentiable is

• A) A step function representing temperature changes over time
• B) A linear function representing distance traveled over time
• C) A quadratic function representing projectile motion
• D) An exponential function representing population growth

Q12: 12. At which point is the function f(x) = x^2 - 4x continuous but not differentiable?

• A) x = 0
• B) x = 2
• C) x = 4
• D) x = -2

Q13: 13. The function f(x) = sqrt(x) is differentiable for

• A) x > 0
• B) x < 0
• C) x = 0
• D) x ≠ 0

Q14: 14. A real-life application of differentiability is

• A) Calculating the slope of a hill for construction purposes
• B) Determining the area of a painting
• C) Finding the solution to a system of equations
• D) Measuring the length of a fence

Q15: 15. The function f(x) = |x^2 - 4x| is continuous at x = 2

• A) True
• B) False
• C) Not enough information given
• D) Undefined

Q16: 16. If a function is differentiable at a point, it must be

• A) Continuous at that point
• B) Discontinuous at that point
• C) Constant at that point
• D) Increasing at that point

Q17: 17. A real-life problem involving continuity and differentiability is

• A) Estimating the speed of a car at a specific time
• B) Calculating the area of a circle
• C) Solving a trigonometric equation
• D) Deriving the equation of a line

Q18: 18. The function f(x) = x^3 is differentiable

• A) Everywhere
• B) Only at x = 0
• C) Only at x = 1
• D) Nowhere

Q19: 19. Which of the following functions is continuous but not differentiable at x = 2?

• A) f(x) = |x - 2|
• B) f(x) = x^2
• C) f(x) = sqrt(x)
• D) f(x) = sin(x)

Q20: 20. The function f(x) = 1/x is differentiable for

• A) x > 0
• B) x < 0
• C) x = 0
• D) x ≠ 0

Q21: 21. A real-life example of a function that is continuous but not differentiable is

• A) A step function representing changes in stock prices
• B) A linear function representing time and distance
• C) A quadratic function representing acceleration of an object
• D) An exponential function representing interest growth

Q22: 22. At which point is the function f(x) = x^3 - 3x^2 continuous but not differentiable?

• A) x = 0
• B) x = 1
• C) x = 2
• D) x = 3

Q23: 23. The function f(x) = sqrt(x) is differentiable for

• A) x > 0
• B) x < 0
• C) x = 0
• D) x ≠ 0

Q24: 24. A real-life application of differentiability is

• A) Calculating the gradient of a terrain for road construction
• B) Finding the solution to a quadratic equation
• C) Measuring the area of a room
• D) Estimating the speed of a moving object

Q25: 25. The function f(x) = |x^2 - 2x| is continuous at x = 2

• A) True
• B) False
• C) Not enough information given
• D) Undefined