1.07g Differentiation from first principles Quiz: OCR Maths, Unit 7

The 20 questions

1. In differentiation from first principles, what quotient is evaluated in the limit?

   • f(x+h) / h
   • (f(x) - f(h)) / h
   • (f(x+h) - f(x)) / h
   • (f(x+h) + f(x)) / h

2. Using first principles, what is the derivative of x^2?

   • x
   • x^2 + h
   • 2
   • 2x

3. Expand (x + h)^2 - x^2.

   • 2xh + h^2
   • 2x + h
   • 2xh
   • x^2 + h^2

4. Using first principles, what is the derivative of x^3?

   • 3x^2 + h
   • 3x^2
   • 3x
   • x^2

5. What is the derivative of sin x from first principles?

   • tan x
   • sin x
   • -cos x
   • cos x

6. Find the derivative of x^2 + 3x from first principles.

   • 2x
   • x + 3
   • 2x^2 + 3
   • 2x + 3

7. Find f'(2) for f(x) = x^3 from first principles.

   • 12
   • 6
   • 8
   • 4

8. Evaluate lim (h tends to 0) ((2 + h)^2 - 4)/h.

   • 2
   • 0
   • 4
   • 8

9. Using first principles, what is the derivative of 4x^2 at x = 1?

   • 16
   • 8
   • 2
   • 4

10. In differentiation from first principles, why can h not simply equal 0?

    • Undefined numerator
    • Infinite gradient
    • Negative result
    • Division by zero

11. After dividing (x + h)^4 - x^4 by h, what does the expression tend to as h tends to 0?

    • 12x^2
    • 4x^4
    • x^3
    • 4x^3

12. Which expression is (x + h)^3 - x^3 divided by h, simplified?

    • 3x^2 + h
    • 3x + h^2
    • 3x^2 + 3xh + h^2
    • x^2 + 3xh + h^2

13. Which expansion is required when differentiating sin x from first principles?

    • cos x cos h - sin x sin h
    • sin x cos h - cos x sin h
    • sin x cos h + cos x sin h
    • cos x cos h + sin x sin h

14. What is the limit of (cos h - 1)/h as h tends to 0?

    • 0
    • 1
    • -1
    • does not exist

15. The gradient of the chord of y = x^2 between x = 1 and x = 1 + h is:

    • 1 + h^2
    • 2h + 1
    • 1 + h
    • 2 + h

16. What is the derivative of a constant c from first principles?

    • 0
    • 1
    • c
    • undefined

17. What is the derivative of x^3 at x = 1 from first principles?

    • 6
    • 3
    • 2
    • 1

18. After cancelling h, ((x + h)^2 - x^2)/h gives:

    • x + h
    • 2x
    • 2x + h
    • 2xh + h^2

19. What is the compound angle expansion for cos(x + h)?

    • sin x cos h - cos x sin h
    • cos x cos h + sin x sin h
    • sin x cos h + cos x sin h
    • cos x cos h - sin x sin h

20. What is the value of the limit of sin(h) / h as h approaches 0?

    • 0
    • cos h
    • infinity
    • 1

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