1.07a Gradients and second derivatives Quiz: OCR Maths, Unit 7

The 20 questions

1. Geometrically, what does the derivative f'(x) represent on a curve?

   • Area under curve
   • Normal gradient
   • Tangent gradient
   • Chord gradient

2. What quantity does the second derivative d^2y/dx^2 represent?

   • Rate of x change
   • Tangent slope value
   • Gradient change rate
   • Chord length rate

3. If f''(x) > 0 on an interval, the curve on that interval is:

   • Convex, i.e. concave up
   • Concave, i.e. concave down
   • Linear
   • Periodic

4. If f''(x) < 0 on an interval, the curve on that interval is:

   • Concave, i.e. concave down
   • Convex, i.e. concave up
   • Increasing only
   • Constant

5. What is the gradient of y = x^2 at x = 3?

   • 6
   • 9
   • 12
   • 3

6. What is the gradient of y = x^3 - 3x at x = 2?

   • 6
   • 9
   • 3
   • 12

7. For y = x^3 - 3x, what is f''(1)?

   • 0
   • 6
   • 3
   • 9

8. What is the tangent to y = x^2 + 1 at x = 1?

   • y = 2x - 1
   • y = 2x
   • y = 2x + 1
   • y = x + 1

9. What is f''(0) for f(x) = x^4?

   • 24
   • 0
   • 4
   • 12

10. What is the minimum value of the gradient of y = x^3?

    • 1
    • 3
    • -3
    • 0

11. For y = 2x^3 - 6x^2, on which set is the curve convex?

    • x > 1
    • x > 0 only
    • x < 1
    • all real x

12. A stationary point has f'(2) = 0 and f''(2) = -4. What type of point is it?

    • Cannot be classified
    • A maximum
    • A point of inflection
    • A minimum

13. For y = x^3, what is f''(x), and where does it change sign?

    • 3x^2, changing sign at x = 0
    • 6x, changing sign at x = 0
    • x^2, changing sign at x = 1
    • 6x, never changing sign

14. What gradient is obtained as the chord interval h approaches zero?

    • Normal gradient
    • Secant gradient
    • Asymptote gradient
    • Tangent gradient

15. In the derivative dy/dx, which variable's rate of change is being measured?

    • t
    • y
    • x
    • dx

16. For f(x) = x^2 + x, what is f'(0)?

    • 1
    • -1
    • 0
    • 2

17. Because its second derivative is always negative, the curve y = -x^2 is:

    • Periodic
    • Concave everywhere
    • Linear
    • Convex everywhere

18. Which condition makes the function f(x) = e^x convex for all real x?

    • f''(x) < 0
    • f''(x) > 0
    • f''(x) = 0
    • f'(x) = 0

19. What must f''(x) do across a point for it to be a point of inflection?

    • Equal one
    • Remain positive
    • Tend to infinity
    • Change sign

20. For f(x) = x^3 - 3x, at which x values is f'(x) = 0?

    • x = 1 only
    • x = 3 and x = -3
    • x = 1 and x = -1
    • x = 0 only

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