8.05a-d Surfaces, sections, contours and partial derivatives Quiz: OCR Further Maths, Unit 5

The 20 questions

1. What equation represents a level curve or contour line for a surface z = f(x, y)?

   • f(x, y) = z
   • f(x, y) = 0
   • f(x, y) = k
   • z = x + y

2. What plane cross-section is obtained by setting x = c on a surface z = f(x, y)?

   • Section y = c
   • Section z = c
   • Section x = c
   • Section x = y

3. What geometric feature do closely spaced contour lines on a surface map represent?

   • Steep gradient
   • Saddle point
   • Gentle slope
   • Flat terrain

4. What is the term for a point on a surface where both first partial derivatives are zero?

   • Stationary point
   • Discontinuity
   • Inflection point
   • Asymptote

5. Assuming continuous second derivatives, how do the mixed partial derivatives f_xy and f_yx compare?

   • They are reciprocals
   • f_xy = -f_yx
   • They are equal
   • f_xy + f_yx = 1

6. What is the intersection curve of a surface z = f(x, y) with the plane z = 0 called?

   • xz-trace
   • xy-trace
   • Normal line
   • yz-trace

7. What geometric object is formed by holding y constant in a surface equation z = f(x, y)?

   • Space curve
   • Tangent plane
   • Normal vector
   • Contour line

8. For the surface z = x^3 y^2, what is the partial derivative ∂z/∂x?

   • 6x^2 y
   • 3x^2 y
   • 3x^2 y^2
   • 2x^3 y

9. For the surface z = x^3 y^2, what is the partial derivative ∂z/∂y?

   • x^3 y
   • 6x^2 y
   • 2x^3 y
   • 3x^2 y^2

10. What shape are the contours z = k (where k > 0) for the surface z = x^2 + y^2?

    • Parabolas
    • Straight lines
    • Circles
    • Hyperbolas

11. What shape are the contours z = k (where k is a non-zero constant) for z = x^2 - y^2?

    • Ellipses
    • Hyperbolas
    • Circles
    • Parabolas

12. If z = sin(xy), what is the partial derivative ∂z/∂x?

    • y cos(xy)
    • x cos(xy)
    • cos(xy)
    • -y cos(xy)

13. What is the equation of the section of z = x^2 + y^2 cut by the plane x = 3?

    • z = x^2 + 9
    • z = y^2 + 3
    • z = y^2 + 9
    • z = y^2 - 9

14. If f(x, y) = e^(2x + 3y), what is the second partial derivative ∂²f/∂x²?

    • 4 e^(2x + 3y)
    • 6 e^(2x + 3y)
    • 2 e^(2x + 3y)
    • 9 e^(2x + 3y)

15. Evaluate the partial derivative ∂z/∂x for z = x^2 y at the point (2, 3).

    • 12
    • 6
    • 8
    • 4

16. For f(x, y) = x^3 y^2, what is the mixed second partial derivative ∂²f/∂x∂y?

    • 6x y^2
    • 6x^2 y
    • 2x^3
    • 3x^2 y

17. What standard 3D surface is described by the equation z = x^2 + y^2?

    • Ellipsoid
    • Cone
    • Elliptic paraboloid
    • Hyperbolic paraboloid

18. Which condition must hold for (a, b) to be a stationary point of z = f(x, y)?

    • f_xx = 0
    • f_x = 0 and f_y = 0
    • f_xy = 0
    • f_x = f_y ≠ 0

19. If z = ln(x^2 + y), what is the value of ∂z/∂y at the point (1, 1)?

    • 0.5
    • 2
    • 1
    • 0

20. What shape are the vertical cross-sections parallel to the z-axis for the surface z = x^2 - y^2?

    • Hyperbolas
    • Parabolas
    • Circles
    • Straight lines

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