Elementary Mathematics II (Calculus) — 2014 Past Questions

Lagos State University MAT102: 50 questions.

Topics: Derivatives as limits of rate of change, differentiation techniques, maxima and minima, Integration, definite integrals, reduction formulae, Applications to areas, volumes (Trapezium and Simpson's rules), Functions of a real variable, graphs, limits, and continuity, Curve sketching, integration, definite integrals, reduction formulae

1. The rate of change of the area of a circle with respect to its radius is
   1. Diameter
   2. Perimeter
   3. Area × radius
   4. 2 × diameter
2. Let F(x) = ∫₁ˣ (t³ - 4t² + t + 2) dt. Find F′(4).
   1. 4
   2. 5
   3. 6
   4. 7
3. Compute ∫₄⁹ (1/√t) dt.
   1. 1
   2. 2
   3. 3
   4. 4
4. Evaluate limₙ→∞ (10n² - 3n + 7)/(2n + 6)².
   1. 1
   2. 1.5
   3. 2
   4. 2.5
5. Compute [f(2 + h) - f(2)]/h, where f(x) = 2x² - x + 3.
   1. 7 - h
   2. 7 + h
   3. 7 - 2h
   4. 7 + 2h
6. Compute f′(1) where f(x) = 2√(3x² - 4x + 5).
   1. 2
   2. -2
   3. 1
   4. 3
7. Find the average rate of change of g(t) = t² - 3t + 4 on the interval [2,4].
   1. 4
   2. 6
   3. 8
   4. 10
8. Suppose g(4) = 4, g′(4) = -2, and f(x) = g(x)/x. Find f′(4).
   1. -5/8
   2. -7/8
   3. -3/4
   4. -5/4
9. The second derivative of (x + 4)(11x + 1)(3x - 2) is
   1. 198x + 226
   2. 226x² + 198x
   3. 226x + 198
   4. 198x² - 226x
10. If y = f(x), then at turning points
    1. f(x) = 0
    2. dy/dx = 0
    3. dy/dx > 0
    4. dy/dx = 1
11. Which of these is not correct about the first principle?
    1. dy/dx = [f(x+h) - f(x)]/x
    2. dy/dx = [f(x-h) + f(x)]/h
    3. dy/dx = lim_{h→0} {[f(x+h) - f(x)]/h}
    4. dy/dx = lim_{x→0} {[f(x+h) - f(x)]/x}
12. What is the limit of (x³ - 1)/(x² - 1) as x → 1?
    1. ∞
    2. -∞
    3. 3/2
    4. 1
13. Evaluate lim_{x→π/2} (sin x)/(cos x).
    1. Not possible
    2. π/2
    3. 0
    4. ∞
14. If y = 112, then dy/dx is
    1. 112
    2. 112y
    3. 112x
    4. 211
15. If f(x,y) = 1 + x²/y + y³/x, calculate f(20,14).
    1. 5837/35
    2. 20378/35
    3. 697/7
    4. 10009/5
16. Given g(x) = 2 + 3x - 8x² and h(x) = (x - 1)², then g(h(x)) is
    1. Very difficult
    2. 3 + 26x - 45x² - 8x⁴
    3. -3 + 26x - 45x² + 32x³ - 8x⁴
    4. 3 - 26x - 45x² - 32x³ - 8x⁴
17. At minimum points of y = f(x), which of the following is correct?
    1. d²y/dx² = 1
    2. d²y/dx² < 0
    3. d²y/dx² = 0
    4. d²y/dx² > 0
18. The equation y = x³(x - 4) has ______ number of inflexion.
    1. 1
    2. 2
    3. 3
    4. 4
19. If y = θⁿ, where n ∈ ℤ⁺, which of the following is not correct?
    1. θ²y″ - n²y′ + ny = 0
    2. θy′ = ny
    3. θ²y″ = n(n-1)y
    4. y + y′ = nθⁿ(n⁻¹ + θ⁻¹)
20. Differentiate sin²(x² + 1).
    1. 4x cos²(x² + 1)
    2. 2 cos(x² + 1)
    3. 4x sin(x² + 1) cos(x² + 1)
    4. 2x sin²(x² + 1) cos(x² + 1) + 2x
21. The first derivative of cosᵐ ψ with respect to ψ is
    1. -m cosᵐ⁻¹ ψ sin ψ
    2. m cosᵐ⁻¹ ψ sinᵐ ψ
    3. -m sinᵐ⁻¹ ψ cosᵐ ψ
    4. m sinᵐ ψ
22. Find d/dx [arcsin((1 - x²)/(1 + x²))].
    1. Not possible
    2. 2(1 + x²)
    3. (1 - x²)/(1 + x²)
    4. -2/(1 + x²)
23. The demand and cost functions of Cyntax Nig. Ltd are p = 22500 - 3q², C = 5000 + 14400q. The marginal revenue function is
    1. 22500 - 3q²
    2. -6q
    3. 22500² - 3q³
    4. 22500 - 9q²
24. The quantity for maximum revenue is
    1. 49
    2. 50
    3. 51
    4. 52
25. The price for maximum revenue is
    1. 13000
    2. 14500
    3. 15000
    4. 13500
26. The profit function is
    1. 5000 - 8100q + 3q³
    2. 8100q - 3q³ + 5000
    3. -8100q - 5000 + 3q³
    4. 8100q - 5000 - 3q³
27. The marginal profit function is
    1. 8100 - 9q²
    2. -8100 + 9q²
    3. 8100 + 9q²
    4. -8100 - 9q²
28. The maximum profit is
    1. 175000
    2. 150070
    3. 157000
    4. 751000
29. The price for maximum profit is
    1. 19800
    2. 8100
    3. 5000
    4. 22500
30. The average cost function is
    1. 5000q + 14400q²
    2. 14400q
    3. 5000q⁻¹ + 14400q²
    4. 5000q + 7200q²
31. The marginal cost function is
    1. 5000
    2. 14400
    3. q² + 1
    4. 5000q - q²
32. At what quantity is the marginal cost a minimum?
    1. 0
    2. Any quantity
    3. 50
    4. 28
33. Find the equation of the tangent at the point (2,2) on the curve y² = 2x.
    1. x - 2y + 2 = 0
    2. y - 2x - 2 = 0
    3. x - 2y - 2 = 0
    4. y + 2x - 2 = 0
34. The equation of the normal at the point (3, 4/3) on xy = 4 is
    1. 12x - 27y - 65 = 0
    2. 27x + 12y + 65 = 0
    3. 12x + 27y + 65 = 0
    4. 27x - 12y - 65 = 0
35. The minimum value of y = x³ - 3x² - 9x + 27 is
    1. 54
    2. 0
    3. 27
    4. -27
36. The maximum value of y = x³ - 3x² - 9x + 27 is
    1. 36
    2. 34
    3. 32
    4. 30
37. The integral ∫ (5x + 3)^0.5 dx evaluates to
    1. (5x + 3)^1.5/1.5
    2. (3/2)(5x + 3)^(-0.5)
    3. (3/10)(5x + 3)^0.5
    4. (2/15)(5x + 3)^1.5
38. ∫₂^{2√3} dx/(x² + 4) =
    1. π/4
    2. 1/2 arctan(1/2 x)
    3. π/3 - π/4
    4. π/4 - π/3
39. The best substitution to evaluate ∫ x√(a² - x²) dx is
    1. x = a cos θ
    2. x = a sin θ
    3. u = a² - x²
    4. x = a tan θ
40. Evaluate ∫₀² √(x/(4 - x)) dx.
    1. π + 2
    2. 2 - π
    3. π - 2
    4. 2
41. Integrate cos² ax with respect to x.
    1. cot ax + c
    2. tan² ax + c
    3. (1/a)cot ax + c
    4. (-1/a)cot ax + c
42. Evaluate ∫ ln x dx.
    1. Not Possible
    2. ln x + c
    3. ln x - x + c
    4. x ln x - x + c
43. The expression for ∫ x e^{3x} dx is
    1. (1/3)x e^{3x} - (1/9)e^{3x} + c
    2. (1/3)e^{3x} - (1/9)x e^{3x} + c
    3. (1/3)x e^{3x} + (1/9)e^{3x} + c
    4. (-1/3)e^{3x} + (1/9)x e^{3x} + c
44. A rule f which assigns to each element x of a set X a unique element y of a set Y is called
    1. Equation
    2. Differential
    3. Function
    4. Integration
45. If y(x) = u(x)v(x)w(x), then dy/dx is
    1. uvw + uvw + uvw
    2. does not exist
    3. uvw + uvw + uvw
    4. uvw + uvw
46. Given that the acceleration of an electron is y = ∫ e^{cos x} ln x arcsin x dx, then the velocity of this electron is
    1. e^{sin x} ln x arccos x
    2. ln x sin x
    3. e^{cos x} e^x
    4. e^{cos x} ln x arcsin x
47. Evaluate ∫_a^a e^{sin x} ln x arccos x dx.
    1. 0
    2. 1
    3. 2
    4. 2a
48. Evaluate ∫ dx/(x ln x).
    1. ln x + c
    2. ln(x ln x) + c
    3. ln(ln x) + c
    4. x ln x + c
49. Integrate x with respect to x.
    1. -x + c
    2. x²/2 + c
    3. x + c
    4. 1 + c
50. Using the substitution u = t - 1/t, the integral ∫ (t² + 1)/(t⁴ + 1) dt reduces to
    1. ∫ du/(u² + 1)
    2. ∫ du/(u² + 2)
    3. ∫ du/(u² + 3)
    4. ∫ du/(u² + 4)

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