MTH102 Past Questions PDF (2017/2018)

# FEDERAL UNIVERSITY DUTSE ## FACULTY OF SCIENCE ### DEPARTMENT OF MATHEMATICS #### 2017/2018 SECOND SEMESTER EXAMINATION Course Title: Elementary Mathematics II (Calculus) Course Code: MTH 102 (3 Credits) Level: 100 Date: 9th July, 2018 Time Allowed: 3 Hours **READ THE INSTRUCTIONS BELOW CAREFULLY...

# FEDERAL UNIVERSITY DUTSE ## FACULTY OF SCIENCE ### DEPARTMENT OF MATHEMATICS #### 2017/2018 SECOND SEMESTER EXAMINATION Course Title: Elementary Mathematics II (Calculus) Course Code: MTH 102 (3 Credits) Level: 100 Date: 9th July, 2018 Time Allowed: 3 Hours **READ THE INSTRUCTIONS BELOW CAREFULLY!!!** **Instructions:** 1. DO NOT write ANYTHING on this paper EXCEPT your REG. NO in the space provided above. 2. All your rough work should be done inside the answer booklet provided. 3. Answer All Questions. **1. All of the following are true about real-valued functions EXCEPT.** * i. All relations are functions but not all functions are relations. * ii. Every element of the domain has an image in the co-domain. * iii. A function has an inverse if it is bijective. **2. Determine the domain of f: x → 2x² - 1, if the range is {1, 7, 17}.** **3. Determine the composite functions g[f(x)] and f[g(x)] if f(x) = 4x - 1 and g(x) = 2x + 3.** **4. Evaluate lim_(x→0) (tan x)/(sin x).** **5. Find the inverse of the function h(x) = (2x - 1)/(3 - x), where x is a real number.** **6. Use implicit differentiation to find dy/dx if y = (x + y)² + 10** **7. Find the derivative of f(x) = 5^(x^2).** **8. Find d/dx if y = ln(4x - 1)²** **9. Find d/dx if x = (1 + t)/(1 + t²) and y = (1 + t²)/(1 + t).** **10. Find the fourth derivative of the function y = 3x³ - 2x² + x² + 1** **11. Find the slope of x²y + xy² + 3x - 13 = 0 at (1, 2)** **12. If y is the total cost of manufacturing x units of an item and is related by y = (x² + 20)/4x find dy/dx.** **13. All the following statements are TRUE about continuity of function at a point c EXCEPT** * i. f(c) is defined * ii. lim_(x→c) f(x) exists * iii. lim_(x→c) f(x) = f(c) **14. Find dy/dx of the function y = sin(x²).** **15. Use the slope obtained in Question 14 to find the equation of the tangent to the same curve at point (1, 2).** **16. If the equation of a tangent to a curve is (15/13)y + (41/13)x = 0, find the equation of the normal to the curve at (1,2).** **17. Find the critical values of the function y = x³ - 6x² + 9x + 6** **18. Use the critical values obtained in Question 17 to find the maximum and minimum points of the function.** **19. Evaluate lim_(x→0) (1 - cos x)/x²** **20. Evaluate lim_(x→-2) (x² - 2x + 3)/(x² + 4x + 4)** **21. Find the derivative of the function f(x) = cot x** **22. Evaluate the indefinite integral ∫√(x²/3 - 1) dx** **23. Evaluate the integral ∫sin³x dx** **24. Evaluate the indefinite integral ∫ x⁵e^x dx** **25. Evaluate the integral ∫(3x - 4)/(x² + x - 5) dx** **26. If f(x) = x² - 4x find f’(a) given that a = -1** **27. The velocity of a particle moving in a straight line at time t is given by v = 2t² - 3t. Find an expression of the distance (s) travelled, if s = 0 when t = 0.** **28. Evaluate the integral ∫(2x)/(x + 1)² dx** **29. Evaluate ∫(1/√(1+√y)) dy** **30. Find ∫(3x - 5)² dx** **31. Find the derivative with respect to x of y = x^(x²sin2x)** **32. Evaluate ∫(5x²/√(x² + 5y)) dx** **33. Find ∫(5x² + 5)/(x² + 5)² dx** **34. If y = sin θ/(1 + cos θ) find dy/dx** **35. Evaluate lim_(x→0) sin x/x** **36. Find dy/dx if x = sin²t and y = cos²t** **37. Given y = cos 3x sin²3x find y'(x)** **38. Evaluate lim_(x→-2) ((x + 3)(2x - 1))/(x² + 3x - 2)** **39. If y = ln(sin 2x), find dy/dx** **40. The function f over the set of real numbers is defined by f(x) = (x² - 3), find f’(x).** **41. Evaluate the integral ∫ cos²xdx** **42. Evaluate ∫ dx/x^4** **43. Evaluate ∫(3sin x + 5cos x)dx** **44. Find ∫(3x - 5)² dx** ## Page 2 **45. Evaluate the integral ∫e^sinx dx** **46. Evaluate ∫3^x dx** **47. Evaluate lim_(x→1)((x² - 1)/(x - 1))** **48. Evaluate lim_(x→0) (3/x - 9/x²)** **49. Evaluate ∫sin θ/cos² θ dθ** **50. Evaluate ∫√(x²-3x) dx**