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Differentiation I · CAPE Pure Mathematics Unit 1

204 past-paper questions on Differentiation I, part of Calculus I, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

Practise Differentiation I
  1. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The function f(x) is decreasing for the range
  2. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1Given that f(x) = (2x + 1)^3, then f'(2) equals
  3. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The first derivative of -\frac{1}{x^2 - 1} is
  4. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The stationary point of the function y = (x - 1)^2 is
  5. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The gradient of the normal to the curve y = \ln x at x = 2 is
  6. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1A curve is defined by the equation y = -5(2x - 1)^2. Given that x increases at a rate of 1\text{ unit per second} when x = 1, what is the corresponding rate of change for y?
  7. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1From the diagram above, which of the following statements are true? I. f'(1) < 0 II. f(1) > k III. f(2) = 0 IV. f'(2) = k
  8. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Given that f(x) = (2x + 1)^3, f'(2) equals
  9. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Given f(x) = x^3, f'(x) is
  10. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If y = \frac{(x^2 + 8)^4}{3} then \frac{\mathrm{d}y}{\mathrm{d}x} =
  11. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The radius of a circle is increasing at a rate of 0.1\text{ cm s}^{-1}. At the instant when the radius is 3\text{ cm}, the rate of increase of the area in \text{cm}^2\text{ s}^{-1} is
  12. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1At x = 0, the gradient of the function x^3 - 2x^2 is
  13. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The number of stationary points of the function g(x) = \frac{2}{x - 3}, x \in \mathbb{R}, and x \ne 3, is
  14. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Which of the following is true given that f'(x) and f''(x) can be expressed in terms of x?
  15. 5(b)7 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2A ladder which is 10\text{ metres} long is leaning against a wall. The bottom of the ladder is sliding away from the base of the wall at a rate of 4\text{ m/s}. Determine the rate at which the top of the ladder is…
  16. 5(c)(i)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2A function f is given as f(x) = 4x^3 - 3x^2 + 1, for -1 \le x \le 1. Determine the coordinates of the stationary points of the function f.
  17. 5(c)(ii)4 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Determine the nature of these stationary points.
  18. 5(c)(iii)3 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Determine the absolute maximum and minimum values of the function f.
  19. 11(b)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Deduce, from first principles, the derivative with respect to x of y = sqrt(x).
  20. 13(a)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the value of the constant k.
  21. 13(b)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the value of d^2y/dx^2 at P.
  22. 13(c)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the equation of the normal to the curve at P.
  23. 14(a)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the coordinates of the stationary points of the function f.
  24. 14(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Determine the nature of the stationary points of f.
  25. 15(a)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the coordinates of EACH of the points P, Q and R.
  26. 12(a)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Determine the nature of the critical value(s) of f(x).
  27. 12(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Differentiate, with respect to x, f(x) = sin^2(x^2).
  28. 13(a)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the coordinates of each of the stationary points, A and B.
  29. 13(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the equation of the normal to the curve f(x) = x(x^2 - 12) at the origin.
  30. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The number of visas V(x) issued by an embassy annually, is given by V(x) = 7x^2 - 42x + 72. The LEAST number of visas issued in a particular year, x, is
  31. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If y = \tan 6x then \frac{dy}{dx} is
  32. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If y = \frac{x - 6}{3 - 4x} then \frac{dy}{dx} is
  33. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  34. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The gradient at x = \frac{\pi}{6} on the curve y = \sin x is
  35. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1A curve is defined by the equation y = -5(1 - 2x)^2. Given that x increases at a rate of 1 unit per second when x = 1, what is the corresponding rate of change for y?
  36. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1A rod is heated and its length at time t seconds is given by L = 5t^2 + 100\text{ centimetres}. When t = 3, the rate of increase of L, in \text{cm s}^{-1}, is
  37. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If a rope of length k\text{ metres} forms three sides of a rectangle of width x\text{ metres} then the area, R, in square metres, of the rectangle is given by R = x(k - 2x). The MAXIMUM value of R is
  38. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1An arch may be modelled by the Cartesian equation y = -2x^2 + 4x + 1, where x and y represent respectively horizontal and vertical distances. The coordinates of the HIGHEST point on the arch are
  39. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\frac{\mathrm{d}}{\mathrm{d}r}(\pi r^2) is equal to
  40. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1A curve is defined for x > 0 by the equation y = x + \frac{2}{x}. The gradient of the curve at x = 2 is
  41. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\frac{\mathrm{d}}{\mathrm{d}x}(x^3 \sin x) may be expressed as
  42. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Given y = 3x^2 + 5\sin 2x, then \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} is equal to
  43. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The displacement, s metres, of a marble moving along a board at time t minutes is given by s(t) = 4t^3 - 30t^2 + 72t + 7 for t \ge 0. For what values of t is the displacement of the marble increasing?
  44. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1A curve has a stationary point at (2, 4). The equation of the normal at (2, 4) on the curve is
  45. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If y = 3x^2 + 5\sin 2x, then \frac{d^2y}{dx^2} is equal to
  46. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If y = \tan 6x then \frac{dy}{dx} is
  47. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1A curve has a stationary point at (2, 4). The equation of the normal to the curve at (2, 4) is
  48. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Given \frac{dy}{dx} = 2x, then a sketch of the graph of y is
  49. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1A rod is heated and its length at time t seconds is given by L = 5t^2 + 100 centimetres. When t = 3, the rate of increase of L, in \text{cm s}^{-1}, is
  50. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The radius of a circle is increasing at a rate of 0.1\,\text{cm s}^{-1}. At the instant when the radius is 3\text{ cm}, the rate of increase of the area in \text{cm}^2\text{ s}^{-1} is
  51. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The function f(x) is decreasing for the range
  52. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1An arch may be modelled by the Cartesian equation y = -2x^2 + 4x + 1, where x and y represent, respectively, horizontal and vertical distances. The coordinates of the HIGHEST point on the arch are
  53. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The first derivative of \frac{-1}{x^2 - 1} with respects to x is
  54. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1\frac{d}{dr}(\pi r^2) is equal to
  55. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1If y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  56. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1Given y = 3x^2 + 5\sin 2x, then \frac{d^2y}{dx^2} is equal to
  57. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1If y = \tan 6x then \frac{dy}{dx} is
  58. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The displacement, s metres, of a marble moving along a board at time t minutes is given by s(t) = 4t^3 - 30t^2 + 72t + 7 for t \ge 0. For what values of t is the displacement of the marble increasing?
  59. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1Water is leaking from a tank. The rate of change in volume of water in the tank with respect to time, t, is inversely proportional to the volume, V, of water in the tank. If k is a positive constant of…
  60. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1\frac{d}{dx}(x^3 \sin x) may be expressed as
  61. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If y = \frac{x - 6}{3 - 4x}, then \frac{dy}{dx} is
  62. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  63. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If y = \tan 6x then \frac{dy}{dx} is
  64. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The path of an object is given parametrically as x = \sin t + 2, y = \cos t + 1. The slope of the tangent at t = \frac{\pi}{4} is
  65. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The gradient of the normal to the curve y = 3x^2 - 2x + 1 at x = 1 is
  66. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Water is leaking from a tank. The rate of change in volume of the water in the tank with respect to time, t, is inversely proportional to the volume, V, of water in the tank. If k is a positive constant of…
  67. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The radius of a circle is increasing at a rate of 0.1\text{ cm s}^{-1}. At the instant when the radius is 3\text{ cm}, the rate of increase of the area in \text{cm}^2\text{ s}^{-1} is
  68. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1From the diagram above, which of the following statements are true? \begin{align*} \text{I.} & \quad f'(1) < 0 \\ \text{II.} & \quad f(1) > k \\ \text{III.} & \quad f(2) = 0 \\ \text{IV.} & \quad f'(2) = k \end{align*}
  69. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1\frac{d}{dr}(\pi r^2) is equal to
  70. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1A curve is given parametrically by the equations x = t^2 - 2t, y = t^2 + 2t. The simplest expression for \frac{dy}{dx} is given by
  71. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1If y = \sqrt{2x + 1}, then \frac{d^2y}{dx^2} is
  72. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1If \frac{dy}{dx} = 2xy, then the value of \frac{d^2y}{dx^2} at the point (1, 2) is
  73. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The gradient at x = \frac{\pi}{6} on the curve y = \sin x is
  74. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Water is leaking from a tank. The rate of change in volume of water in the tank with respect to time, t, is inversely proportional to the volume, V, of water in the tank. If k is a positive constant of…
  75. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1\frac{d}{dx}(x^3 \sin x) may be expressed as
  76. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1Given that f(x) = (2x + 1)^3 then f''(2) equals
  77. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1\frac{d}{dx}(x^3 \sin x) may be expressed as
  78. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  79. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The gradient of the normal to the curve y = \ln x at x = 2 is
  80. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The gradient of the normal to the curve y = 3x^2 - 2x + 1 at x = 1 is
  81. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1Given \frac{dy}{dx} = 2x then a sketch graph of y may be represented by I. II. III. IV.
  82. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The radius of a circle is increasing at a rate of 0.1\text{ cm s}^{-1}. At the instant when the radius is 3\text{ cm}, the rate of increase of the area in \text{cm}^2\text{ s}^{-1} is
  83. 5(b)8 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2How fast is the water in the cup rising when the height is 4 cm?
  84. 5(c)(ii)9 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Using the method of the second derivative, determine the height, x, that will maximize the volume of the container.
  85. 5(c)(iii)1 mark· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Determine the maximum volume of the container.
  86. 6(c)7 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Determine the velocity and height of the rocket 5 seconds after launch.
  87. 5(c)(i)9 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Determine the stationary points of g and the nature of the stationary points.
  88. 5(c)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Hence, or otherwise, sketch the graph of g showing its intercepts, stationary points and any other important features on the provided grid.
  89. 5(c)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Given that y = (2x³ + 4)⁷ and x = 1 - 2t, determine dy/dt.
  90. 5(d)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine f'(x), using first principles, given that f(x) = sin x.
  91. 6(a)(ii)a)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the coordinates of the stationary points on y = f(x).
  92. 6(a)(ii)b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Classify EACH of the stationary points on y = f(x) as a maximum point, minimum point or point of inflection.
  93. 6(a)(iii)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Hence, sketch a carefully labelled graph of y = f(x).
  94. 2(a)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Hence, or otherwise, find the turning point of the graph and determine whether it is a maximum or a minimum.
  95. 2(a)(iv)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(iv)By sketching the graph of y = f(x), or otherwise, state the turning point P of the graph and indicate whether P is a maximum or a minimum.
  96. 3(a)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Find dy/dx in terms of t.
  97. 4(a)(ii)2 marks· Pure Mathematics · Unit 1 Q4 4(a)(ii)hence, find ONE value of x between 0° and 360° for which the curve y = 4 sin x - cos x has a stationary point.
  98. 4(b)(ii)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)The viewing angle of the painting, (α – β), is at a maximum when x = √h (d+h). Calculate the maximum viewing angle, in radians, when d = 3h.
  99. 5(a)5 marks· Pure Mathematics · Unit 1 Q5 5(a)Using differentiation, determine the range of real values of x for which the function f: x → 12 + 6x² - x³ is decreasing.
  100. 5(a)5 marks· Pure Mathematics · Unit 1 Q5 5(a)Given that y = tan⁻¹ (1 − x²), find d²y/dx².
  101. 5(a)(i)4 marks· Pure Mathematics · Unit 1 Q5 5(a)(i)Obtain dy/dx.
  102. 5(a)(ii)2 marks· Pure Mathematics · Unit 1 Q5 5(a)(ii)Show that y dy/dx = 5x.
  103. 5(a)(iii)4 marks· Pure Mathematics · Unit 1 Q5 5(a)(iii)Hence, or otherwise, show that y d^2y/dx^2 + (dy/dx)^2 = 5.
  104. 5(a)(iii)7 marks· Pure Mathematics · Unit 1 Q5 5(a)(iii)Hence, or otherwise, differentiate with respect to x, from first principles, the function y = sin 2x.
  105. 5(b)6 marks· Pure Mathematics · Unit 1 Q5 5(b)Using first principles, determine the derivative of f(x) = sin (2x).
  106. 5(b)6 marks· Pure Mathematics · Unit 1 Q5 5(b)Find the values of u and v.
  107. 5(b)6 marks· Pure Mathematics · Unit 1 Q5 5(b)Differentiate x³ with respect to x from first principles.
  108. 5(b)5 marks· Pure Mathematics · Unit 1 Q5 5(b)Show that dy/dx = (−4x³ – 10x² – 14x + 4) / (x² + 2)⁴
  109. 5(b)1 mark· Pure Mathematics · Unit 1 Q5 5(b)Find
  110. 5(b)4 marks· Pure Mathematics · Unit 1 Q5 5(b)If y = A/x + Bx, where A and B are constants, show that x(d²y/dx²) + x(dy/dx) = y.
  111. 5(b)(i)3 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Find dy/dx in terms of t.
  112. 5(b)(i)4 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Determine dy/dx in terms of t.
  113. 5(b)(i)3 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Obtain an expression for f'(x).
  114. 5(b)(i)6 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Find the coordinates of the stationary points of f(x).
  115. 5(b)(i)8 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)Let y = 1/√x. Using first principles, find dy/dx.
  116. 5(b)(i)5 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)the values of the constants h and k
  117. 5(b)(i)a)4 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)a)Find dy/dx.
  118. 5(b)(i)b)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(i)b)Show that x² dy/dx = y².
  119. 5(b)(ii)6 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Hence, determine all points of C such that dy/dx = 0.
  120. 5(b)(ii)5 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Find the second derivative of f(x), and hence, determine which stationary point is a local maximum and which is a local minimum.
  121. 5(b)(ii)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Find the stationary point(s) of f.
  122. 5(b)(ii)3 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)Hence, or otherwise, show that x² d²y/dx² + 2(x - y) dy/dx = 0.
  123. 5(b)(ii)4 marks· Pure Mathematics · Unit 1 Q5 5(b)(ii)If y = x/√(1+x), determine an expression for dy/dx. Simplify the answer FULLY.
  124. 5(b)(iii)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(iii)Find the time at which the rate of decrease vanishes.
  125. 5(b)(iii)5 marks· Pure Mathematics · Unit 1 Q5 5(b)(iii)Determine the nature of the stationary point(s) of f.
  126. 5(b)(iii)5 marks· Pure Mathematics · Unit 1 Q5 5(b)(iii)Determine the rate, in metres per minute, at which the tide is falling 75 minutes after high tide.
  127. 5(b)(iv)5 marks· Pure Mathematics · Unit 1 Q5 5(b)(iv)Sketch the curve.
  128. 5(b)(iv)2 marks· Pure Mathematics · Unit 1 Q5 5(b)(iv)Show that the rate of decrease is never negative.
  129. 5(c)7 marks· Pure Mathematics · Unit 1 Q5 5(c)If y = 1/(x² + 2), show that d²y/dx² = 2(3x² - 2)y³.
  130. 5(c)5 marks· Pure Mathematics · Unit 1 Q5 5(c)Find dy/dx in terms of θ.
  131. 5(c)4 marks· Pure Mathematics · Unit 1 Q5 5(c)Find dy/dx in terms of θ. Simplify the answer as far as possible.
  132. 5(c)6 marks· Pure Mathematics · Unit 1 Q5 5(c)Differentiate from first principles, with respect to x, the function y = 1/x².
  133. 5(c)8 marks· Pure Mathematics · Unit 1 Q5 5(c)find the values of u and v.
  134. 5(c)(i)5 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)Find dy/dx in terms of x.
  135. 5(c)(i)1 mark· Pure Mathematics · Unit 1 Q5 5(c)(i)Determine f'.
  136. 5(c)(i)1 mark· Pure Mathematics · Unit 1 Q5 5(c)(i)Simplifying your answers where possible, find expressions in terms of x for the length of TS.
  137. 5(c)(i)7 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)Show that x dy/dx = y/(1 + x²).
  138. 5(c)(i)6 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)Show that f(x) has stationary points at x = -1, x = 0 and x = 2.
  139. 5(c)(i)4 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)Show that h = 20/x - 3x/5.
  140. 5(c)(i)6 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)Find the values of the constants p and q
  141. 5(c)(i)(a)9 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)(a)Show that y(dy/dx) - 2x = 0.
  142. 5(c)(i)(b)9 marks· Pure Mathematics · Unit 1 Q5 5(c)(i)(b)Show that d²y/dx² - 4/y³ = 0.
  143. 5(c)(ii)8 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Show that d²y/dx² + (3y)/(1 + x²)² = 0.
  144. 5(c)(ii)4 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Simplifying your answers where possible, find expressions in terms of x for the length of RS.
  145. 5(c)(ii)1 mark· Pure Mathematics · Unit 1 Q5 5(c)(ii)Determine f''.
  146. 5(c)(ii)7 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Find the coordinates of the stationary points of C and determine the nature of EACH point.
  147. 5(c)(ii)5 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Show that x(x² + 1)dy/dx - 4y = -4/(x² + 1).
  148. 5(c)(ii)4 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Determine the nature of these stationary points.
  149. 5(c)(ii)3 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Hence, find the value of d²y/dx² when x = 0.
  150. 5(c)(ii)9 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Find the height of the box for which its volume V cm³ is a maximum.
  151. 5(c)(ii)3 marks· Pure Mathematics · Unit 1 Q5 5(c)(ii)Find the equation of the normal to the curve at T
  152. 5(c)(iii)11 marks· Pure Mathematics · Unit 1 Q5 5(c)(iii)Calculate the x coordinates of the stationary points of f(x) = 3x⁴ - 2x³ - 6x² + 6x and determine the nature of these stationary points.
  153. 5(c)(iii)5 marks· Pure Mathematics · Unit 1 Q5 5(c)(iii)Sketch the graph of C and label the x-intercepts.
  154. 5(c)(iii)2 marks· Pure Mathematics · Unit 1 Q5 5(c)(iii)Simplifying your answers where possible, find expressions in terms of x for the area, A, of PRST.
  155. 5(c)(iv)5 marks· Pure Mathematics · Unit 1 Q5 5(c)(iv)Simplifying your answers where possible, find expressions in terms of x for the stationary value of x and show that it is a maximum.
  156. 5(c)(v)2 marks· Pure Mathematics · Unit 1 Q5 5(c)(v)Simplifying your answers where possible, find expressions in terms of x for the maximum area of the sheep enclosure.
  157. 5(c)a)4 marks· Pure Mathematics · Unit 1 Q5 5(c)a)Determine whether the function f has turning points.
  158. 5(d)8 marks· Pure Mathematics · Unit 1 Q5 5(d)By investigating the sign of f'(x), determine the range of real values of x for which x³ - 5x + 3 is decreasing.
  159. 5(d)(i)7 marks· Pure Mathematics · Unit 1 Q5 5(d)(i)Solve an appropriate differential equation to show that the number of bacteria present at any time can be modelled by the equation y = 10,000 e⁰.⁰²ᵗ.
  160. 6(a)4 marks· Pure Mathematics · Unit 1 Q6 6(a)Find the equation of the tangent to the curve f(x) = 2x³ + 5x² – x + 12 at the point where x = 3.
  161. 6(a)6 marks· Pure Mathematics · Unit 1 Q6 6(a)Differentiate, with respect to x, (x² + 7)³ + sin 3x.
  162. 6(a)6 marks· Pure Mathematics · Unit 1 Q6 6(a)Find the stationary point(s) of the curve, f(x) = x³ - 3x + 2.
  163. 6(a)6 marks· Pure Mathematics · Unit 1 Q6 6(a)Show that d²y/dx² + 4y = 0.
  164. 6(a)(i)8 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Find the coordinates of each of the stationary points A and B
  165. 6(a)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Differentiate with respect to x: y = sin (3x + 2) + tan 5x.
  166. 6(a)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Given that y = √4x² – 7, show that y dy/dx = 4x.
  167. 6(a)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Differentiate with respect to x: x√(2x - 1).
  168. 6(a)(i)b)8 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)b)Find the coordinates of the stationary points and determine their nature.
  169. 6(a)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Differentiate with respect to x: sin²(x³ + 4).
  170. 6(a)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Differentiate with respect to x: y = (x² + 1) / (x³ - 1).
  171. 6(a)(ii)3 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Hence, or otherwise, show that y d²y/dx² + (dy/dx)² = 4.
  172. 6(a)(ii)2 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Find the equation of the normal to the curve f(x) = x(x² - 12) at the origin, O
  173. 6(a)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Sketch the curve in (a) (i) a) above, clearly marking ALL stationary points and intercepts.
  174. 6(b)2 marks· Pure Mathematics · Unit 1 Q6 6(b)Obtain an expression for the derivative g'(x) of g at x ∈ R.
  175. 6(b)3 marks· Pure Mathematics · Unit 1 Q6 6(b)Determine the nature of the stationary point(s).
  176. 6(b)6 marks· Pure Mathematics · Unit 1 Q6 6(b)Find the stationary points of h.
  177. 6(b)4 marks· Pure Mathematics · Unit 1 Q6 6(b)Determine dy/dx in terms of θ.
  178. 6(b)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Determine the values of x for which the function has stationary points.
  179. 6(b)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Show that for f(x) = x/(x² + 4), f'(x) = (4 - x²)/(x² + 4)².
  180. 6(b)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Show that for f(x) = 2x/(x²+4), f'(x) = (8-2x²)/(x²+4)².
  181. 6(b)(i)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(i)Determine (d/dθ)g(x), in terms of θ.
  182. 6(b)(ii)2 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Determine the values of x for which the function is increasing.
  183. 6(b)(ii)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Find the coordinates of the stationary points of C.
  184. 6(b)(ii)8 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Find the coordinates and nature of the stationary point of the curve in (b) (i) above.
  185. 6(b)(ii)6 marks· Pure Mathematics · Unit 1 Q6 6(b)(ii)Hence, determine the length, x, that maximises the area enclosed by the track.
  186. 6(b)(iii)6 marks· Pure Mathematics · Unit 1 Q6 6(b)(iii)If the value of b = 3, determine a such that f'(0) = lim (t→0) (f(0 + t) - f(0)) / t.
  187. 6(b)(iii)2 marks· Pure Mathematics · Unit 1 Q6 6(b)(iii)Determine the values of x for which the function is decreasing.
  188. 6(b)(iii)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(iii)Determine the nature of EACH stationary point.
  189. 6(b)(iii)3 marks· Pure Mathematics · Unit 1 Q6 6(b)(iii)Sketch the curve in (b) (i) by clearly labelling the stationary points.
  190. 6(c)4 marks· Pure Mathematics · Unit 1 Q6 6(c)Find the stationary points of g.
  191. 6(c)6 marks· Pure Mathematics · Unit 1 Q6 6(c)Use first principles to differentiate f(x) = √x with respect to x.
  192. 6(c)(i)2 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)Determine the value(s) of x where h has a local maximum.
  193. 6(c)(i)4 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)Show that y'' = x sin x.
  194. 6(c)(i)a)3 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)a)Show that h = 45/r² - 2r/3.
  195. 6(c)(i)b)3 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)b)Show that A = 5πr²/3 + 90π/r, where A units is the external surface area of the can.
  196. 6(c)(ii)2 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Determine the value(s) of x where h has a local minimum.
  197. 6(c)(ii)8 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Hence, find a possible value of x such that V is a maximum.
  198. 6(c)(ii)4 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Hence, determine the specific solution of the differential equation y'' = x sin x, given that when x = 0, y = 1 and when x = π, y = 6.
  199. 6(c)(ii)5 marks· Pure Mathematics · Unit 1 Q6 6(c)(ii)Hence, find the value of r for which A is a minimum and the corresponding minimum value of A.
  200. 6(d)6 marks· Pure Mathematics · Unit 1 Q6 6(d)Sketch the curve, f(x) = x³ - 3x + 2, for -2 ≤ x ≤ 2.
  201. 6(d)5 marks· Pure Mathematics · Unit 1 Q6 6(d)Using the above, and any other information, sketch the graph of h.
  202. 6(d)(i)2 marks· Pure Mathematics · Unit 1 Q6 6(d)(i)Determine the value(s) of x where g has a local maximum.
  203. 6(d)(ii)2 marks· Pure Mathematics · Unit 1 Q6 6(d)(ii)Determine the value(s) of x where g has a local minimum.
  204. 6(e)7 marks· Pure Mathematics · Unit 1 Q6 6(e)Using the above, and any other information, sketch the graph of g.

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