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Differentiation practice

QCE Mathematical Methods · Original practice questions with worked solutions

All differentiation practice questions

98 original questions · Page 2 of 5

  1. Q78 · Original practice · 3 marks
    For f(x)=−2x2+12x+5f(x)=-2x^2+12x+5 on 0≤x≤50\le x\le5, determine the maximum value of ff.
    Further applications of differentiation
  2. Q86 · Original practice · 4 marks
    For f(x)=ln⁡xxf(x)=\dfrac{\ln x}{x}, x>0x>0, determine the stationary point and classify it.
    Differentiation of exponential and logarithmic functions
  3. Q87 · Original practice · 5 marks
    For f(x)=sin⁡x+12cos⁡(2x)f(x)=\sin x+\dfrac12\cos(2x) on 0≤x≤π0\le x\le\pi, determine all interior stationary points.
    Differentiation of trigonometric functions and differentiation rules
  4. Q88 · Original practice · 5 marks
    A closed cylindrical can has volume 500π cm3500\pi\text{ cm}^3. Let its radius be rr cm and height be hh cm. Determine the radius and height that minimise its total surface area.
    Further applications of differentiation
  5. Q101 · Original practice · 1 mark
    For f(x)=7e−3x+4f(x)=7e^{-3x}+4, determine f′(0)f'(0).
    Differentiation of exponential and logarithmic functions
  6. Q102 · Original practice · 1 mark
    Solve ln⁡(x−2)=1\ln(x-2)=1 for x>2x>2.
    Differentiation of exponential and logarithmic functions
  7. Q103 · Original practice · 1 mark
    For f(x)=exln⁡xf(x)=e^x\ln x, x>0x>0, determine the gradient of the tangent at x=1x=1.
    Differentiation of exponential and logarithmic functions
  8. Q104 · Original practice · 3 marks
    A population is modelled by P(t)=1200e−0.08tP(t)=1200e^{-0.08t}, where tt is measured in years. Determine P′(5)P'(5) and interpret its meaning in context.
    Differentiation of exponential and logarithmic functions
  9. Q105 · Original practice · 5 marks
    The concentration of a substance is modelled by C(t)=t2e−t/3C(t)=t^2e^{-t/3} for t≥0t\ge0. Determine the time at which CC is maximised. Then determine C(3)C(3) as a percentage of this maximum value.
    Differentiation of exponential and logarithmic functions
  10. Q106 · Original practice · 1 mark
    For y=sin⁡2(2x)y=\sin^2(2x), determine dydx\dfrac{dy}{dx} at x=π8x=\dfrac{\pi}{8}.
    Differentiation of trigonometric functions and differentiation rules
  11. Q107 · Original practice · 1 mark
    For x≠0x\ne0, if y=cos⁡xxy=\dfrac{\cos x}{x}, then dydx\dfrac{dy}{dx} is
    Differentiation of trigonometric functions and differentiation rules
  12. Q108 · Original practice · 1 mark
    For f(x)=x+2cos⁡xf(x)=x+2\cos x on 0<x<π0<x<\pi, the stationary points occur at
    Differentiation of trigonometric functions and differentiation rules
  13. Q109 · Original practice · 3 marks
    Determine the equation of the tangent to y=cos⁡(2x)y=\cos(2x) at x=π6x=\dfrac{\pi}{6}.
    Differentiation of trigonometric functions and differentiation rules
  14. Q110 · Original practice · 5 marks
    A particle has displacement s(t)=tsin⁡ts(t)=t\sin t for 0≤t≤π0\le t\le\pi. Determine, to three decimal places, the time at which the velocity is maximised and the corresponding maximum velocity.
    Differentiation of trigonometric functions and differentiation rules
  15. Q111 · Original practice · 1 mark
    The graph shown represents f′(x)f'(x). On which interval is ff decreasing?
    Further applications of differentiation
  16. Q112 · Original practice · 1 mark
    If f′′(x)=12x2−12f''(x)=12x^2-12, the possible points of inflection occur at
    Further applications of differentiation
  17. Q113 · Original practice · 1 mark
    For f(x)=x3−3x2−9x+5f(x)=x^3-3x^2-9x+5, the local maximum point is
    Further applications of differentiation
  18. Q114 · Original practice · 4 marks
    Analyse the turning points of f(x)=x4−8x2f(x)=x^4-8x^2: find their coordinates and state the nature of each turning point.
    Further applications of differentiation
  19. Q115 · Original practice · 6 marks
    A rectangular poster must contain a printed area of 600 cm2600\text{ cm}^2. The side margins are each 22 cm wide and the top and bottom margins are each 33 cm wide. Determine the dimensions of the printed area that minimise the total area of the poster, and state the minimum total area.
    Further applications of differentiation
  20. Q151 · Original practice · 1 mark
    For f(x)=5e2x−3f(x)=5e^{2x}-3, determine f′(0)f'(0).
    Differentiation of exponential and logarithmic functions
  21. Q152 · Original practice · 1 mark
    Solve e2x=7e^{2x}=7.
    Differentiation of exponential and logarithmic functions
  22. Q153 · Original practice · 1 mark
    For x>14x>\dfrac14, ddxln⁡(4x−1)\dfrac{d}{dx}\ln(4x-1) is
    Differentiation of exponential and logarithmic functions
  23. Q154 · Original practice · 4 marks
    The temperature of a drink is modelled by T(t)=18+72e−0.2tT(t)=18+72e^{-0.2t} degrees Celsius, where tt is the number of minutes after the drink is poured.
    (a) Determine the instantaneous rate of change of the temperature at t=4t=4.
    (b) Determine when the temperature first reaches 30∘C30^\circ\mathrm C.
    Differentiation of exponential and logarithmic functions
  24. Q155 · Original practice · 5 marks
    A contaminant concentration is modelled by C(t)=Ae−ktC(t)=Ae^{-kt}, where A>0A>0, k>0k>0 and tt is measured in hours. Measurements give C(2)=36C(2)=36 and C(8)=15C(8)=15.
    Determine AA and kk, and hence determine the instantaneous rate of change C′(5)C'(5).
    Differentiation of exponential and logarithmic functions