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Differentiation practice

QCE Mathematical Methods · Original practice questions with worked solutions

All differentiation practice questions

98 original questions · Page 4 of 5

  1. Q270 · Original practice · 4 marks
    The scent concentration from a diffuser is modelled by C(t)=Ae−ktC(t)=Ae^{-kt}, where CC is measured in arbitrary units and tt in hours. Measurements give C(0)=80C(0)=80 and C(3)=20C(3)=20.
    Differentiation of exponential and logarithmic functions
  2. Q271 · Original practice · 4 marks
    The curve y=ln⁡(2x+1)y=\ln(2x+1) is defined for x>−12x>-\dfrac12.
    Differentiation of exponential and logarithmic functions
  3. Q272 · Original practice · 5 marks
    A pop-up shop models its hourly sales rate by R(t)=60te−tR(t)=60te^{-t} for 0≤t≤50\leq t\leq5, where tt is hours after opening and RR is items per hour.
    Further applications of differentiation
  4. Q273 · Original practice · 5 marks
    Let f(x)=xx2+4f(x)=\dfrac{x}{x^2+4}.
    Differentiation of trigonometric functions and differentiation rules
  5. Q274 · Original practice · 3 marks
    For f(x)=3sin⁡(2x)+xf(x)=3\sin(2x)+x,
    Differentiation of trigonometric functions and differentiation rules
  6. Q275 · Original practice · 5 marks
    The height of a decorative wave is modelled by h(x)=2sin⁡2xh(x)=2\sin^2x for 0≤x≤π0\leq x\leq\pi.
    Differentiation of trigonometric functions and differentiation rules
  7. Q276 · Original practice · 6 marks
    Let f(x)=x3−3xf(x)=x^3-3x.
    Further applications of differentiation
  8. Q277 · Original practice · 5 marks
    A rectangular herb garden is built against a straight wall. A total of 3636 m of fencing is used for its other three sides. Let xx metres be the side length perpendicular to the wall.
    Further applications of differentiation
  9. Q278 · Original practice · 6 marks
    Squares of side xx cm are cut from each corner of a 3030 cm by 1818 cm sheet. The sides are folded up to make an open box.
    Further applications of differentiation
  10. Q279 · Original practice · 5 marks
    A cup of tea has temperature T(t)=22+68e−0.12tT(t)=22+68e^{-0.12t} degrees Celsius, tt minutes after pouring.
    Differentiation of exponential and logarithmic functions
  11. Q280 · Original practice · 4 marks
    Let f(x)=excos⁡xf(x)=e^x\cos x.
    Differentiation of trigonometric functions and differentiation rules
  12. Q332 · Original practice · 8 marks
    A rectangular logo has its lower edge on the xx-axis, left edge on the yy-axis and upper-right corner on y=e−xy=e^{-x}. Its width is b>0b>0 units.
    Further applications of differentiation
  13. Q338 · Original practice · 10 marks
    A festival entrance has upper profile y=4sin⁡(πx/12)y=4\sin(\pi x/12) metres for 0≤x≤120\leq x\leq12. A rectangular screen stands on the baseline and is centred at x=6x=6. Its top corners touch the profile, and its height is hh metres, with 0<h<40<h<4.
    Further applications of differentiation
  14. Q339 · Original practice · 1 mark
    Determine lim⁡h→0e2h−1h\displaystyle\lim_{h\to0}\frac{e^{2h}-1}{h}.
    Differentiation of exponential and logarithmic functions
  15. Q340 · Original practice · 1 mark
    The line LL is perpendicular to the tangent to y=ln⁡(3x+1)y=\ln(3x+1) at x=1x=1. Determine the gradient of LL.
    Differentiation of exponential and logarithmic functions
  16. Q341 · Original practice · 1 mark
    The graph of f′(x)=(x−1)2(x−3)f^{\prime}(x)=(x-1)^2(x-3) is shown for 0<x<40<x<4. Which statement describes the local extrema of ff on this interval?
    Further applications of differentiation
  17. Q350 · Original practice · 1 mark
    The graphs of y=e−xy=e^{-x} and y=xy=x intersect for x>0x>0. Determine the x-coordinate of their intersection to three decimal places.
    Differentiation of exponential and logarithmic functions
  18. Q351 · Original practice · 5 marks
    The graph of f(x)=aln⁡(x+2)+bf(x)=a\ln(x+2)+b has vertical asymptote x=−2x=-2 and passes through (0,1)(0,1) and (2,3)(2,3), as shown.
    Differentiation of exponential and logarithmic functions
  19. Q352 · Original practice · 5 marks
    Let f(x)=e2xsin⁡(3x)f(x)=e^{2x}\sin(3x).
    Differentiation of trigonometric functions and differentiation rules
  20. Q353 · Original practice · 5 marks
    Consider g(x)=ex/(x+2)g(x)=e^x/(x+2) for x>−2x>-2.
    Differentiation of trigonometric functions and differentiation rules
  21. Q354 · Original practice · 3 marks
    A student claims that the graph of f(x)=x4/12f(x)=x^4/12 has a point of inflection at x=0x=0 because f′′(0)=0f^{\prime\prime}(0)=0. Evaluate the reasonableness of the claim.
    Further applications of differentiation
  22. Q355 · Original practice · 6 marks
    The graph of f′(x)=x2−4f^{\prime}(x)=x^2-4 is shown on the left. It is known that f(0)=1f(0)=1. Use the grid on the right for your sketch.
    Further applications of differentiation
  23. Q356 · Original practice · 4 marks
    The derivative of a function is f′(x)=ex(x−2)f^{\prime}(x)=e^x(x-2). Determine the interval on which ff is both decreasing and concave up. Justify your answer.
    Further applications of differentiation
  24. Q357 · Original practice · 6 marks
    A closed cylindrical container must hold 32π cm332\pi\ \mathrm{cm}^3. Its radius is rr cm and height is hh cm. Material for its two circular ends costs 33 cents per cm2\mathrm{cm}^2, while material for its curved side costs 11 cent per cm2\mathrm{cm}^2. Ignore seams and waste.
    Further applications of differentiation