Math topic中文

Graphing calculator

Use of a graphing calculator or a graphing software to graph a given function.

Practice questions

  1. 1Sketch, on the same axes, the graphs of y=5−1x−1y=5-\dfrac{1}{x-1} and y=1(x−5)2y=\dfrac{1}{(x-5)^2}. Solve the inequality 5−1x−1<1(x−5)25-\dfrac{1}{x-1}<\dfrac{1}{(x-5)^2}…
  2. 2Sketch the graphs of y=3exy = 3e^x and y=x+3y = x + 3 on the same diagram. Indicate clearly the coordinates of the points of i…
  3. 3Sketch the graph of y=7−2x3x2+x−14y = \frac{7-2x}{3x^2 + x - 14}. Give the equations of the asymptotes and the coordinates of the tu…
  4. 4The following diagram shows the graphs of two differentiable functions y=f(x)y=f(x) and y=g(x)y=g(x). At x=cx = c, the distance b…
  5. 5The curves C1C_1 and C2C_2 have equations y=2x2+9x2−4y = \frac{2x^2 + 9}{x^2 - 4} and y225−x29=1\frac{y^2}{25} - \frac{x^2}{9} = 1 respec…
  6. 6Sketch the curve with equation y=ln⁡(x+1),x>−1y = \ln(x+1), x > -1, stating the equation of asymptote and intercepts with the axes. O…
  7. 7The function f is defined by f:x↦4+1(x−1)2+1,x≥0f : x \mapsto 4 + \dfrac{1}{(x-1)^2 + 1},\quad x \ge 0. The function g is defined by g :…
  8. 8The curve CC has equation y=xcos⁡2xy = x\cos 2x, where 0≤x≤π0 \le x \le \pi. Find the exact xx-coordinates of the points where CC…
  9. 9Sketch, on the same diagram, the curves with equations y=∣x+1∣∣x−2∣y = \frac{|x + 1|}{|x - 2|} and y=ln⁡(1−x2)+12y = \ln\left(1 - \frac{x}{2}\right) + \tfrac{1}{2}…
  10. 10The function f\mathrm{f} is defined by f:x↦2xx2−1,x∈R, x>1\mathrm{f} : x \mapsto \frac{2x}{x^2 - 1}, \quad x \in \mathbb{R}, \ x > 1. Sk…
  11. 11An astroid is a curve with parametric equations x=acos⁡3t, y=asin⁡3t.x = a \cos^3 t, \ y = a \sin^3 t. Sketch an astroid with a=2a = 2 a…
  12. 12In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
  13. 13A family of curves is given by the differential equation xdydx−y=x(x−y)x\dfrac{dy}{dx}-y=x(x-y). By substituting y=vxy=vx, show that t…
  14. 14The parametric equations of a curve CC are x=t+2et,y=2t+e−t,t>−2.x = t + 2e^t, \quad y = 2t + e^{-t}, \quad t > -2. Sketch CC. Find the…
  15. 15A function f\mathrm{f} is said to be self-inverse if f(x)=f−1(x)\mathrm{f}(x) = \mathrm{f}^{-1}(x) for all xx in the domain of…
  16. 16A curve GG has equation y=ax+b+c(x+1)2y = ax + b + \dfrac{c}{(x+1)^2}, where aa, bb and cc are positive integers. Let RR be the…
  17. 17Sketch the graphs of y=3exy=3e^{x} and y=x+3y=x+3 on the same diagram. Indicate clearly the coordinates of the points of inter…
  18. 18The function f\mathrm{f} is defined by f:x↦ln⁡x2−(ln⁡x)2,x>0.\mathrm{f} : x \mapsto \ln x^2 - (\ln x)^2, \quad x > 0. Sketch the graph of…
  19. 19The function f is defined as follows. f : x−>−1x2+1x -> \displaystyle -\frac{1}{x^{2}+1} for x∈Rx \in \mathbb{R}. Sketch the gra…
  20. 20A curve CC has parametric equation x=t2−1,y=4t−t3+15,x = t^2 - 1,\qquad y = 4t - t^3 + 15, where t∈Rt \in \mathbb{R}. Show that the cu…
  21. 21The diagram above shows the curve C1C_1 with equation y=ln⁡xx2y = \frac{\ln x}{x^2}, where x≥1x \ge 1. Show that the exact coor…
  22. 22The parametric equations of a curve C1C_1 are x=3t3+2t2x=3t^3+2t^2 and y=−t3+2t2+1y=-t^3+2t^2+1 for −1≤t≤1-1 \le t \le 1. Another curve C2C_2…
  23. 23A curve CC has parametric equations x=3t3, y=31+t2,t∈R.x = 3t^3,\ y = \frac{3}{1+t^2},\quad t \in \mathbb{R}. Find dydx\frac{\text{d}y}{\text{d}x}…
  24. 24In this question you may use expansions from the List of Formulae (MF27). Find the Maclaurin expansion of ln⁡(1+cos⁡3x)\ln(1 + \cos 3x)…
  25. 25The curve CC is defined by the parametric equations x=1−cos⁡t,y=sin⁡2t,where 0≤t≤π2.x = 1 - \cos t, \quad y = \sin 2t, \quad \text{where } 0 \le t \le \frac{\pi}{2}.…
  26. 26The functions f\text{f} and g\text{g} are defined as follows f:x↦ln⁡(3x−2)x,x>23\text{f} : x \mapsto \frac{\ln(3x - 2)}{x}, x > \frac{2}{3}…
  27. 27A curve CC has parametric equation x=t2−1,y=4t−t3+15,where t∈R.x = t^2 - 1, \quad y = 4t - t^3 + 15, \quad \text{where } t \in \mathbb{R}. Show…
  28. 28The curve C is defined by the parametric equations x=cos⁡t+12cos⁡5t,y=sin⁡t+12sin⁡5tx = \cos t + \dfrac{1}{2} \cos 5t, \quad y = \sin t + \dfrac{1}{2} \sin 5t \quad…
  29. 29Sketch, on the same diagram, the curves with equations y=∣x+1x−2∣y = \left|\frac{x + 1}{x - 2}\right| and y=ln⁡(1−x2)+12y = \ln\left(1 - \frac{x}{2}\right) + \frac{1}{2}…
  30. 30A curve CC has parametric equations x=3cos⁡θ−2cos⁡3θ,x = 3\cos\theta - 2\cos 3\theta , y=9sin⁡θ−sin⁡3θ,y = 9\sin\theta - \sin 3\theta , for 0≤θ≤2π0 \le \theta \le 2\pi…
  31. 31A curve CC has equation y=x−3(x−2)(x+5)y = \dfrac{x-3}{(x-2)(x+5)}. Sketch CC, giving the equations of the asymptotes, the coordina…