New Zealand Scholarship · Calculus and Advanced Mathematics

Differential and Integral Calculus, Real-World Modelling and Differential Equations

40 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 40 questions

  1. Evaluate the indefinite integral integral( 2x * cos(x^2) dx ) using substitution.
  2. Using implicit differentiation, find dy/dx for the curve x^2 + y^2 = 25 at the point (3, 4).
  3. What is the general solution of the separable differential equation dy/dx = 3x^2 * y for y > 0?
  4. Using integration by parts, evaluate integral( x * e^x dx ).
  5. What is the volume of the solid generated when the region under y = sqrt(x) from x = 0 to x = 4 is revolved 360 degrees about the x-axis?
  6. Differentiate y = ln(x^2 + 1) with respect to x.
  7. What is the derivative of f(x) = x^x for x > 0 using logarithmic differentiation?
  8. Newton's law of cooling states dT/dt = -k(T - T_env). If a pie cools from 100 deg C to 60 deg C in 10 minutes in a room at 20 deg C, what is the value of the cooling constant k?
  9. Find the horizontal and vertical asymptotes of the curve y = (2x^2 - 8) / (x^2 - 1).
  10. A 5-metre ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at 2 m/s, how fast is the top sliding down when the bottom is 3 metres from the wall?
  11. Evaluate the definite integral integral_0^1 ( x / (x^2 + 3x + 2) ) dx using partial fraction decomposition.
  12. A conical water tank with vertex pointing downwards has height 6 metres and radius 3 metres at the top. Water flows in at a constant rate of 2 m^3/min. At what rate is the water level rising when the…
  13. Find the coordinates of the point on the parabola y = x^2 closest to the point (0, 4.5).
  14. Solve the differential equation dy/dx + 2y = 4 with initial condition y(0) = 5.
  15. Evaluate the improper integral integral_0^inf ( x * e^(-x) dx ).
  16. Find the area enclosed between the cubic curve y = x^3 - 4x and the line y = 0 from x = -2 to x = 2.
  17. A particle moves in a straight line with acceleration a(t) = 6t - 12 for t >= 0. If initial velocity v(0) = 9 and initial position s(0) = 2, what is the position function s(t)?
  18. What is the maximum value of the function f(x) = sin(x) + cos(x) on the real line?
  19. Using the integrating factor method, find the general solution to dy/dx + (2/x)*y = 4x for x > 0.
  20. What is the limit as x approaches 0 of (sin(3x) - 3x) / x^3?
  21. A cylindrical tank of radius 2 m is draining such that depth h changes as dh/dt = -0.1*sqrt(h). What is dV/dt when h = 4 m?
  22. Evaluate the improper integral integral from 1 to infinity of 1 / (x^2 + 1) dx.
  23. Find the arc length of the parametric curve x = cos^3(t), y = sin^3(t) for 0 <= t <= pi/2.
  24. Using partial fractions, evaluate integral of (5x - 2) / ((x - 2)(x + 2)) dx.
  25. What is the first non-zero term in the Maclaurin series expansion of f(x) = ln(1 + x) - sin(x)?
  26. What is the acute angle of intersection between the curves y = x^2 and y = sqrt(x) at (1, 1)?
  27. Find the coordinates of the inflection point for the function f(x) = x^3 - 6x^2 + 9x + 2.
  28. Evaluate the limit as x approaches infinity of (1 + 3/x)^(2x).
  29. Using the substitution x = 2*sin(theta), evaluate integral of sqrt(4 - x^2) dx.
  30. A particle moves with acceleration a(t) = -4*v(t). If initial velocity v(0) = 16 m/s, what is v(t)?
  31. Evaluate the definite integral integral from 0 to pi/4 of tan^2(x) dx.
  32. What is the maximum area of a rectangle inscribed in the semi-circle y = sqrt(16 - x^2) above the x-axis?
  33. Find the slope of the normal line to the curve x^3 + y^3 = 9 at the point (1, 2).
  34. Evaluate the limit as x approaches 0 of (e^(2x) - 1 - 2x) / x^2.
  35. Find the particular solution of dy/dx = y^2 with initial condition y(1) = 1.
  36. Find the indefinite integral of x e^x with respect to x.
  37. Solve the differential equation dy/dx = 2y given that y = 3 when x = 0.
  38. Differentiate y = ln(sin x).
  39. Evaluate the integral of sin^2 x from x = 0 to x = pi/2.
  40. Evaluate the integral of 1 / (1 + x^2) from x = 0 to x = 1.

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