Tawjihiya Bahrain (Secondary School Certificate) · Advanced Mathematics

Differential and Integral Calculus

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

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All 31 questions

  1. Evaluate the limit: lim (x -> 3) [ (x^2 - 9) / (x - 3) ].
  2. Find the derivative dy/dx of the function f(x) = 5x^4 - 3x^2 + 7x - 12.
  3. What is the derivative of g(x) = sin(4x)?
  4. Evaluate the indefinite integral: integral of (6x^2 - 4x + 5) dx.
  5. Find the slope of the tangent line to the curve y = x^3 - 2x + 4 at x = 2.
  6. Evaluate the definite integral: integral from 0 to 2 of (3x^2 + 2x) dx.
  7. Determine the critical points of the function f(x) = 2x^3 - 3x^2 - 12x + 5.
  8. What is the second derivative f''(x) of f(x) = e^(3x)?
  9. Using L'Hopital's Rule, find lim (x -> 0) [ sin(5x) / x ].
  10. Evaluate the integral: integral of (1 / (2x + 1)) dx.
  11. Find the inflection point of the cubic curve y = x^3 - 6x^2 + 9x + 2.
  12. Evaluate the integral using integration by parts: integral of x * e^x dx.
  13. What is the derivative of y = ln(x^2 + 4)?
  14. Find the absolute maximum value of f(x) = -x^2 + 4x + 1 on the closed interval [0, 3].
  15. What is the area of the region bounded by the curve y = 4 - x^2 and the x-axis (y = 0)?
  16. Evaluate lim (x -> infinity) [ (4x^3 - 2x + 1) / (7x^3 + 5x^2 - 3) ].
  17. Find dy/dx by implicit differentiation for the equation x^2 + y^2 = 25.
  18. Evaluate the definite integral: integral from 0 to pi/2 of cos(x) dx.
  19. A particle moves along a straight line with position s(t) = t^3 - 6t^2 + 9t. At what times is the particle at instantaneous rest?
  20. What is the average value of f(x) = 2x + 3 on the interval [1, 5]?
  21. Evaluate the integral: integral of (x / sqrt(x^2 + 9)) dx.
  22. What is the volume of the solid generated by revolving the region under y = sqrt(x) from x = 0 to x = 4 about the x-axis?
  23. If f'(x) = 4x^3 - 2x and f(1) = 5, find the function f(x).
  24. Differentiate y = tan(3x) with respect to x.
  25. Evaluate lim (x -> 0) [ (e^(2x) - 1) / x ].
  26. Find the total distance traveled by an object with velocity v(t) = 6 - 2t over the interval [0, 4].
  27. What is the area of the region in the first quadrant bounded by the curve , the line , and the x-axis?
  28. Evaluate the limit: .
  29. What is the volume of the solid generated by revolving the region enclosed by , , , and about the x-axis?
  30. Find the distance between the two parallel planes given by and .
  31. Evaluate the definite integral: .

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