General Secondary Certificate (Tawjihiya Kuwait) · Pure Mathematics

Calculus and Mathematical Analysis

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

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All 31 questions

  1. Evaluate the trigonometric limit: lim (x -> 0) [sin(5x) / (3x)].
  2. Find the derivative of the rational function f(x) = (2x - 3) / (x + 4) for x != -4.
  3. What is the slope of the tangent line to the curve y = x^3 - 3x^2 + 2 at the point where x = 3?
  4. Given the function f(x) = 2x^3 - 9x^2 + 12x - 5, find the x-coordinates of all local extrema (critical points).
  5. Determine the point of inflection for the cubic curve f(x) = x^3 - 6x^2 + 9x + 1.
  6. Evaluate the indefinite integral: integral of (6x^2 - 4x + 5) dx.
  7. Evaluate the definite integral: integral from 0 to 2 of (3x^2 + 2x) dx.
  8. Using the method of integration by parts, evaluate the integral: integral of x * e^(2x) dx.
  9. Find the area of the region enclosed by the parabola y = 4 - x^2 and the x-axis (y = 0).
  10. Find the volume of the solid generated by revolving the region bounded by y = sqrt(x), the x-axis, and x = 4 around the x-axis.
  11. Solve the separable differential equation: dy/dx = 2x * y for y > 0 with initial condition y(0) = 3.
  12. If f(x) = ln(x^2 + 1), what is f'(x)?
  13. A rectangular garden is to be fenced along three sides, with the fourth side bounded by an existing stone wall. If 120 meters of fencing material is available, what dimensions maximize the enclosed ar…
  14. Evaluate the limit: lim (x -> infinity) [(3x^2 - 5x + 2) / (7x^2 + 4x - 1)].
  15. What is the derivative of y = sin^3(4x) with respect to x?
  16. Evaluate the integral: integral of [x / (x^2 + 9)] dx.
  17. Find the equation of the normal line to the curve y = x^2 - 4x + 3 at the point (4, 3).
  18. Evaluate the definite integral: integral from 0 to pi/4 of [sec^2(x)] dx.
  19. If a particle moves along a straight line with position function s(t) = t^3 - 6t^2 + 9t + 2 (in meters, t in seconds), at what times is the particle momentarily at rest?
  20. Evaluate the integral using partial fractions: integral of [1 / (x^2 - 1)] dx for x > 1.
  21. Find the absolute maximum value of f(x) = x^3 - 3x on the closed interval [0, 2].
  22. Evaluate the limit: lim (x -> 0) [(e^(3x) - 1) / x].
  23. Given y = x^x for x > 0, find dy/dx using logarithmic differentiation.
  24. Evaluate the improper integral: integral from 1 to infinity of (1 / x^2) dx.
  25. What is the average value of the function f(x) = 3x^2 - 2x on the interval [0, 2]?
  26. A 5-meter ladder leans against a vertical wall. If the bottom of the ladder slips away from the wall at a rate of 1.5 m/s, how fast is the top of the ladder sliding down the wall when the bottom is 3…
  27. What is the area of the region bounded by the curves and ?
  28. Evaluate the limit: .
  29. What is the volume of the solid generated by revolving the region bounded by , , and about the x-axis?
  30. If vectors and , what is the cross product ?
  31. Solve the differential equation with initial condition . What is ?

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