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Algebra Challenge Geometry Challenge Calculus Challenge Physics I Challenge
Do not worry about your difficulties in Mathematics.
I can assure you mine are still greater.
Albert Einstein
I can assure you mine are still greater.
Albert Einstein
Calculus Challenge
1. Difficulty: Beginner 
Find the limit:
Find the limit:
5/8
2. Difficulty: Beginner
Find the limit:
Find the limit:
1/10
3. Difficulty: Beginner
Find the limit:
Find the limit:
-1/3
4. Difficulty: Beginner
Find the derivative of the function:
Find the derivative of the function:
5. Difficulty: Beginner
Find the derivative of the function:
Find the derivative of the function:
6. Difficulty: Beginner
Evaluate the integral:
Evaluate the integral:
7. Difficulty: Easy 
Find the limit:
Find the limit:
8. Difficulty: Easy
Find the limit:
Find the limit:
1
9. Difficulty: Easy
Evaluate the integral:
Evaluate the integral:
10. Difficulty: Easy
Find the derivative of the function:
Find the derivative of the function:
11. Difficulty: Easy
Evaluate the integral:
Evaluate the integral:
12. Difficulty: Easy
Evaluate the integral:
Evaluate the integral:
13. Difficulty: Easy
Find an equation of the tangent line to the curve
at the point (0,1).
Find an equation of the tangent line to the curve
at the point (0,1).
y = (1/2)x + 1
14. Difficulty: Medium 
Find the equation of the tangent line to the curve
at the point
Find the equation of the tangent line to the curve
at the point
15. Difficulty: Medium
Sand is being dumped from a conveyor belt at a rate of 30 cubic feet per minute and its coarseness is such that it forms a pile in the shape of a cone whose height is always two times larger than its base radius. How fast is the height of the pile increasing when the pile is 10 feet high?
Sand is being dumped from a conveyor belt at a rate of 30 cubic feet per minute and its coarseness is such that it forms a pile in the shape of a cone whose height is always two times larger than its base radius. How fast is the height of the pile increasing when the pile is 10 feet high?
16. Difficulty: Medium
A 13-foot ladder leans against the wall of a house. Someone pulls the base of the ladder away from the house at the rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when it is 5 feet from the ground?
A 13-foot ladder leans against the wall of a house. Someone pulls the base of the ladder away from the house at the rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when it is 5 feet from the ground?
24/5 ft/s
17. Difficulty: Medium
Find the point on the line y = 3x - 2 that is closest to the origin.
Find the point on the line y = 3x - 2 that is closest to the origin.
(3/5, -1/5)
18. Difficulty: Medium
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
65 mi/h
19. Difficulty: Hard 
A steel pipe is being carried down a hallway 16 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 2 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?
A steel pipe is being carried down a hallway 16 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 2 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?
20. Difficulty: Hard
Find equations of the lines tangent to both curves
Find equations of the lines tangent to both curves
y = -2x - 6 and y = 6x - 6
21. Difficulty: Hard
A runner sprints around a circular track of a radius 100m at a constant speed of 7 m/s. The runner’s friend is standing at a distance 200m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200m?
A runner sprints around a circular track of a radius 100m at a constant speed of 7 m/s. The runner’s friend is standing at a distance 200m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200m?
22. Difficulty: Hard
Find an equation of the line through the point (3,5) that cuts off the least area from the first quadrant.
Find an equation of the line through the point (3,5) that cuts off the least area from the first quadrant.
y = -(5/3)x + 10
23. Difficulty: Hard
At what distance from point A should the point P be chosen on the line segment AB so as to maximize the angle θ?
At what distance from point A should the point P be chosen on the line segment AB so as to maximize the angle θ?
24. Difficulty: Expert 
Find the function f(x) that satisfies the following equation:
Find the function f(x) that satisfies the following equation:
25. Difficulty: Expert
Find the function f(x) that satisfies the following equation:
Find the function f(x) that satisfies the following equation:
26. Difficulty: Expert
Evaluate the integral (a > 0, b > 0) :
Evaluate the integral (a > 0, b > 0) :
27. Difficulty: Expert
Find the limit:
Find the limit:
28. Difficulty: Expert
Find the limit:
Find the limit:
29. Difficulty: Expert
Show that the following integral does not depend on the value of the parameter a. Evaluate the integral.
Show that the following integral does not depend on the value of the parameter a. Evaluate the integral.
30. Difficulty: Expert
Find the limit:
Find the limit:
2
31. Difficulty: Expert
Find the limit:
Find the limit:
0
32. Difficulty: Expert
Find the limit:
Find the limit:
-1
33. Difficulty: Expert
Find the error in the following line of reasoning:
Find the error in the following line of reasoning:
34. Difficulty: Expert
Evaluate the integral:
Evaluate the integral:
1/2
35. Difficulty: Expert
Evaluate the integral:
Evaluate the integral:
36. Difficulty: Expert
Evaluate the integral:
Evaluate the integral:
37. Difficulty: Expert
Evaluate the integral:
Evaluate the integral:
1
38. Difficulty: Expert
Evaluate the integral:
Evaluate the integral:
39. Difficulty: Expert
Evaluate the integral:
Evaluate the integral:
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