Calculus
Problems
1. What are the break-even odds for rolling a single dice to obtain a
6? (Note the game is played up to the given number of rolls or
until a 6 appears.)
2. Name several games that are played today for which the division
of stakes theorem is applicable. Explain how.
3. The normal to the curve C at the point (3; 2) is ¡1. What is the
subnormal?
4. The normal to the curve C at the point (4; 5) is 2. What is the
tangent?
5. Find the tangent of following functions using the adequality method
of Fermat.
(a) f (x) = 3x3 + 2x ¡ 2 at x = 2
1
(b) f (x) =
at x = 1
x+2
(c) f (x) = (x ¡ 2)2 at x = 0
6. Find the Fermat extreme points of the following.
(a) f (x) = x2 + 2x + 4
(b) f (x) = x3 ¡ 3x + 4
7. Find the subtangent line to yx2 + x4 = y 2 ¡ 2xy using the method
of de Sluse. Find the tangent for the same problem, using a similar
method.
8. Using Web or other resources write a report on the life of Roberval.
In it, be sure to give his mathematical accomplishments.
9. Trace the use of power series in the development of calculus.
10. Give two good reasons that power series were used in lieu of other
functions, such as sin x.
11. What are the Cartesian ovals, the Cissoid, the Conchoid, the Cycloid, the Epicycloid? What other mathematicians studied these
curves and for what reasons?
Calculus
2
12. What are the Epitrochoid, the Hypocycloid, the Hypotrochoid, the
Kappa curve and the Serpentine? What other mathematicians studied these curves and for what reasons?
13. Cite several applications to which Leibniz applied his calculus.