If a problem is solved. It is not 'the' answer.
No attempt is made to search for the most elegant answer.
I highly recommend that you at least try to solve the
problem before you read the solution.

Find the number of vertical asymptotes of f(x) = tan(x) + cot(x) in the interval [10,100]. 
Prove that there is a real number r such that for all x _______  a + x 1 x arctan(   ) =  . arcsin() + r \ a  x 2 a The number a is a real strictly positive number. Then calculate the value of r. 
Calculate the condition for the real value of the parameter h
such that the following function has a minimum and a maximum.
h + 3x  x^{2} f(x) =  x  4 
Recall previous problem and take h < 4. Investigate the
number of intersection points of f(x) with the line y = k.
Deduce from this the difference between the maximum and minimum value
of f(x). 
Calculate the inflection points of y = (x + 1)/(x^{2} + 1) 
Calculate the inflection points of y = sin(2x)+3sin(2x/3) in [0,3pi/2]. 3 hint : sin(3t) = 3.sin(t)  4 sin (t) 
Calculate the inflection points of y = e^{2x}  5 e^{x} +6 
Study the curve f(x) = (x^{2}  x)^{2} Give the number of intersection points of that curve with the line y = m. 
f(x) = arcsin(tan(x/pi))Find

Choose an xaxis and a yaxis (orthonormal) and let O be the origin. C_{1} is a fixed circle, through O, with center (m,0) with m > 0. C_{2} is a fixed circle, through O, with center (n,0) with n > 0. The line d has y = x tan(t) as equation and t is variable in (0,pi/2). The line d' has y = x tan(t+a) as equation and a is a constant in (pi/2,pi). The line d intersects C_{1} a second time in point A and the line d' intersects C_{2} a second time in point B.

The curve F is a circle with radius r and center O.
AB is a central line with points A and B on F.
The line CD is parallel to AB with points C and D on F, such that ACDB
is a isosceles trapezium ( AC = DB ). The angle(DOC) = 2t radians. Calculate the area of the trapezium as a function of t. Calculate the value of t such that this area is maximum. 
A function y = f(x) is implicitly defined by
y x  sqrt(y.y + 2(x  1)y + 4x) = 0

Define all real values of m such that the asymptotes of the curve 2(m1)x  m + 1 y =  (m+3)x + m intersect in a point above the line y = 2x1 
Investigate if there is a vertical asymptote at x = 0 for the function arcsin(2x)  2 arcsin(x) y =  x^{3} 
Calculate the horizontal asymptote to the function sin(1/x) y =  arctan(1/x) 