Majmaah University
College of Engineering
Math. ‘105’
Total Marks: 40
Final Exam.
Examiner: Dr. Tarek Nabil Ahmed
Time: 3 hour , 1/7/1433
Answer All Questions
Question 1: Evaluate the following limits.
e x  e x  x 2  2
2 - lim
,
2
2
x 2
sin x  x
1 - lim (e x  1) cotx,
x 0
1
1
5 - lim ( ),
x 0 x e x  1
4  lim (cosecx - cotx),
x 0
x 3  64
3 - lim
x  4 x 2  16
x  sin 2 x
6 - lim
x 0 x  sin 2 x
)12 marks)
Question 2: : Find the first derivative of the following functions.
(1) y  tan -1(sinh x)  ln(sin x)
(4) y  a(1 - cost),
(2) y  tan (3 x  e x ),
x  a(t - sint)
(5) y  x x ,
(3) tan y  xy  sinx ,
(6) y  sec x 2  5
)12 marks)
Question 3: Find the Local extrema of F. Find the intervals of convexity and
concavity ,the points of inflection and sketch the curve.
F ( x)  x 3  2 x 2  x  1
)6 marks)
Question 4: Apply the mean value theorem of F.
on
[0,1] F ( x)  x 2  2 x  1
)3 marks)
Question 5: Find equations of tangent and normal lines to the graph of the equation.
F ( x)  3x  sin 3x
at P(0,0)
)3 marks)
Question 6: If
y  x 2e x
then prove that
y  (x 2  6 x  6) e x
)4 marks)
With my best wishes