Majmaah University
College of Engineering
Math. ‘105’
Total Marks: 40
Final Exam.
Examiner: Dr. Tarek Nabil Ahmed
Time: 3 hour , 8/7/1433
Answer All Questions
Question 1: Evaluate the following limits.
x 4  256
1 - lim
,
2
x  4 x  16
4  lim
x 0
tan2x
,
x
x2  x  6
2 - lim
,
2
x 2 x  4
5 x  2 ln x
3 - lim
x  x  3 ln x
5 - lim xcosecx,
x 0
6 - lim (sec x  tan x)
x 0
)12 marks)
Question 2: : Find the first derivative of the following functions.
(1) y  ln( x 4  2)  7sinx
3
(2) y  x 2  2 x  1,
(4) y  e - x cosx  e tanx
(5) y  x sin x ,
(3) tan y  xy ,
(6) y  cosh3x  tanh -1 sinh x
)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  6 x 2  9 x  1
)6 marks)
Question 4: Apply the Roll's theorem of F.
on
[-1,3] F ( x)  x 2  2 x  2
)3 marks)
Question 5: Find equations of tangent and normal lines to the graph of the equation.
F ( x)  3 x 2  2x
at P(2,8)
)3 marks)
Question 6: If
y 2  3x 2  2 x
then prove that
y y  (y) 2  3  0
)4 marks)
With my best wishes