SAMPLE QUESTIONS FOR THE FINAL EXAMINATION PART 1
In Exercises 1-6, find the domain of the given function.
√
1. f (x) = x2 − 2x − 8.
√
2. f (x) = 3 2x + 4.
3. f (x) = ln(4x − 3 − x2 ).
4. f (x) =
5. f (x) =
x2 +x−2
.
x2 −x−2
q
(x−2)(x+1)
.
x−4
6. f (x) = ln (x+2)(x−5)
.
(x+3)(x−3)
x3 + x2 + 3x − 5
.
x→1
x3 − 1
7. Evaluate lim
x2 − x − 6
.
x→3 x4 − x3 − 6x2 − 4x + 12
8. Evaluate lim
x3 − x2 − x − 2
√
.
√
x→2
x− 2
√
3
x−2
√ .
10. Evaluate lim √
x→8
x−2 2
9. Evaluate lim
x2 + 2x + 5
.
x→1 x2 − 3x + 2
11. Evaluate lim
12. Evaluate lim
x→
x2 + x − 2
.
x4 − 2x3 + 2x − 1
3x3 − 2x2 + 1
.
x→∞ 2x2 − 5x − x3 + 2
13. Evaluate lim
2x2 − x − 1
.
x→−∞ 5x2 − 3x
√
15. Evaluate lim x2 + x + 6 − x + 2.
14. Evaluate lim
x→∞
16. Evaluate lim
√
x→−∞
x2 + 2 − 3 + x.
In questions 17-24, do not use L’Hb
opital’s Rule
 x2 −2x
 x2 −3x+2 if x < 2
a
if x = 2 . Determine whether f (x) is continuous at x = 2.
17. Let f (x) =
 x2 +bx+c
if
x>2
x2 +2x−8
18. Let f (x) =


x3 −5x+4
x2 −4x+3

x2 +bx+c
x2 −1
a
if x < 1
if x = 1 . Determine whether f (x) is continuous at x = 1.
if x > 1
19. Let f (x) =
20. Let f (x) =
21. Let
22. Let
23. Let
24. Let


x3 +3x2 +7x+5
x2 +x−2

x2 +bx+c
x2 −1
if x < −1
if x = −1 . Determine whether f (x) is continuous at x = −1.
if x > −1
a



x2√
−5x+4
x−2


x2 +bx+c
x−4
if x < 4
if x = 4 . Determine whether f (x) is continuous at x = 4.
if x > 4
a
 2
 x − 7x + 20 if
8
if
f (x) =
√

4 x
if

 x2 + 2x + 5
4
f (x) =
 3
2
x +x −x+3

10x + 26

4
f (x) =
 3
x − 3x2 + x + 1
 2
 x − 2x − 30 if
5
if
f (x) =

12x − 79
if
x<4
x = 4 . Find f 0 (4) if it exists.
x>4
if x < −1
if x = −1 . Find f 0 (−1) if it exists.
if x > −1
if x < 3
if x = 3 . Find f 0 (3) if it exists.
if x > 3
x<7
x = 7 . Find f 0 (7) if it exists.
x>7
25. Write an equation of the tangent line drawn to the graph of f (x) = x3 − 2x2 + 3x − 5 at x = 2.
26. Write an equation of the tangent line drawn to the graph of f (x) = ln(x2 + 1) at x = 1.
2
27. Write an equation of the tangent line drawn to the graph of f (x) = ex −3x+2 at x = 1.
√
28. Write an equation of the tangent line drawn to the graph of f (x) = x2 + 3x at x = 1.
2
29. Find f 0 (x) if f (x) = ex + ln(2x + 3).
30. Find f 0 (x) if f (x) = (x − 2) ln x.
31. Find f 0 (x) if f (x) =
ex
.
ln x
32. Find f 0 (x) if f (x) =
xex
.
x2 + 1
33. Find y 0 if x2 + y 2 = 5.
34. Find y 0 if 3xy 2 + 2xy + y 3 + x2 = 0.
35. Find y 0 if xexy + ln(x2 y) + xy = 2.
2
36. Find y 0 if xy 3 + ln xy + exy = 3.
37. Two nonnegative numbers have sum 9. What is the largest possible value for their product?
38. Two nonnegative numbers have product 16. What is the smallest possible value for their sum?
39. Two nonnegative numbers have sum 60. What are the numbers if the product of one of them
and the square of the other is maximal?
40. Among all rectangles with perimeter 8, find the dimensions of the rectangle with largest area.
39. Among all rectangles with area 16, find the dimensions of the rectangle with smallest perimeter.
40. Find the largest possible area for an isosceles triangle if the length of each of its two equal sides
is 10 m.
41. Find the largest possible perimeter of a rectangle inscribed in a semicircle of radius 8 if one
side of the rectangle lies along the diameter of the semicircle.




2 3 −1
1 −2 3
42. Let A = 1 −1 2  and B =  2 −1 0
4 2
5
−2 1 3
a) Find A2 .
b) Find AB.
c) Find BA.
d) Find AT B.
e) Find B T AT .
f) Find 3A + B.




1 2
2
3 −2 1
43. Let A = 2 −1 1  and B = 0 −2 1
2 3 −4
3 1 2
a) Find A3 .
b) Find AB.
c) Find BA.
d) Find AT B.
e) Find B T A.
f) Find A + 2B.
2 2
4a + b c + d
44. Find a, b, c, d if
=
3 −4
2d + c b − 2a
1 2
a−b c+d
44. Find 3a + b + 17c + 10d if
=
−2 3
3c − d a + b
45. Find the trace of the following matrices, it possible.


1 2
2
a) 2 −1 1 .
2 3 −4

T
3 −2 1
b) 0 −2 1 .
3 1 2



T
3 −2 1
3 −2

c) 0 −2 1 +
0 −2 1
3 1 2
d)


3 −2
3 −2 1 
0 −2.
0 −2 1
3 1
46. Let A, B be two 3 × 3 matrices. If T r(A) = 4 and T r(B) = 2, evaluate the followings if
possible.
a)
b)
c)
d)
T r(A + B).
T r(3A − 2B).
T r(AT ).
T r(AT + AB + 3B − BA).
Questions will be similar to above questions.