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Instructions for answering this paper:
RELAX!!
ALL questions must be answered.
Marks denote percentages.
Read each question carefully!
Number your answers clearly.
Make neat skecthes of any graphs in your examination book and state all reasoning clearly.
QUESTION 1
Determine the following limits:
x 2 C x 20
(a) lim (3)
x! 5 3 .x C 5/
sin 5t
(b) lim (3)
t!0 t 2 C 4t
3 jxj
(c) lim : (4)
x! 1 2 jxj C 1
[10]
QUESTION 2
Use the Sandwich Theorem to determine
5
lim x 2 cos : .x 6D 0/ [5]
x!0 x
QUESTION 3
(a) Determine whether the function f is continuous at x D 2:
( 2
x x 2
x 2
if x D
6 2
f .x/ D (3)
1 if x D 2
, 4
(b) Let the function f be de ned as:
8
< 4a
> if x 2
f .x/ D 3x 2 if 2 < x 1 (5)
>
:
x Cb if x > 1
Determine the values of the constants a and b so that f is continuous at x D 2 and x D 1: [8]
QUESTION 4
Differentiate the following functions. You don't have to simplify:
(a) f .x/ D ln sec2 e3x (3)
2
(b) g . / D .cos 5 /sin (3)
3
C1
(c) h . / D (4)
j 1j
p
(d) k .y/ D tan4 2y 3
1 cot y: (4)
[14]
QUESTION 5
Find the equations of the tangent and the normal lines to the curve of
2x y C sin y D 2 x
at the point 1; 2
: [7]
QUESTION 6
Determine the area of the region enclosed by the curves
f .x/ D 8 x2 and g .x/ D jx 2j : [6]
QUESTION 7
dy
Use the Fundamental Theorem of Calculus to evaluate dx
if
Z sec
dt
yD
cos x 2 C t3
where is a constant. [4]
, 5 MAT112-P/103
QUESTION 8
Use substitution to obtain the value of the de nite integrals:
Z 9p
p
(a) 2C xd x (Leave your answer in root form.) (5)
0
Z e3
ln .ln t/
(b) dt: (5)
e t ln t
[10]
QUESTION 9
Determine the following inde nite integrals:
Z
x 4 2x 3
(a) dx (5)
x2 C 1
Z
4x 5x
(b) dx (3)
7x
Z
ln x
(c) d x: (4)
x ln2 x C 9
[12]
QUESTION 10
Solve the following initial value problem for y as a function of x:
dy
D x 3 cos x 4 C 2 I y .0/ D 1: [4]
dx
QUESTION 11
dy
p
Find dx
if 1 C x 2 y 2 D 2x y: [5]
QUESTION 12
Use logarithmic differentiation to differentiate g .x/
3p 2 10
4
xe x x 2 C 1
g .x/ D : [5]
.ln .x//2 C sin2 x
, 6
QUESTION 13
Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference
between the object and its surroundings. Suppose that a roast turkey is taken from an oven when its temperature has
reached 185o F and is placed on a table in a room where the temperature is 75o F. If u .t/ is the temperature of the
turkey after t minutes, the Newton's Law of Cooling implies that
du
D k .u 75/ :
dt
This could be solved as a separable differential equation. Another method is to make the change of variable y D
u 75:
(a) What initial–value problem does the new function y satisfy? What is the solution?
(b) If the temperature of the turkey is 150o F after half an hour, what is the temperature after 45 minutes?
(c) When will the turkey have cooled to 100o F? [5]
QUESTION 14
For a real gas, van der Waal's equation states that
n2a
PC .V nb/ D n RT:
V2
Here P is the pressure of the gas, V is the volume of the gas, T is the temperature (in degrees Kelvin), n is the
number of moles of gas, R is the universal gas constant and a and b are constants. Compute and interpret
@P @T
and : [5]
@V @P
c
UNISA 2004
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