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[Marks]
- Use the graph of the function f(x) to answer each question. Where appropriate use ,or “does
not exist”. [3]
a.
lim f ( x ) x
b.
lim ( )
4
f x
x
c.
lim ( ) 4
f x x
d.
lim ( ) 6
f x x
e. f ( 6 )
f. List the x-value(s) at which the f(x)
is continuous but non differentiable
- Evaluate the following. Where appropriate use ,or “does not exist”.
a. 4
lim 2
2
x
x x
x
[2]
b.
3 27
lim 2
2
6 x x
x
x
[3]
c.
2 0
lim
x x x
[2]
d.
lim 3
2
x
x x
x
[3]
- Find all the x-values at which f ( x ) is discontinuous, and determine the type of discontinuity at each
value. [5]
2
if x x x
x
if x
x
x if x
f x
x
y
- Let f(x) be a continuous and differentiable function over [0,3]. If f (^ x ) 2 for all values of x and
f (0) = 4. What is the maximum possible value of f (3)? [3]
- Given the function x
f x
( ) , find f ( x )using the LIMIT DEFINITION of the derivative. [4]
- Find dx
dy for each of the following:
a.
2
cot log 2
3
2
3
(^5 )
3 e x x
x
y x
x
[3]
b.
tan 3 ( ) 2
2
x
e g x
x
[3]
c.
csc( ) 2 1
x y x [3]
d.
3 2 7
4 3 2
ln x x x
x x x y [3]
- The equation of motion of a particle is
3 2 (^) s t 3 t , where s is in meters and t is in seconds. [6]
a. Find the velocity and acceleration as functions of t.
b. When is the particle at rest? What is the acceleration at that moment?
c. Find the velocity after 4 s.
d. When is the acceleration zero?
- Find the
th 25 derivative of f ( x )sin( 2 x ) [3]
- For which values of x does the graph of
x y xe
2 have a horizontal tangent line? [4]
- Given the curve x y 9 xy
3 3 ( folium of Descartes) find the following:
a. dx
dy [2]
b. The equation of the normal line to the curve at the point (2,4) [2]
- 3 meters above the ground a fly is flying horizontally at a rate of 4 meters per minute. It passes over a
small rock at noon. How fast is the distance between the fly and the rock increasing one minute later?
[5]
- Given
3 2
2
2 2 2 1
x
x f x
x
x f x x
f x ,find all : [10]
a. The x and y intercepts.
b. The vertical and horizontal asymptotes.
c. The interval of increase and decrease.
d. The local (relative) extrema( if any).
e. The interval of upward and downward concavity.
f. The inflection point(s) (if any).
g. On the next page sketch the graph of f(x). Label all intercepts, asymptotes, extrema and points of
inflection.
Answers
a) 3 b) c) Does Not Exist d) 1 e) 3 f) 0, 3
a)
4
b) 11
c) d) – 4
Infinite discontinuity at x = 2 , and Jump Discontinuity at x = - 1
f ( 3 ) 10 5)
2 3
x
- a)
ln 2
5 ln 5 csc 2
3 2 85
2
x
x x
x dx
dy (^) x
- b)
2 2
2 2 2
2 3 5 tan 3 sec 3 6 tan 3 ( )
x
e x e e x e g x
x x x x
6)c)
1 csc cot ln 1 csc 2
2 csc 2
x
x y x x x x x
x
- d)
x x
x
x x x x
x y 2
3 2
2
- a) v ( t ) 3 t 6 t
2 ; a ( t ) 6 t 6 b)at t= 0 s. or t= 2 s.
2 a ( 0 ) 6 m / s ;
2 a ( t 2 ) 6 m / s
c) v ( 4 ) 24 m / s d) at t = 1 s.
- ( ) 2 cos( 2 )
25 25 f x x
th 9) 2
x 10) a) y x
y x
dx
dy
2
2
b) 2
y x
- /min
5
m 12)a) 0 , 1 , no x intercept b)vertical Asymptote at x 1 ; horizontal Asymptote at y = 0
c) f(x) increases on 0 ,, and decreases on , 0 d) Local minimum at 0 , 1
e) f(x) is concave down on , 1 1 ,and concave up on 1 , 1 f) no inflection point
g)
x
y
- absolute max. 2 , 2 , absolute min 4 , 2 and 2 , 2 14)
2
24 m
- a) x x x
x x x x x
x x x x x c dx
d sec sec tan
tan sec sec sec tan
sec tan sec lnsec tan
2
b)
ln 1 16) a) 2
ln 2 4
b) x c
x 3 2
2
c) x x c
x
ln 5
5 ln 6
- (^ )^2 cos^2
3 f x x x x
- a)
x
y
b) 4 1 ln 5 c) 15
19) tan x
