Limits and Derivatives
2006 FAMAT State Convention
For all questions, NOTA means None Of The Aforementioned is correct.
1. Find
d
log a u  .
du
a) u ln a
b)
1
u ln a
c)
1
log u a
d) log u a
e) NOTA
2. When determining the limit of the quotient of two functions, which of the following is considered an
indeterminate form?
I.
0
0
a) I, III, IV only


II.
III.
b) I, II, IV only

0
IV.
c) I, III only
0

d) I, II only
e) NOTA
3. Newton’s second law of motion states that F t  
d
m(t )v(t )  . A snowball begins at rest with
dt
negligible mass and accelerates down a slope at 2 m/s2 while gaining mass at the rate of 3 kg/s.
Determine the force (in Newtons) exerted if it were to hit a stationary object 10 seconds after it begins to
move.
a) 90
b) 60
c) 30
d) 10
e) NOTA
c) 1
d) does not exist
e) NOTA
 ln x! 
.
4. Evaluate lim 
x x ln  x   x 


a) -1
b) 0
5. Laguerre polynomials are used in quantum mechanics as part of determining the position of a single
d n 1
orbiting electron. Let Ln 1 (  )  e 
 n 1e   . Determine L2   .
n 1
d

a) 1

c) 2  4    2
b) 1  
d)   3  9 2  18  6
e) NOTA
6. Which of the following properties are true laws for calculating limits?
Note: Suppose that c and a are constant and f and g are continuous for all x.
I. lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)
x a
x a
x a
II. lim c f ( x)  c lim f ( x)
x a
x a
III. lim
x a
f ( x)
f ( x) lim
 x a
g ( x) lim g ( x)
x a
a) I only
b) II only
c) I, II only
d) I, II, III
e) NOTA
2006 Limits and Derivatives, page 2
x  b;

x 2  a,
7. Let f ( x)   2
 x  5 x  12, x  b.
Find a such that f (x) is continuous and differentiable (i.e. “smooth”) at x = b.
a) 21/8
b) 71/8
c) 12
8. Find the equation of the tangent line to the curve
a) x  4 y  20  0
b) x  4 y  20  0
d) 121/8
e) NOTA
x  y  5 at the point (16, 1).
c) 4 x  y  20  0
d) 4 x  y  20  0
e) NOTA
9. Use two iterations of Newton’s method to approximate the root of the equation x 3  2 x  6  0 .
Begin with an initial guess x1  2 .
a) -4
b) 0
c) 1
d) 2
e) NOTA
10. Which of the following commonly identifies the theorem:
“If f ( x)  g ( x)  h( x) for all x in an open interval that contains a (except possibly at a) and
lim f ( x)  lim h( x)  L then lim g ( x)  L .”
x a
x a
x a
a) Central Limit Theorem
c) Mean Value Theorem
b) Intermediate Value Theorem
d) Sandwich Theorem
e) NOTA
11. Let f ( x)  x 3  7 x  4 for  3  x  2 . How many values of x satisfy Rolle’s Theorem for this interval?
a) 3
b) 0
c) 1
d) 2
e) NOTA
12. Let f ( x)  x . Find the tangent line approximation of f near 2025.
a) 45 
x  2025
90
b) 45 
x  2025
90
c) 45 
x  2025
45
d) 45
e) NOTA
13. A water tank is in the shape of an inverted regular tetrahedron. Water is pumped out of the tank at rate
or 1 cfm (ft3 per minute). If the tank holds a maximum of 100 ft3, at what rate is the height of the water
falling when the tank is half full? Round to 4 decimal places.
3 3
h
Hint: V 
8
a) 0.0033
b) 0.0133
c) 0.0267
d) 0.0409
e) NOTA
2006 Limits and Derivatives, page 3
sin 2 5t
.
x0
t2
14. Evaluate lim
a) 5
b) 12.5
15. Evaluate lim
x 
a)
 3x  2 x  1 
2
b)
3/2
c) 25
d) D.N.E.
e) NOTA
c) 3 / 2
d) 1
e) NOTA

3x 2  x .
6 /3
x4
d
16. Find
sect dt
dx 1
 
a) 4 x 3 sec x 4  1
 
b) 4 x 3 sec x 4  C
 
c) 4 x 3 sec x 4
   
d) x 4 sec x 4 tan x 4
e) NOTA
17. Sum the absolute maximum and absolute minimum of the function f ( x)  3x 4  16 x 3  18x 2 over the
closed interval  1  x  4 .
a) -22
b) 10
c) 12
d) 25
e) NOTA
c) 0
d) DNE
e) NOTA
 1  Ct 
18. Find lim 1    , where C  0 .
t 
 t  
a) C exp t 
b) exp Ct 
19. How many points of inflection does f have in the figure below?
Figure 1: f (x) plotted over the interval a  x  b .
a) 4
b) 3
c) 2
d) 1
e) NOTA
2006 Limits and Derivatives, page 4
20. Let f ( x)  sin x . Which of the following represents
n 

a) sin  x 

4 

n 

b) sin  x 

4 

dn
f (x) , the nth derivative of f with respect to x?
dx n
n 

c) cos x 

2 

n 

d) cos x 

2 

e) NOTA
21. Let f (x) be a one-to-one, differentiable function, where f 1 ( x)  g ( x) .
Given the following information, evaluate g ' ( ) .
g ( )  2
a) e-1
f ( )  e
f ' ( )  1
g 1 (2)  
f ' (1)  0
f ' (2)  2
b) -1
c) -1/2
d) undefined
e) NOTA
b) 0
c) 1
d) D.N.E.
e) NOTA
22. Evaluate lim x x
x0
a)  
23. Which of the following is a solution to the differential equation y" ( x)  2 y( x) , where λ is a constant
and   0 .
a) y( x)  sin x
n
24. Evaluate lim
n 
a) 1
b) y ( x)  0
c) y( x)  cosx
d) y  C x
e) NOTA
b) 2
c) 3
d) 7
e) NOTA
 4i 
2

  n
i 1
25. Which of the following is not equivalent to df dx ?
a) lim
f ( x  h)  f ( x )
h
b) lim
f ( x  h)  f ( x  h)
2h
c) lim
f ( x )  f ( x  h)
h
d) lim
f ( x  h)  f ( x  h)
2h
h 0
h 0
h 0
h 0
e) NOTA
a 
26. Find lim  n1  , where an is the nth term of the Fibonacci sequence.
n  a
 n 
a) 8/5
 
b) e ln 
c) π / 2


d) 1  5 / 2
e) NOTA
2006 Limits and Derivatives, page 5
27. Suppose you are driving on the interstate. Your clock and odometer are functioning properly, but your
speedometer is broken. At 1:00, you are interested to know your instantaneous speed. Which of the
following will give the best approximation? Assume you’re looking for a theoretical definition more
than for a practical method.
a)
b)
c)
d)
e)
Divide distance traveled from 1:00 to 1:01 by one minute.
Divide distance traveled from 1:00 to 1:10 by ten minutes.
Divide distance traveled from 1:00 to 1:30 by thirty minutes.
Divide distance traveled from 1:00 to 2:00 by one hour.
NOTA
28. The designed range, s, of an aircraft is given by the equation below, where C is constant and all
variables are positive. Which of the following corresponds to maximum range?
 C L  1 
 
s  C
 C D   


a) minimum
CL
, minimum ρ
CD
b) minimum
CL
, maximum ρ
CD
c) maximum
CL
, minimum ρ
CD
d) maximum
CL
, maximum ρ
CD
e) NOTA
29. Legendre polynomials are used to describe the conduction of heat in a sphere. They are mathematically
defined by the following formula:
Pn ( x) 


n
1 dn 2
x 1
n
n
2 n! dx
Find P2(x).
a) 12 x 4  4
b) 3 x 2  1
c) 2x
d) x
e) NOTA
30. Chris Leak can throw a football with initial velocity v0  25 yards per second and at an inclined angle
. Upon leaving his hand, the ball travels a horizontal distance according to the range equation below.
yard
What is the maximum distance (in yards) Leak can throw the ball? Approximate gravity as 10 2 .
s
R 
a) 75
b) 62.5
2v02 sin  cos
g
c) 50
d) 37.5
e) NOTA