Calculus Extra Credit day
1) If f ( x) 
(C)
1
x
2) If
ln x
, what is f '( x) ?
x
(D)
1  ln x
x
(E)
1  ln x
x
(F)
1  ln x
x2
(G)
1  ln x
x2
dy
 2 xy , then a possible solution for y is
dx
(L) 3e x
2
(N) x 2  4
(M) e x  5
2
(O) ln( x 2  1)
2 x2  7
(P)
3) What is the equation of the line normal to the curve y  e2 x ln x where x = 1?
(M) y  e2 ( x  1)
(O) y  e( x  1)
(N) y  e2 ( x  1)
(P) y  e2 ( x  1)
(Q) y  e2 ( x  1)
4)
What is the value of lim
x 0
(B) 
(A) 0
5)
tan x
?
x
(C) 
(D) 1
(E)
1
2
For xy 2  3x  y 3 , find y ' when y  1.
(A) 2
(B) 1
(C) 
1
2
(D) 0
(E)
4  bx 2 ,  1  x  2
6) Given the piecewise function f ( x)  
abx , 2  x  4
1
2
with a and b as non-zero constants,
what are all possible values of b that will make f(x) continuous and differentiable?
(R) only 1
(S) 1 or  4
(T) 1 or  1
(U) only 1
(V) none of the above
7) The position of a particle moving in a line is s(t )  t 3  5t 2  2t  13 . What is the speed of the
particle at t  2 ?
(Q) 21
(R) 6
(S) 6
(T) 10
(U) 32
1
3 x 2 ( x  2)( x  5) 3
8) The derivative of f ( x ) is given by f '( x) 
. In which of the following open
( x  6)
intervals is f ( x ) decreasing?
(T) (, 5)  (2, 6)
(U) (5, 0)  (2, 6)
(W) (0, 2) only
9) If f ( x)  ln x and
1
2
(I)  ln ln x
(V) (5, 0)  (0, 2)
(X) (, 0)  (0, 2)  (5, 6)
f '( x)
 g '( x) , which of the following could be g ( x ) ?
[ f ( x)]2
1
4
(J)  (ln x) 2
(K)
1
ln x
(L)
1
ln x
(M)
1
(ln x) 2
10) The line y  ax  k is tangent to the circle x 2  ( y  4)2  20 at the point (4, 6) . What is the
value of a  k ?
(L) 14
(M) 12
(N) 12
(O) 14
(P) None of these
11) A young girl, 5 ft tall, is walking away from a lamppost which is 12 ft tall. She walks at a
constant rate of 2 ft/sec. How fast is the length of her shadow increasing in ft/sec when she is 20 feet
from the post?
(G)
7
ft / sec
10
(H)
10
ft / sec
7
(I) 2 ft / sec
(J)
34
ft / sec
7
(K)
27
ft / sec
10
12) What is the power-reducing formula for cos 2 x ?
U.
1  cos 2 x
2
V.
1  cos 2x
x
W.
1  cos 2 x
2
X.
1  sin 2x
x
Y.
1  sin 2 x
2
13. If f ( x)  x5  3x 2  2 x  7 and g ( x)  3x, then g  f ( x)  
E. x 6  9 x3  2 x 2  7 x
F. 3 x 5  9 x 2  6 x  21
H. 243x5  27 x3  6 x  21
I. 3x 6  9 x3  6 x 2  21x
14. Find all vertical asymptotes for f ( x) 
4
3
K. x   , x  2
G. 3x
3x 2  x  4
.
3x 2  2 x  8
M. x  2
L. x  1, x  2
N. x  
4
3
O. x 
4
3
15. If f ( x ) is continuous at x  a, which of the following could be false?
A.
lim f ( x ) exists
B. f ( a ) exists
xa
C. lim f ( x)  lim f ( x)
x a
x a
E. lim f ( x)  f (a)
D. lim f ( x)  a
x a
xa
16. Suppose that f is a continuous function defined for all reals and f (5)  3 and f (1)  2.
If f ( x)  0 for one and only one value of x, then which of the following could be x?
A. 7
B. 2
C. 0
D. 1
E. 2
17. Graph a function with the following properties:
lim f ( x)  2
x 
lim f ( x)  1
lim f ( x)  0
x 
lim f ( x)  1
x  3
x  3
f (3)  0
f (3) is not defined
lim f ( x)  DNE()
x  5
lim f ( x)  DNE( )
x  5
x
Use this chart for problems 18 – 19.
f ( x) g ( x)
f ( x) g ( x)
2
1
4
3
5
3
7
2
8
6
18) Find the derivative of f ( x)  g ( x) at x  2.
19) Find the derivative of
f ( x)
at x  3.
g ( x)
20) If f ( x)  2 x 2  3x  5 is defined on [1, 4], at what value of x is the average rate of change equal
to the instantaneous rate of change?
 
 is:
9
21. If f ( x)  tan 3x, then the value of f  
A.
4
B.
6
C. 12
D.
4
3
E. 6 3
22. If f ( x)  x 4 x  1, then f ( x ) is:
A.
6x 1
4x 1
B.
2x
4x 1
C.
1
4x 1
D.
6 x  2
4x 1
E.
9x  2
2 4x 1
23. Where (ordered pair) on the curve f ( x)  3x 2  5x  6 is the slope of the tangent line equal to 7?
24. If x3  2 x 2 y  4 y  7, then when x  1,
A. 
9
2
B.
7
2
D. 3
C. 0
25. Given x cos y  x 2  y 3 , what is
dy
must equal what?
dx
2x
3 y  sin y
B.
2x
3 y  sin y
C.
2 x  cos y
3 y 2  sin y
D.
2 x  cos y
3 y 2  x sin y
E.
2 x  cos y
3 y 2  x sin y
B.
x y
xy
26. Find
8
dy
?
dx
A.
2
E.
2
dy
for e y  xy.
dx
A. ln x  ln y
C.
xy
x y
D.
xy  x
y
E.
y
xy  x
27. Give the formulas for volume of a
a) sphere
b) cylinder
c) cone
d) trough