BC 1
Sample Quiz
Show all appropriate work clearly for full credit.
Name:_________________
NO CALCULATORS
Skills:
1.
For each function below, find its first derivative. Do not simplify the result.

g  x   sinh(2 x)  sin 1 ( x 2 )
a.

5

4 
1
g   x   5 sinh(2 x)  sin 1 ( x 2 )   2 cosh(2 x) 
 2x 


1  x4



k  x   tan 1  cosh x 
b.
k x 
2.

sinh( x)
1   cosh x 
2
Suppose f  2  4, f   2  6, f  4  8, f   4  1, f 8  2, and f  8  3. If f is invertible and
everywhere differentiable, then what is the value of derivative of f 1  x  at x = 4?
 f   4  f ( f 1 (4))  f 1(2)  16
1
1
3.
dk
1
If k ( x)  tan 1  x   tan 1   , determine
, simplifying completely.
dx
 x
dk
1
1
 1 



2
2  2 
dx 1  x
1  x 
1  
x
1
1


1
2
1  x 1  x2
0
 if x  0

1 
Note: This makes if we notice k ( x)  tan 1  x   tan 1     2
 x 
 2 if x  0
BC 1
4.
Sample Quiz
Name:_________________
Consider the curve defined by the equation 3 y3  12 x2 y  16 x3  16 .
dy
a.
Use implicit differentiation to determine
.
dx
d
dy
dy
3 y 3  12 x 2 y  16 x3  0  9 y 2
 24 xy  12 x 2
 48 x 2  0
dx
dx
dx


dy 24 xy  48 x 2

dx 9 y 2  12 x 2
b. Find any point(s) on the curve where the tangent line to the curve is horizontal.
dy
 0  24 xy  48 x 2  0
dx
 x( y  2 x)  0  x  0 or y  2 x

If x = 0, 3 y  16  y  3
3
16
3
If y  2 x , 24 x3  24 x3  16 x3  16  x  1 and then y  1 .

16 
So ( x, y )   0, 3
 or (1, 2)
3 

c.
Find the equations of all the tangent lines to the curve at x = 1.


x  1  3 y3  12 y  0  y  y 2  4  0  y  0, 2, 2
dy 24 xy  48 x 2
Since
,

dx 9 y 2  12 x 2
At (1,0): y  4 x  4
At (1, 2) : y  4 x  2
At (1,2): y = 0
BC 1
Sample Quiz
Name:_________________
5. Use the graph below to estimate  f 1  (3) .
 f  (3) 
1
1
1
2


1
f ( f (3)) f (2) 3
dy
if sin(3x  y)  y 2 cos x .
dx
d
d
 sin(3x  y )    y 2 cos x  
dx
dx
dy 
dy

cos(3 x  y )  3    y 2 sin x  cos( x)2 y 
dx 
dx

2
dy y sin( x)  3cos(3 x  y )

dx cos(3x  y )  2 y cos( x)
6.
Find
7.
The length of a rectangle is increasing at the rate of 4 cm/sec and the width is decreasing at the
rate of 2 cm/sec. What is the rate of change of the area of the rectangle when the area is 15 and
the width is 3 cm? Include units in your final answer.
If l = length, w = width, and a = area, then A  lw .
dA
dl
dw
 w l
Then
. When A  15 and w  3, l  5 . So,
dt
dt
dt
dA
dl
dw
dA
 w l

 3(4)  5(2)  2 cm 2 / sec
dt
dt
dt
dt
BC 1
8.
Sample Quiz
Name:_________________
At a certain moment, one bicyclist is 4 miles east of an intersection, traveling toward the
intersection at the rate of 9 miles/hour. At the same time, a second bicyclist is 3 miles south of
the intersection traveling away from the intersection at the rate of 10 miles/hour. Is the distance
between the bicyclists increasing or decreasing at that moment? At what rate?
Given:
dx
 9 . Bicyclist is heading toward intersection means x is decreasing.
dt
dy
ds
 10 . We want
when x = 4, y = 3, (and then s = 5).
dt
dt
x
dx
dy
ds
 2y
 2s . So, when x = 4,
dt
dt
dt
ds
2  4(9)  2  3 10  2  5
dt
ds 6
Thus, 2  4(9)  2  3 10  2  5 
miles/hour. Since the sign is
dt
5
Negative, the bicyclists are getting closer together.
Now, x 2  y 2  s 2  2 x
y
s
BC 1
9.
Sample Quiz
Name:_________________
2
dV
and V is a function of h with graph shown below, estimate the value of
when
dt
t
h  1 . (Show work and/or explain reasoning).
If h(t ) 
2 dh 1
dh
 1 .

,
. When h = 1, t = 1, so
3
2
dt
t dt
t
dV dV dh


Now,
(Ahhh, the chain rule).
dt
dh dt
dV
 6  (1)  6 .
So, when h = 1.
dt
Since h(t ) 