CALCULUS BC
WORKSHEET 1 ON IMPLICIT DIFFERENTIATION
Work the following on notebook paper. Show all work, and circle your answers.
On problems 1 – 4, find
dy
.
dx
1. x3 y 4  5 y  2 x
3. y  sin  xy 
2. sin x  2cos  3 y   1
4. y 2  cos  x  y   3x
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dy
5. (1998 Mult. Ch.) If x 2  xy  10, then when x  2,

dx
7
2
3
7
(A) 
(B)  2
(C)
(D)
(E)
7
2
2
2
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6. (2003 Mult. Ch.) What is the slope of the line tangent to the curve 3 y 2  2 x 2  6  2 xy at the point (3, 2)?
5
7
6
4
(A) 0
(B)
(C)
(D)
(E)
3
7
9
9
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7. (2008 AB 6 – Form B)
Consider the closed curve in the xy-plane given by x2  2 x  y 4  4 y  5.
(a) Show that
dy   x  1

.
dx 2 y 3  1


(b) Write an equation for the line tangent to the curve at the point   2, 1 .
(c) Find the coordinates of two points on the curve where the line tangent to the curve is vertical.
(d) Is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis?
Explain your reasoning.
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8. (2004 BC 4)
Consider the curve given by x2  4 y 2  7  3xy.
dy 3 y  2 x

.
(a) Show that
dx 8 y  3x
(b) Show that there is a point P with x-coordinate 3 at which the line tangent to the curve at P is
horizontal. Find the y-coordinate of P.
d2y
(c) Find the value of
at the point found in part (b).
dx 2
Answers to Worksheet on Implicit Differentiation
dy 2  3x 2 y 4
1.

dx 4 x3 y 3  5
2.
dy
cos x

dx 6sin  3 y 
3.
y cos  xy 
dy

dx 1  x cos  xy 
4.
dy 3  sin  x  y 

dx 2 y  sin  x  y 
5.
dy 2 x  y
dy
7

. At  2,3 ,  
dx
x
dx
2
6.
dy 4 x  2 y 2 x  y
dy 4


. At  3, 2  , 
dx 6 y  2 x 3 y  x
dx 9
A
B
7. 2008 BC 6 – Form B
(a)
  x  1
dy  2 x  2  x  1
 3
 3

dx 4 y  4 2 y  2 2 y 3  1

(b) At   2,1 ,

dy 1

dx 4
1
 x  2
4
(c) Vertical tangents at   4, 1 and
Tangent line: y  1 
 2, 1
(d) No, the curve cannot have a horizontal tangent where it crosses the x-axis.
8. 2004 BC 4
(a)
dy 3 y  2 x

dx 8 y  3x
(b) y = 2
(c) 
2
7
CALCULUS BC
WORKSHEET ON DEFINITION OF THE DERIVATIVE AND IMPLICIT DIFFERENTIATION
Work these on notebook paper. Show all work, and circle your answers.
On 1 – 4, evaluate. Show the steps that lead to your answer.
2
1. lim
h 0
2. lim
   h  1

4

tan 
5  x  h   5x2
3. lim
h 0
h
tan  3  x  h    tan  3x 
4. lim
h 0
h
 h  2
6
 64
h 0
h
h
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5. If f is a differentiable function, then f   a  is given by which of the following?
I. lim
f  a  h  f  a
h 0
II. lim
x a
h
f  x  f a
f  x  h  f  x
III. lim
x a
xa
h
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
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dy
6. Find
given x + tan(xy) = 0
dx
dy
2
7. If y  xy  x  1, find
when x   1.
dx
8. Find
dy
dx
2
and
d y
dx
2
2
, given y  2 y  2 x  1 .
9. Consider the curve defined by the equation y  cos y  x  1 for 0  y  2 .
dy
(a) Find
in terms of y.
dx
(b) Write an equation for each vertical tangent to the curve.
2
(c) Find
d y
dx
2
in terms of y.


10. The function f is differentiable for all real numbers. The point  3,
and the slope at each point (x, y) on the graph is given by


it at the point  3,
dy
dx
y
2
1
 is on the graph of y  f  x  ,
4
6  2x  .
2
Find
d y
dx
2
, and evaluate
1
.
4
2
3
11. Consider the curve given by xy  x y  6 .
dy
(a) Find
.
dx
(b) Find all points on the curve whose x-coordinate is 1, and write an equation for the tangent line to the
curve at each of these points.
(c) Find the x-coordinate of each point on the curve where the tangent line is vertical.
Answers to Worksheet on Definition of the Derivative and Implicit Differentiation
1. 10x
2. 3sec2  3x 
3. 2
4. 192
5. C
6.
1  y sec2  xy 
x sec2  xy 
7. 
8. 
1
2
1
 y  1
3
dy
1

dx 1  sin y

(b) x   1
2
2
d y
cos y
(c)

3
2
dx
1  sin y 
9. (a)
10.
2
d2y
1
2
 1 d y
2
3


2
y

2
y
6

2
x
.
At
3,
,





2
2
dx
8
 4  dx
dy 3x 2 y  y 2

dx 2 xy  x3
(b) The points are 1, 3 and 1,  2  .
11. (a)
At (1, 3) , the tangent line is y  3.
At 1,  2  , the tangent line is y  2  2  x  1 .
(c) x  5  24
CALCULUS
WORKSHEET 1 ON RELATED RATES
Work the following on notebook paper.
1. A ladder 15 feet tall leans against a vertical wall of a house. If the bottom of the ladder is pulled away
horizontally from the house at 4 ft/sec, how fast is the top of the ladder sliding down the wall when the
bottom of the ladder is 9 feet from the wall?
2. A particle moves along the curve y  x 2  1 in such a way that
dx
dt
 4. Find
dy
dt
when x = 3.
3. Two automobiles start from a point A at the same time. One travels west at 72 miles per hour; the other
travels north at 54 miles per hour. How fast is the distance between them increasing 3 hours after they
start?
4. An observer stands 25 feet from the base of a flagpole and watches a flag being
lowered at a rate of 5 ft/sec. Determine the rate at which the angle of elevation from
the observer to the flag is changing at the instant that the flag is 25 feet above eye-level.
25 ft.
5. A man 6 ft tall walks at a rate of 5 feet per second away from a light that is 15 ft above the ground.
When he is 10 feet from the base of the light,
(a) at what rate is the tip of his shadow moving?
(b) at what rate is the length of his shadow changing?
6. The length L of a rectangle is decreasing at the rate of 2 cm/sec while the width W is increasing at the
rate of 2 cm/sec. When L = 12 and W = 2, find the rate of change of:
a) the area of the rectangle
b) the perimeter of the rectangle
c) the length of a diagonal of the rectangle.
7. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic ft per
minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the
1
height of the pile changing when the pile is 15 ft high? (The volume of a cone is given by V   r 2 h .)
3
8. A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the
tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water
is 8 ft deep.
Answers to Worksheet 1 on Related Rates
1.  3
2.
ft
sec
12 12 10 6 10


10
5
10
3. 90
4. 
mi
hr
1 rad
10 sec
5. (a)
25 ft
3 sec
cm 2
6. (a) 20
sec
(b)
10 ft
3 sec
(b) 0
cm
sec
7.
40
8
ft

 .006
2025 405
min
8.
90
9
ft

 .286
100 10
min
(c) 
10 cm
37 sec
CALCULUS BC
WORKSHEET 2 ON RELATED RATES
Work these on notebook paper, showing all steps.
1. A paper cup, which is in the shape of a right circular cone, is 16 cm deep and has a radius of 4 cm. Water is
cm3
draining out of the cup at a constant rate of 2
. When the radius has a length of 3 cm, what is the rate of
sec
1


change of the radius?  Volume of a cone   r 2 h 
3


_____________________________________________________________________________________________
in 3
. How fast is the
2. A snowball is in the shape of a sphere. Its volume is increasing at a constant rate of 10
min
4


radius increasing when the volume of the sphere is 36 in3 ?  Volume of a sphere   r 3 
3


_____________________________________________________________________________________________
3. (1985) The balloon shown is in the shape of a cylinder with hemispherical ends of the
same radius as that of the cylinder. The balloon is being inflated at the rate of
cm3
261
. At the instant that the radius of the cylinder is 3 cm, the volume of the
min
cm
balloon is 144 cm3 , and the radius of the cylinder is increasing at the rate of 2
.
min
4 3

2 
 Volume of a sphere   r . Volume of a cylinder   r h. 
3


(a) At this instant, what is the height of the cylinder?
(b) At this instant, how fast is the height of the cylinder increasing?
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4. (1988) The figure shown represents an observer at point A watching balloon B
as it rises from point C. The balloon is rising at a constant rate of 3 m/sec, and
the observer is 100 m from point C.
(a) Find the rate of change in x at the instant when y = 50.
(b) Find the rate of change in the area of right triangle BCA at the instant
when y = 50.
(c) Find the rate of change of  at the instant when y = 50.
_____________________________________________________________________________________________
5. (1994) A circle is inscribed in a square, as shown in the figure. The circumference
of the circle is increasing at a constant rate of 6 in/sec. As the circle expands, the
square expands to maintain the condition of tangency.
(a) Find the rate at which the perimeter of the square is increasing.
(b) At the instant when the area of the circle is 25 in 2 , find the rate
of increase in the area enclosed between the circle and the square.
TURN->
Multiple Choice. All work should be shown.
6.
The sides of the rectangle above increase in such a way that
dz
 1 and
dx
3
dy
.
dt
dt
dt
dx
At the instant when x = 4 and y = 3, what is the value of
?
dt
1
(A)
(B) 1
(C) 2
(D) 5
(E) 5
3
_____________________________________________________________________________________________
7. (1997) If  increases at a constant rate of 3 rad/min, at what rate is x increasing
in units/min when x = 3 units?
(A) 3
(B)
15
(C) 4
4
(D) 9
(E) 12
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8. (1998) If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at
a rate of 3 inches per minute, which of the following must be true about the area A of the triangle?
(A) A is always increasing.
(C) A is decreasing only when b  h.
(E) A remains constant.
(B) A is always decreasing.
(D) A is decreasing when b  h.
_______________________________________________________________________________________
Answers
1 cm
1. 
18 sec
2.
5
in
18 min
3. (a) 12 cm
4. (a)
5. (a)
6. B
7. E
8. D
3 m
5 sec
24 in
 sec
(b) 5
cm
min
(b) 150
m2
(c)
sec
2
 120
 in
 30 
 
 sec
(b) 
3 rad
125 sec