CALCULUS 2
Name: _____________________________
WORKSHEET 0.1
1. find y  if y 
2. find
1 4
5
x  3x 3  x 2  5x  6
6
2
dy
 8x
if y  2
dx
7x  8
3. find f x  if f x   3x 4 tan x  5
4. find
dy
if y  sin 7 x  cos3 x
dx
5. find
dy
if 4xy  3y5  5x 7  9y
dx
6. find the x-coordinate of any local max and min of f x   6x 3  12x 2  64x  6
7. find the intervals of concavity of y  x 3  3x 2  4
CALCULUS 2
Name: _____________________________
WORKSHEET 0.2
Evaluate.
1. sin
5
6
2. cos
5. sec

4
6. cot
 11
6
10
3
3
2
3. tan
 4
3
4. csc
7. sec
 7
2
8. tan 
7
3
Draw one period of the following curves. Label both axes.
9. y  4 sin x
10. y  3 cos x
11. y  2 tan x
Verify the following identities.
2
2
2
12. sec x  sin x sec x  1
15.
1  2 csc x
 tan x  2 sec x
cot x
13.
cos x
 sec x
1  sin 2 x
16.
14. csc x  sin x  cot x cos x
csc x cos 2 x  sin x  csc x
CALCULUS 2
Name: _____________________________
WORKSHEET 3.2/3.3
Use the definition of the derivative, f ' x   lim
h 0
f (x  h)  f (x )
, to find f ' x  for each function.
h
1. f x   x 2  x  3
2. f x  
3
x 1
Find the derivative of each of the following.
3. y = −3x6 + 2x4
6. y = 1 
1 2
3
 2  3
x x
x
9. y = 3x2 +
2
3
x4
4
4. y = 9 x 3
5. y = 2x−4
7. y = 3
8. y = 83 x 2 
10. y = 24 x 7 
64 3
x
5
10
x5
11. y =  1  x  x 
1
1

x
x
12. Find the equation of the tangent to y = 2x2 + 3 at the point where x = 1.
13. Find the equation of the horizontal tangent to y = x2 + 2x – 3.
14. At which point(s) does the graph of the equation y  13 x 3  32 x 2  2x have a horizontal tangent line?
2
15. Find the equation of the tangent to y = 2 x 3  4x
 12
at (1, −2).
16. Find the equation of the tangent to y  34 x 2  2 x at x = 4.
CALCULUS 2
Name: _____________________________
WORKSHEET 3.4
Find f (x) for each of the following.
1. f(x) = 2 cos x – 3 sin x
2. f(x) = sec x −
3. f(x) = x – 4 csc x + 2 cot x
4. f(x) = cos 2 x  sin 2 x
5. f(x) =
4 cos x  sin x
cos x
6. f(x) =
2 tan x
cos x
1  sin 2 x
7. f(x) = x 2 csc 2 x  x 2 cot 2 x
8. f(x) = csc xsin x  cos x 
9. f(x) = sec x  tan x sec x  tan x 
10. f(x) =
7  3 sin x
11. f(x) =
2 cos x
5 cos 2 x  5 sin 2 x
12. f(x) =
sec x
1
cot x
Find the equation of the line tangent to the graph of y = sin x at the given x-coordinate.
13. x = 0
14. x = 
15. x =
π
2
CALCULUS 2
Name: _____________________________
WORKSHEET 3.3/3.4
Find the derivative of each of the following.
1. y = (3x2 + 6)(2x – 1)
2. y = (2 – x – 3x3)(7 + x5)
1 1 
4. y =   2 3x 3  27 
x x 
5. y =
1
7. y =
5x  3
3.
x2 1
3x
y = (x3 + 7x2 – 8)(2x−3 + x−4)
6. y =
x3  2
8. y =
6
9. y =
2x  1
x3
4x  1
x2  5
11. y = (x – 2)−1(2x3 – 1)
 x 1
12. y = (2 x 7  x 2 )

 x  1
14. y = x tan x
15. y =
17. y = sin 2 x
18. y = x2 cos x
19. y = x3 sin x – 5 cos x
20. y = (x2 + 1) sec x
21. y = sec x tan x
22. y = csc x cot x
23. y =
10. y =
1
(2x  3)( x  1)
13. y = sin x cos x
16. y =
sin x
1  cos x
sin x sec x
1  x tan x
24. y =
d2y
Find
for each of the following.
dx 2
26. y = x2 sin x - 4 cos x
25. y = x cos x
27. Find the equation of the tangent to y =
28. At which point(s) does the graph of y =
x
at x = π
cos x
x
have a horizontal tangent line?
x 9
2
sin x
x
sin x cos x
x  x cos 2 x
CALCULUS 2
Name: _____________________________
WORKSHEET 3.5-1
Find the derivative of each of the following.
1. y = (5x 2  2x  4) 4
3. y =
2. y =
2x 3  3x 2
4. y =
5. y = sin 3x + cos 5x
4x
1
3
 5x 2

6
3
25 3x  7 
4
6. y = 4 sec 5x
 

7. y = cos x 3
8. y = 1 cos 2 x
 

6
9. y = tan x 2  tan 2 x
10. y = sin 3 x  cos 3 x
11. y = cot5(4x)
12. y = 3 cos2(2x)
13. y = sin(2x + 4)4
14. y = sin cossin x 
15. y = cos 2 cos 4x 
16. y   sec 3 (2x 2  x  1)
17. Find the equation of the tangent to y =
3

x 2  4 at x = 2.
18. At which point(s) does the graph of y = 2x 3  x 2

4
have a horizontal tangent line?
CALCULUS 2
Name: _____________________________
WORKSHEET 3.5-2
Find the derivative of each of the following.
1. y = 3x  2 4x  1
5
3. y =
5x  34
3x  53
8
 7x 3  1 

2. y =  2
 2x  5 
4. y =
4
2x  3
4x 2  9
5. y = tan4x sec2x
6. y = cot(2x - 1) csc(3x  1)
 1 
7. y = cos

 x  2
 x 
8. y = sin 2 

 x 1 
cos 4x
9. y =
1  sin 4x
 sin x 
10. y  

 1  cos x 
2
CALCULUS 2
Name: _____________________________
WORKSHEET 3.6
Find
dy
for each of the following (by using implicit differentiation).
dx
1. y3  2x  4
4. 2x y  3x  2x
2
7. y 2 
x
x 1
10. y 2  y 3  4x
13.
x  y3
 x2
y  x2
3. 4x 3  2y3  x
2. 2x 4  3y3  14
5
5.
2
3
2
3
6. x  y  4
x  y  100
8. x  sin xy 
9. sin y  3x  2
11. x  y   x  y  1
12. y  x 2 y3  x 3y2
14. y  sin 3x  4y
15. cos 2 x  cos 2 y  cos2x  2y
2
16. Find the equation of the line tangent to the curve x 2  y2  1 at the point where x=1.
17. Find an equation of the line tangent to the graph of x 2  y2   8x 2 y2 at the point (−1, 1) .
3
CALCULUS 2
Name: _____________________________
WORKSHEET 4.2-4.6 -1
Identify the intervals where f is increasing, decreasing, concave upward and concave downward. Find
the coordinates of any local extreme points and points of inflection. Then, graph.
1. f(x) = x3 – 3x + 2
2. f(x) = (x – 5)3 + 2
3. f(x) = x3 –
4. f(x) = 3x4 – 4x3
3
2
x2 – 6x + 2
CALCULUS 2
Name: _____________________________
WORKSHEET 4.2-4.6 -2
Identify the intervals where f is increasing, decreasing, concave upward and concave downward. Find
the coordinates of any local extreme points and points of inflection. Then, graph.
6
1. f(x) = x 7
1
2. f(x) = 3x 3  x
2
3. f(x) = x 3 (x  5)
CALCULUS 2
Name: _____________________________
CHAPTERS 3&4 PRACTICE TEST
Find the derivative of each of the following.
1. y = 4x 5  6x  9  x5  x
2. y = 4π 3  3x
3. y = x 2 x
4. y =
7x 5  2x 3
x6
2x  1
x2  1
6. y =
3
( x  5)(3x  1)
7. y = 2 sin x  6 cot x
8. y =
3 sec 2 x  3 tan 2 x
csc x
sinx  cos 2 x
9. y =
cos x
10. y =
cosx
1  cosx
11. y = 3x  x csc x
12. y = (1  x2 ) 5
13. y = sin 3 x  6x 4
14. y = cot( 4x )
15. y = (2x  5) 3 ( x 2  5x ) 6
 3x  2 
16. y = 

 x 1 
17. y = cos 5 (4x 2  1)
18. sec y  x
19. 5x 2  xy  y  2x
20. cos( x 2 y)  2y  x
21. Find the equation of the tangent
to y2  2y  2x  1 at (7, 3).
22. Find the equation(s) of the horizontal tangent(s)
to y = x 5  15x 3  2 .
5. y =
3
Identify the intervals on which the graph of f is increasing and decreasing and concave upward and
concave downward, and find the coordinates of any local extreme points and points of inflection.
Then graph the general shape of the function.
23. f(x) = –3x4 – 8x3 – 6
test:
f ( x ) = _____________________
f ( x ) = ________________________
critical points: _______________
critical points: __________________
increasing:
_______________
concave up:
_______________
decreasing:
_______________
concave down:
_______________
local max(s): _______________
f 
f
f
pt(s) of inflection: _______________
local min(s): _______________
24. f(x) = 5x 5  2x
2
test:
f ( x ) = _____________________
f ( x ) = ________________________
critical points: _______________
critical points: __________________
f 
f
f
increasing:
_______________
concave up:
_______________
decreasing:
_______________
concave down:
_______________
local max(s): _______________
local min(s): _______________
pt(s) of inflection: _______________