Multivariable Calculus / Differential Equaitons Differential Equations Welcome

Welcome
to
Multivariable Calculus /
Differential Equations
Equaitons
The Attached Packet is for all students who are planning to take Multivariable
Multibariable Calculus/
Calculus
Differential Equations in the fall. The first quiz will include the materials covered in this packet.
If there are any topics that you do not understand, you are expected to get help on your own.
I look forward to meeting each one of you.
Have a great summer!
Mr. Choi
Have a Great Summer!
Calculus Summer Review Packet
I. Trigonometry Review
o
1. Find the exact radian measure of 210 .
2. Find the exact degree measure of
5π
.
6
3. Find the values of the trigonometric functions if
4. If
θ
is in standard position and
θ
is an acute angle and tan θ =
5
.
12
Q ( 4, −3) is on the terminal side of θ , find the values of the trigonometric
functions.
5.
Find the exact value:
(a)
⎛ 2π ⎞
sin ⎜
⎟
⎝ 3 ⎠
(d)
⎛ π⎞
sec ⎜ − ⎟
⎝ 6⎠
(b)
6. If f ( x ) = cos x , show that
⎛ 5π ⎞
tan ⎜
⎟
⎝ 6 ⎠
⎛ π⎞
sin ⎜ − ⎟
⎝ 3⎠
(c)
f ( x + h) − f ( x)
⎛ cos h − 1 ⎞
⎛ sin h ⎞
= cos x ⎜
⎟ − sin x ⎜
⎟
h
h
⎝
⎠
⎝ h ⎠
7. Verify the identities:
(a)
(1 − sin t )(1 + tan t ) = 1
2
2
csc 2 θ
(b)
= cot 2 θ
2
1 + tan θ
(c)
1
= csc x + cot x
csc x − cot x
8. Find all solutions of the equations:
(a)
2 cos 2θ − 3 = 0
9. Find the solution of the equations for
(a)
2 sin 2 θ = 1 − sin θ
(b)
2sin 3θ + 2 = 0
0 ≤ θ < 2π .
(b)
2 tan θ − sec 2 θ = 0
(c)
sin 2θ + sin θ = 0
II. Limits
Find the limit, if it exists.
10.
lim ( x 2 + 2 )
11.
lim
( x + 3)( x − 4 )
x → − 3 ( x + 3 )( x + 1)
12.
13.
x−4
2
x → −2 x − 2 x − 8
14.
x2 + 2x − 3
x →− 3 x 2 + 7 x + 12
15.
x3 + 8
x →− 2 x + 2
16.
lim
x −5
x −5
17.
lim
lim f ( x )
(c)
lim f ( x )
lim f ( x )
(f)
lim f ( x )
x →3
lim
x →5
lim
x →8
lim
x → 25
x −5
x − 25
lim
1
x −8
Refer to the graph to find each limit:
(a)
(d)
.
lim f ( x )
(b)
lim f ( x )
(e)
x → 2−
x → 0−
x → 2+
x → 0+
x→ 2
x→ 0
Find each limit, if it exists.
22.
(a)
⎧ x2 −1 , x < 1
f ( x) = ⎨
⎩4 − x , x ≥ 1
lim f ( x )
(b)
x →1−
lim f ( x )
(c)
lim f ( x )
lim f ( x )
(c)
lim f ( x )
lim f ( x )
(c)
lim f ( x )
− x3 + 2 x
x →∞ 2 x 2 − 3
27.
lim
x →1+
x →1
⎧3 x − 1 , x ≤ 1
⎩3 − x , x > 1
23. f ( x ) = ⎨
(a)
24.
(a)
lim f ( x )
(b)
x →1−
x →1+
x →1
⎧− x 2 , x < 1
⎪
, x =1
f ( x ) = ⎨2
⎪x − 2 , x > 1
⎩
lim f ( x )
(b)
x →1−
x →1+
x →1
Find the limit, if it exists:
25.
5 x 2 − 3x + 1
x →∞ 2 x 2 + 4 x − 7
lim
lim
26.
2 x2 − x + 3
x →∞
x3 + 1
A function f satisfies the given conditions. Sketch a possible graph of f , assuming that f does not cross a
horizontal asymptote.
28.
29.
lim f ( x ) = 1 , lim f ( x ) = 1 , lim− f ( x ) = −∞ , lim+ f ( x ) = ∞
x→ − ∞
x →∞
x→3
x→3
lim f ( x ) = −2 , lim f ( x ) = −2 , lim− f ( x ) = ∞ , lim+ f ( x ) = −∞ , lim− f ( x ) = −∞ and
x→ − ∞
x →∞
x→3
x→3
lim f ( x ) = ∞
x → −1+
III. DEFINITION OF THE DERIVATIVE
30. Use the definition of the derivative to find
(a)
f ( x ) = 5x2 − 4 x
f ′( x) :
(b)
f ( x ) = 3 − 2 x2
x→ − 1
IV. TECHNIQUES OF DIFFERENTIATION AND TANGENT LINE PROBLEMS.
Find the derivative using the power rule, the product rule or the quotient rule.
31.
f ( x ) = 8x
34.
f ( x) = x
37. Find the
(a)
2
3
3
32.
2
( 3x 2 − 2 x + 5)
35.
f ( x ) = ( x 3 − 7 )( 2 x 2 + 3)
33.
f ( x ) = 12 − 3x 4 + 4 x 6
4x − 5
3x − 2
36.
f ( x) =
f ( x) =
3
x2
3x − 5
x -coordinate of a point on the graph of y = x 3 − 2 x 2 − 4 x + 5 at which the tangent line is:
horizontal
(b)
parallel to the line 2 y + 8 x = 5
38. Find the equation of the line tangent to the curve y =
39. If f and g are functions such that
3 4
at the point P ( −1, 7 ) .
−
x 2 x3
f ( 2 ) = 3 , f ′ ( 2 ) = −1 , g ( 2 ) = −5 and g ′ ( 2 ) = 2 evaluate the
expression:
+ g )′ ( 2 )
(a)
(f
(d)
( f ⋅ g )′ ( 2 )
− g )′ ( 2 )
(b)
(f
(e)
⎛ f ⎞′
⎜ ⎟ ( 2)
⎝g⎠
40. A weather balloon is released and rises vertically such that its distance
10 seconds of flight is given by
(c)
( 4 f )′ ( 2 )
(f)
⎛ 1 ⎞′
⎜ ⎟ ( 2)
⎝f ⎠
s ( t ) above the ground during the first
s ( t ) = 6 + 2t + t 2 where s ( t ) is in feet and t is in seconds.
(a)
Find the velocity of the balloon at t = 1 , t = 4 , t = 8
(b)
Find the velocity of the balloon at the instant the balloon is 50 feet above the ground
Use the power rule, the product rule, the quotient rule and/or the rules for derivatives of trigonometric functions to
find the derivatives:
41.
f ( x ) = 7 tan x
44.
f ( x) =
1
sin x tan x
42.
f ( x ) = 3x sin x
45.
f ( x ) = ( x + csc x ) cot x
43.
f ( x ) = 2 x cot x + x 2 tan x
46. If
f ( x ) = cos x − sin x ; 0 ≤ x < 2π ; find the points where the tangent line is horizontal.
Use the chain rule to find the derivative:
47.
f ( x ) = ( x 2 − 3x + 8)
49.
f ( x ) = ( 5 x 2 − 2 x + 1)
51.
f ( x ) = sin 2 x − cos 2 x
48.
f ( x ) = sin ( 2 x + 3 )
50.
f ( x) =
52.
f ( x ) = ( 6 x − 7 ) (8x 2 + 9 )
3
−3
4
x 4 − 3x 2 + 1
( 2 x + 3)
4
3
2
V. IMPLICIT DIFFERENTIATION
53.
8x 2 + y 2 = 10
56.
x = sin ( xy )
54.
5 x 2 − xy − 4 y 2 = 0
57. Find the slope of the tangent line to the graph of xy + 3 y = 27 at the point
2
58. If y =
(
2 x 2 + 1 ; P −1, 3
55.
y 2 + 1 = x 2 sec y
( 2,3) .
)
(a) Find the equation of the tangent line and the normal line to the graph of the equation at the point P .
(b) Find the
x -coordinate of the graph at which the tangent line is horizontal.
59. Assuming that the equation 3 x + 4 y = 1 determines a function f such that
2
2
60. Suppose f and g are functions such that
y = f ( x ) ; find
f ( 2 ) = −1 ; f ′ ( 2 ) = 4 ; f ′′ ( 2 ) = −2 ; g ( 2 ) = −3 ; g ′ ( 2 ) = 2 and
g ′′ ( 2 ) = 1 . Find the value of each of the following at x = 2 .
− 3 g )′
(a)
(2 f
(d)
( f ⋅ g )′′
d2y
.
dx 2
− 3 g )′′
(b)
(2 f
(e)
⎛ f ⎞′
⎜ ⎟
⎝g⎠
(c)
( f ⋅ g )′
(f)
⎛ f ⎞′′
⎜ ⎟
⎝g⎠
Problems 61- 65 should be completed without the use of a calculator.
61. (1987 AB2) Let f ( x ) = 1 − sin x .
a. What is the domain of f ?
b. Find
c.
f ′( x) .
What is the domain of f ′ .
d. Write an equation for the line tangent to the graph at
62. (1988 AB1) Let f be the function given by
x =0.
f ( x ) = x 4 − 16 x 2 .
a. Find the domain of f .
b. Describe the symmetry, if any, of the graph of f .
c.
Find
f ′( x) .
d. Find the slope of the line normal to the graph of f at
63. (1989 AB-1) Let f be the function given by
x = 5.
f ( x ) = x3 − 7 x + 6
a. Find the zeros of f .
b. Write an equation of the line tangent to the graph of f at
64. (1989 AB-4) Let f be the function given by f ( x ) =
x
x2 − 4
x = −1 .
.
a. Find the domain of f .
b. Write an equation for each vertical asymptote to the graph of f .
c.
Write an equation for each horizontal asymptote to the graph of f .
d. Find
f ′( x) .
65. (1992 AB4, BC1) Consider the curve defined by the equation y + cos y = x + 1 for 0 ≤ y ≤ 2π
a. Find
dy
in terms of y .
dx
b. Write an equation for each vertical tangent to the curve.
d2y
in terms of y .
c. Find
dx 2
SUMMER REVIEW PACKET ANSWERS
7π
6
16. LDNE
2.
150o
18. (a) −1 (b) −2 (c) LDNE
(d) 1 (e) 0 (f) LDNE
3.
sin θ =
1.
17. LDNE
5
13
12
cos θ =
13
5
tan θ =
12
−3
sin θ =
5
4
cos θ =
5
−3
tan θ =
4
4.
5. (a)
3
2
sec θ
cot θ
csc θ
sec θ
cot θ
−1
− 3
(c)
(d)
2
3
11π
+ nπ
12
12
5π 8nπ 7π 8nπ
8. (b)
+
+
;
12 12 12 12
8. (a)
π
(b)
13
5
13
=
12
12
=
5
5
=
−3
5
=
4
4
=
−3
csc θ =
5π 3π
,
6 6 2
π 5π
9. (b)
,
4 4
2π 4π
9. (c) 0 , π ,
,
3 3
9. (a)
π
+ nπ ;
19. (a) 1
(d) 3
(b) 1 (c) 1
(e) 3 (f) 3
20. (a) 3
(d) 2
(b) 1 (c) LDNE
(e) 2 (f) 2
21. (a) 4
(d) 1
(b) 4 (c) 4
(e) 1 (f) 1
22. (a)
0
23. (a) 2
(b)
3 (c) LDNE
(b) 2
(c) 2
24. (a) −1 (b) −1 (c) −1
2
3
25.
5
2
26. LDNE
27.
0
28.
,
29.
10. 11
11.
12.
7
2
1
10
30. (a)
31.
f ′ ( x ) = 12 x1 2
32.
f ′ ( x ) = 10 x 4 + 9 x 2 − 28 x
33.
f ′ ( x ) = −12 x3 + 24 x5
13. LDNE
14. −4
15. 12
10 x − 4 (b) −4x
34. f ′ ( x ) = 8 x
53
10 x 2 3 10
−
+ 13
3
3x
35. f ′ ( x ) =
7
( 3x − 2 )
36. f ′ ( x ) = −
37. (a) x = −
−3 (10 x − 2 )
49. f ′ ( x ) =
2
3 x + 10
3 x1 3 ( 3 x − 5 )
50. f ′ ( x )
2
( 5 x − 2 x + 1)
2 ( 6x + 6 x − 9x − 4)
=
3
38. y = 18 x + 25
2
( 2 x + 3)
5
cos 2 x + sin 2 x
sin 2 x − cos 2 x
51. f ′ ( x ) =
2
4
or 2 (b) x = 0 or
3
3
4
2
52.
f ′ ( x ) = 2 ( 6 x − 7 ) ( 8 x 2 + 9 )(168 x 2 + 112 x + 81)
2
−3
−1
(e)
25
39. (a) 1
(c) −4
(b)
(d) 11
(f)
1
9
53.
dy −8 x
=
dx
y
54.
dy 10 x − y
=
dx x + 8 y
55.
dy
−2 x sec y
= 2
dx x sec y tan y − 2 y
v (1) = s′ (1) = 4 ft / sec
40. (a) v ( 4 ) = s′ ( 4 ) = 10 ft / sec
v ( 8 ) = s′ ( 8 ) = 18 ft / sec
(b) t ≈ 5.708; v ( 5.708 ) ≈ 13.42 ft sec
41.
f ′ ( x ) = 7 sec2 x
42.
f ′ ( x ) = 3x cos x + 3sin x
56.
or
43.
f ′ ( x ) = −2 x csc2 x + 2 cot x + x 2 sec2 x + 2 x tan x
44.
dy
1
y
=
−
dx x cos ( xy ) x
f ′ ( x ) = − csc x + 2 cot 2 x csc x
or
f ′( x) =
− ( sin x sec 2 x + tan x cos x )
3
( x − 2)
5
57.
3
21
or y = − x +
5
5
y −3 = −
58. (a)
sin 2 x tan 2 x
45.
f ′ ( x ) = − x csc x + − csc x + cot x − csc x cot x
2
3
2
⎛ 3π
⎞
⎛ 7π
⎞
46. ⎜
, − 2 ⎟ or ⎜
, 2⎟
⎝ 4
⎠
⎝ 4
⎠
(
47. f ′ ( x ) = 3 x − 3 x + 8
2
(b)
) ( 2 x − 3)
dy −2
=
dx
3
Tangent Line : y − 3 =
−2
( x + 1)
3
Normal Line : y − 3 =
3
( x + 1)
2
x=0
2
48. f ′ ( x ) = 8 ( 2 x + 3) cos ( 2 x + 3)
3
dy 1 − y cos ( xy )
=
dx
x cos ( xy )
59.
4
dy −3 x d 2 y −12 y 2 − 9 x 2
=
; 2 =
dx
4
dx
16 y 3
60. (a) 2
(b)
−7
(c) −14
−10
−19
(f)
9
27
61. a. D f = set of all real numbers OR { x x ∈
(d) 21 (e)
b.
f ′( x) =
}
− cos x
2 1 − sin x
c. D f ′ = set of all real x ≠
d. y − 1 = −
π
2
+ 2π n, n = any integer
1
1
( x − 0 ) OR y = − x +1
2
2
62. a. D f : x ≥ 4 , x ≤ −4 , x = 0
b. Graph is symmetric with respect to the y-axis.
c. f ′ ( x ) =
4 x 3 − 32 x
2 x 4 − 16 x 2
d. Slope of the normal =
=
2 x 3 − 16 x
x 2 ( x 2 − 16 )
−3
34
63. a. Zeros: 1, 2, -3
b.
y − 12 = −4( x + 1) OR y = −4 x + 8
64. a. D f : x < 2 or x > 2
b. Vertical asymptotes: x = 2 , x = −2
c. Horizontal asymptotes: y = 1 , y = −1
d.
f ′( x) =
x2 − 4 −
x ⋅ 2x
2
2
−4
2 x2 − 4 = x − 4 − x =
3
2
32
2
x −4
( x2 − 4) ( x2 − 4)
dy
1
π
, 0 ≤ y ≤ 2π and y ≠
=
dx 1 − sin y
2
π
b. Equation of vertical tangent: x = − 1
2
2
d y
cos y
c.
=
3
2
dx
(1 − sin y )
65. a.