Section 6.1 Integration by Parts
Product Rule for Derivatives: (uv)'
Ex1) Using integration by parts for find the integral:
x
a) xe dx
2
b) x ln xdx
Practice) Using integration by parts for find the integral:
3x
a) xe dx
3
b) x ln xdx
Ex2) Using integrat
DEFINITION OF THE DERIVATIVE
1%!) =
ljrn x + h) x)
hb h
DIFFERENTIATION FORMULAS
Power Rule:
Constant Multiple Rule:
Sum-Difference Rule:
Product Rule:
Quotient Rule:
Generalized Power Rule:
Chain Rule:
Logarithmic Formulas:
Exponential Formulas:
AREA AND
Section 8.1 Triangles, Angles, and Radian Measure
The Circumference of a Circle C =
Ex1) Convert Degrees to Radians.
a) 45o
b) 150o
Ex2) Convert Radians to Degrees.
a)
radian
6
b)
3
radian
4
c) 60o
c)1 radian
Common Measures that should be Memorized:
Ex3
Section 7.2 Partial Derivatives
Ex1) Find the partial derivative:
3 4
3 4
x y
x y
a)
b)
y
x
d)
6
y x
x
e)
6
y x
y
Ex2) Find the partial derivative:
(2 x 4 3x 3 y 3 y 2 4 x 1)
a)
x
c)
4
y
x
f)
7
3y
x
b)
(2 x 4 3 x3 y 3 y 2 4 x 1)
y
Ex3) Find both par
Section 7.1 Function of Several Variables
Ex1) For the function f x, y
a) the domain
x
, find:
y2
b) f 9, 1
uv
Ex2) For the function g u, v e ln u , find:
a) the domain
b) g 1,2
Practice) For the function f x, y
a) the domain
ln x
e
y
, find:
b) f e,4
Section 7.3 Optimizing Functions of Several Variables
Ex1) Find all critial points of f ( x, y ) 3 x 2 y 2 3xy 3x y 6
Practice) Find all critial points of f ( x, y ) x 2 2 y 2 2 xy 2 x 4 y 7
Ex2) Find the relative extreme values of f ( x, y ) 3 x 2 y 2 3
Section 5.2 Integration of Logarithmic and Exponential Functions
Ex1) Find the indefinite integral.
1
x
a) e 2 dx
b)
6
e
3x
t
c) e dt
dx
Practice) Find the indefinite integral.
d) 12e dx
4x
Integrating
e)
2
e
1
x
3
dx
t /3
f) e dt
1
x
Ex2) Find the indefi
Section 9.3 First-Order Linear Differential Equations
Ex1) Write each expression as the derivative of a product.
a) x 5 y ' 5 x 4 y
b) e2 x y ' 2e2 x y
Ex2) Solve the differential equation:
a) y '
2
yx
x
b) y '
4
yx
x
Ex3) Solve the differential equation:
Section 5.6 Integration by Substitution
Ex1) Find the differential df:
b) f( x ) ln x 3
a) f( x) x
c) f( x) e x
2
d) f( x) x 4 5 x 2
Try) Find the differential du:
e 5t
a) u
3
5
b) u ln 2 x
5
b) u 2 x 4 7
Ex2) Integrate by Substitution:
2
3
a) ( x 1) 2 x
Section 8.2 Sine and Cosine Functions
On the Unit Circle, the radius is 1.
Ex1) Evaluate sin and cos for the following values of :
a)0
b)
2
c)
d)
3
2
Ex2) Evaluate the following:
a) sin
and cos
6
6
b) sin
and cos
3
3
b) sin
3
3
and cos
4
4
Ex3) Evaluate t
Section 7.7 Multiple Integrals
Ex1) Evaluate the iterated integral:
a)
1
2
0 0
(3x 2 6 xy 2 )dxdy
b)
1
3
1 0
( x 2 2 y 2 )dxdy
2 x
Ex3) Evaluate the iterated integral: y e dxdy where R ( x, y ) | 0 x 2, 1 y 1
R
Ex4) A dome tent with closed sides is con
Section 6.3 Improper Integrals
Ex1) Evaluate the limit:
1
b b 2
1
3
b) lim
b
b
e 2b 5
c) lim
b
1
x x 3
1
9
e) lim
b
b
1 eb/3
f) lim
b
a) lim
d) lim
Ex2) Evaluate the limit:
b3
a) lim
b
b) lim
b
b 5
d) blim
1
e) blim
1
b
e2a
f) alim
2 e
Section 5.3 Definite Integrals and Areas
Approximating the Area by Rectangles
Ex2) Use the definite integral to find the area under the curve of f ( x) x 2 from 1 to 2.
Ex3) Find the area under each curve.
a) y e 2 x from x = 0 to x = 1.
b) f ( x)
c) y e
Section 5.4 Applications of Definite Integrals: Average Value and Area Between Curves
Ex1) Find the average value of the function f ( x) x from x = 0 to x = 9.
Ex2) The population of the U.S. is predicted to be P(t) 309e0.0087 t million people, where t
is
Section 6.2 Integration using Tables
Ex1) Integrate using an Integration Table:
a)
1
x 2 4 dx
Ex2) A companys sales rate is
the total sales after x weeks.
b)
1
9 x 2 dx
x
sales per week after x weeks. Find a formula for
x9
Ex3) Under certain circumstanc
Section 7.5 Lagrange Multipliers and Constrained Optimization
Ex1) A farmer wants to build a rectangular enclosure along an existing stone wall. If the
side along the wall needs no fence, find the dimensions of the largest enclosure that can be
made using
Section 8.3 Derivatives of Sine and Cosine Functions
Ex1) Differentiate each function:
a) f (t )
sin t
t
c) f (t ) t 2 sin t
b) g (t ) t 3 cos t
d) h(t )
cos t
t
Ex2) Differentiate each function:
a) g (t ) cos(t 2 2)
c) f (t ) sin
t
2
b) h(t ) sin 4t
d)
Section 8.5 Other Trigonometric Functions
Ex1) Evaluate each function:
a) tan
c) tan
3
2
b) cot
6
d) csc
7
6
Ex2) Given the angle in the graph, find the following:
a) tan
b) csc
Try) Given the angle in the graph, find the following:
a) sec
b) cot
Ex3)
Section 9.2 Applications of Differential Equations Growth
Models
Ex1) An art collection, initially worth
$20,000, continuously grows at the rate of
5% a year. Find a formula for the value
of the collection after t years. Estimate its
value after 10 years.
Section 9.1 Separation of Variables
Ex1) Verify that y e 2 x e x 1 is a solution of the differential equation y ' y ' 2 y 2
Try) Verify that y e x e3 x is a solution of the differential equation y ' 2 y ' 3 y 0
Ex2) Solve the differential equation y '
6x
Section 8.4 Integrals of Sine and Cosine Functions
Ex1) Find the area under the curve:
a) y sin t
b) y cosx
2
Ex2) Find the integral (substitution):
2
3
a) x sin x dx
b) cos
2
c) x cos x dx
d) sin
2t
dt
3
t
dt
2
Ex3) Find the integral (substitution):
5
a)
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