Name: 1. Find the domains of the following functions: (a) f (x) = log(x(x + 1)
1
Logs only take positive values, so th domain will be x values such that x(x + 1) > 0. The critical points for this inequality are x = 0 and x = 1. Moreover, the graph
Find the four second order partial derivatives
1.
f ( x , y ) =8 x3 +5 y 38 x 2 y 2 +100
2.
f ( x , y )=
6 x 26 y 2
x2 + y2
Find all relative extreme points
1.
3 2
2
f ( x , y ) =x y +6 x +21 y+ 8
2
2.
f ( x , y ) =2 x 2 +2 y 4 + y10
Constrained Min/Max P
Final Exam Review Fall 2016
1.
a.
x 5 f ( x )
lim
b.
x 5+ f ( x )
lim
c.
x 5
d.
x 1
e.
lim f ( x )
DNE
lim f ( x )
lim f ( x )
x 1
4
1
f. Is f(x) continuous at x = 1? What are rules for something to be continuous?
No. F(a) is defined;
lim f ( x )
x a
is
6.2
Suppose the demand for a product is given by p=d(q)=0.8q+150p=d(q)=0.8q+150 and the
supply for the same product is given by p=s(q)=5.2qp=s(q)=5.2q. For both functions, qq is the
quantity and pp is the price, in dollars.
1.
Find the equilibrium point.
Partial Derivatives
Find
z z z
,
,
x y x

(3,2)
1.
z=8 x4 xy
2.
z=10 x7 y
3.
z=2 x3 +3 xyx
4.
f ( x , y ) = x 2 + y 2
z
y

( 0,4 )
Higher Order Partials
Find the four second order partial derivatives.
1.
f ( x , y ) =7 x y 2 +5 xy2 y
2.
f ( x , y ) =x2
Mixed Review Chapter 4
Take the following derivatives:
1.
f ( x )=ln ( x 2x )
2.
f ( x )=6 x
3.
f ( x )= 2 x ln ( 4 x )
4.
f ( x )= x + ln x
2
2
Antiderivatives
Take the antiderivative of the following functions. Dont forget about +C
55
x
x
1.
2.
6 x x
Warmup: First Derivative Test Quiz
Refer to the figure for Questions 14.
2
1
1
3
1
1.
On what interval is f(x) positive?
a. (0, )
2.
1
b. [1,)
c. [2,)
d. (2,)
e. (,2)
b. 0 and 2
c. 0
d. 0 and 3
e. 3
On what interval(s) is the derivative of f(x) increas
SI Session: Optimization
Optimization Pro Tips!
Word problems are some of the most difficult concepts for students to master, because they
come in so many shapes and forms and dont always have a clear cut way of solving them.
With Optimization problems, t
Chapter 2 Review: Implicit Differentiation Worksheet
dy
x ) for each function.
Find y (aka
2
2
1.
5 y 11 x =9
2.
( 2 y 3+ 5 x 2 ) =6 x 2 +11
3.
6 x4 x5 y 3 + y 5=1
4.
6 x + 4 x y =7 x1
3
2
Find y (aka
5.
2
4
dy
x ) at the given point.
y 2+ y 3=12 x +12 at
Warmup Review of 1.6
Find the Average Rate of Change of f ( x ) with respect to
1)
f ( x )=2 x1; [ 4,9 ]
2)
x 25
f ( x )=
; [ 6,15 ]
x5
2
Find and simplify the difference quotient of:
3)
f ( x )=6 x16
4)
f ( x )=2 x 26
Find the instantaneous rate of chan
Basic Integration Techniques
Find the antiderivative. Dont forget to add C!
(
4 7
+ x
x3 x
)
1.
2.
6 x
3.
( 3 x 24 x+7 ) x
4.
3 3
2 6 x + x 4 x
x
5.
x 4 x x
6.
( 2 t 5 3 ) t 4 t
8x
(
)
5
Solving for C
Find f(x) such that the function meets the given
WarmUp
True/False
1)
2)
3)
4)
5)
6)
The first derivative of an equation is used to determine critical values
Setting f(x) = 0 allows you to determine hypercritical values
If f(x) is defined at x=c, then where f(c) = 0 is where the only critical values ar
4.1 Applications
Find f(x) that satisfies the conditions given. [HINT: Take the indefinite
integral and solve for C]
1.
f ' ( x )=5 x2 +3 x7 ;
2.
f ' ( x )=5 2 x ;
f ( 0 )=
3.
f ' ( x )=x 2+1 ;
f ( 0 )=8
f ( 0 )=9
1
2
4. Solid Rock Industries determines t
Name_
Mixed Review Chapter 4
Take the following derivatives:
1.
f ( x )=ln ( x 2x )
2.
f ( x )=6 x
3.
f ( x )= 2 x ln ( 4 x )
4.
f ( x )= x + ln x
2
2
Name_
Antiderivatives
Take the antiderivative of the following functions. Dont forget about +C
55
x
x
1
1 Practice Test 2 1. Take the derivatives of the following functions: (a) f (x) = 4 3x3 + 2x + x (b) g(x) =
1 (x3 +2x2 +3x1)4
2. Find the equation of the tangent line to F (x) =
2x2  4x + 1 at x = 3.
3. Find the second derivative of the follo
Essay I Assignment: Summary and Response
Your assignment is to write a summary of and response to one of the essays from the Unit I reading assignments.
Steps in the process: Choose one of the essays for your response. Read and annotate the essay
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Harper 1
Bianca Harper Eng 1304.42 Ms Schrock 25 February 2008
Review Questions 1) The five types of claims are claims of fact, claims of definition, claims of cause, claims of value, and claims of policy 2) Claims of Fact Did it happen? Is it tru
Name_
Warmup: Optimization Practice
Solve the following optimization problems:
1. A wholesale store can sell 40 ceramic filters each week when it prices them at $100
each. The manager estimates that for each $0.50 reduction in price, he can sell 20
more f
SI Session Chapter 3.4 and Chapter 3 Review
Warmup
Price Elasticity of Demand
Use your notes to answer the following conceptual questions about price elasticity of
demand.
1. What is the formula for solving price elasticity of demand?
2. If price elastic
Chapter 1 & 2 Review
Warm up
True/false:
1) If
2) If
lim f ( x )
x 5
lim f ( x )
x 2
exists, then f(5) must exist.
= L, then L = f(2)
3) If f is continuous at x = 3, then
lim f ( x ) =f ( 3 )
x 3
.
4) A functions derivative at a point, if it exists, can b
First and Second Derivative Test
Determine the relative min/max, intervals of increase/decrease, concavity and hypercritical
values of each function.
1.
f ( x )=2 x 3+ 2 x 210 x5
2.
4 3
2
f ( x )= x 9 x +4 x +10
3
3. The inventory cost C, in dollars, to a
Final Exam Review Fall 2016
1.
a.
x 5 f ( x )
lim
b.
x 5+ f ( x )
lim
c.
x 5
d.
x 1
e.
lim f ( x )
lim f ( x )
lim f ( x )
x 1
f. Is f(x) continuous at x = 1? What are rules for something to be continuous?
g. Is f(x) differentiable at x = 1? If not, ex
6.1 Area Between Curves
Area Between Two Curves
Find the area between two curves using integration:
1) The area bounded by the functions f ( x )=x +5 , g ( x ) =x2 +2 x+ 2 , and the lines x
= 0 and x = 3.
2) The area bounded by the functions f ( x )=x 4 ,
Warmup Activity
True or false.
1) A limit may exist as x approaches a even though the value f(a) may not exist.
2) A limit as x approaches a cannot be different from the function value f(a).
3) If the lefthand and righthand limits as x approaches a are