MAT 1221
Survey of Calculus
Exam 1 Info
http://myhome.spu.edu/lauw
WebAssign
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Outage yesterday afternoon
HW will be extended to tomorrow.
Quiz: you can
• Do the first problem in the quiz, the score of
•
the quiz is double of your score in problem 1
Do both problems with normal scoring.
Expectations
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(Time = 15 min.)
Use equal signs correctly
𝑑
(… )
𝑑𝑥
and
𝑑
(… )
𝑑𝑡
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Use
notations correctly
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Pay attention to the independent
variables: Is it 𝑥 or 𝑡?
Tutoring Bonus Points
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Turn in your paper today!
Get the new paper for the next exam!
Extra! Extra! Reading Bonus!
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How many reading bonus points did you
get from quiz 0 to quiz 6?
Extra! Extra! Reading Bonus!
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How many reading bonus points did you
get from quiz 0 to quiz 6?
To reward those who have been reading
the textbook, I will match 1 point in the
first exam for each bonus point you got
from the quizzes.
Exam 1
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Date and Time: 4/20 Monday (5:30-6:50
pm)
Section 1.5, 2.1-2.5, B.1, B.2
Total Points: 80 points
Exam 1
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This exam is extremely important.
The second exam is on 5/11. The last
day to withdraw is 5/8. So this exam
gives us the critical info for you to make
a sound decision.
Calculators
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Absolutely no share of calculators.
Bring extra batteries, extra calculators. It
is your responsibility to bring a workable
calculator.
NO cell phone or PDA
Your instructor/TA will not answer any
question related to calculators.
Expectations
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Use equal signs
Simplify your answers.
Provide units.
Check and Double Check your solutions.
Show the “formula” steps.
For word problems in B.2, show all 5
steps
Steps for Word Problems
1. Draw a diagram
2. Define the variables
3. Write down all the information in terms
of the variables defined
4. Set up a relation between the variables
5. Use differentiation to find the related
rate. Formally answer the question.
Major Themes:
Slope of the tangent line

Slope of the tangent line at a point on a
graph can be approximated by a limiting
process. (The same apply to other rate
of change problems in physical
sciences.)
Tangent Lines
y

y  f ( x)

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1
x
3
 1 2
To define the tangent line
at 𝑥 = 1, we pick a point
close by.
We can find the secant
line of the two points
We can move the point
closer and closer to 𝑥 =
1.
Rate of Change
𝑦 = distance dropped (ft)
𝑥 = time (s)
y  f ( x)  16 x 2
Find the average speed from
𝑥 = 2 to 𝑥 = 3.Average  f (3)  f (2)
Speed
3 2
16  32  16  2 2

1
 80 ft/s
Derivative

For a function 𝑦 = 𝑓(𝑥), the derivative at
𝑥 is a function 𝑓’ defined by
f ( x  h)  f ( x )
f ( x )  lim
h 0
h
if it exists. (𝑓 is differentiable at 𝑥
𝑓’(𝑥)=The slope of the tangent line at 𝑥)
Limit Laws Summary
h( x )
h( x) continuous at a h( x) not continuous at a
lim h( x)  h(a)
xa
Other methods
Simplify
x 1
lim
x 1 x  1
2
Multiply by conjugate
2h  2
lim
h 0
h
Differentiation Formulas
Constant Function Rule

d


If y  f ( x)  C , then f ( x)  0   c   0 
 dx


Why?
f ( x  h)  f ( x )
C C
f ( x)  lim
 lim
h 0
h 0
h
h
0
 lim  lim 0  0
h 0 h
h 0
Constant Multiple Rule
If
y  k  u ( x) , then
y  k  u( x)
where k is a constant
d
d
 ku  x    k  u ( x) 
dx
dx
Power Rule
n
y

f
(
x
)

x
If
, then f ( x)  nx n1
(n can be any real number)
 
d n
x  nx n 1
dx
Sum and Difference Rule

If y  u ( x)  v( x) , then y  u( x)  v( x)
Product Rule
If
y  f ( x)  g ( x, )
then y  f ( x) g ( x)  f ( x) g ( x)
Quotient Rule
f ( x)
If y 
g ( x)
f ( x) g ( x)  f ( x) g ( x)
then y 
2
g ( x)
Chain Rule
y  f (u ), u  g ( x)
Therefore,y  ( f g )( x)
dy dy du


dx du dx
y
dy
dx

dy
du
u


du
dx
x
Extended Power Rule
y  g ( x)
n
y  u n , u  g ( x)
dy
n 1 du
 nu 
dx
dx
Important Concepts
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Left-hand limits and right-hand limits
lim f ( x ) exists if lim f ( x)  lim f ( x)
xa
x a
x a
f is continuous at a point if lim f ( x)  f (a)
f is differentiable at a point a if f (a) exists


xa
Important Skills
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Evaluate limits by using algebra.
Finding derivatives using limits and
formula.
Understand and able to perform implicit
differentiation.
Solve word problems.
Remarks
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Portion of points are designated for
simplifying the answers.
Units are required for some answers.
Remarks
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Review quiz solutions.
What is the password?
Finish Practice Problems....
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1 and 2 (a)