mat 1221 survey of calculus exam 1 info
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MAT 1221Survey of Calculus
Exam 1 Info
http://myhome.spu.edu/lauw
Expectations
(Time = 15 min.) Use equal signs correctly Use and notations correctly Pay attention to the independent
variables: Is it or ?
Tutoring Bonus Points
Turn in your paper today! Get the new paper for the next exam!
Exam 1
Date and Time: 4/24 Thursday (5:30-6:50 pm)
Section 1.5, 2.1-2.5, B.1, B.2 Total Points: 80 points
Exam 1
This exam is extremely important. The second exam is on 5/15. The last
day to withdraw is 5/9. So this exam gives us the critical info for you to make a sound decision.
Calculators
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
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.)
x
Tangent Lines
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.
y
1
( )y f x
3
1 2
Rate of Change
y = distance dropped (ft)
x = time (s)
Find the average speed from x=2 to x=3.
2( ) 16y f x x
ft/s 801
216316
23
)2()3(
Speed
Average
22
ff
Derivative
For a function y=f(x), the derivative at x is a function f’ defined by
if it exists. (f is differentiable at x
f’(x)=The slope of the tangent line at x)
0
( ) ( )( ) lim
h
f x h f xf x
h
Limit Laws Summary
lim ( ) ( )x ah x h a
( )h x
( ) continuous at h x a ( ) not continuous at h x a
Other methods
Simplify2
1
1lim
1x
x
x
Multiply by conjugate
0
2 2limh
h
h
Differentiation Formulas
Constant Function Rule
If , then
Why?
Cxfy )( ( ) 0 0d
f x cdx
00lim0
lim
lim)()(
lim)(
00
00
hh
hh
h
h
CC
h
xfhxfxf
Constant Multiple Rule
If , then
where is a constant
( )y xk u
( )y k u x
k
( )d dku x k u x
dx dx
Power Rule
If , then
(n can be any real number)
nxxfy )( 1)( nnxxf
1n ndx nx
dx
Sum and Difference Rule
If , then)()( xvxuy )()( xvxuy
Product Rule
If ,
then
)()( xgxfy
)()()()( xgxfxgxfy
Quotient Rule
If
then
)(
)(
xg
xfy
2)(
)()()()(
xg
xgxfxgxfy
Chain Rule
( ), ( )
Therefore, ( )( )
y f u u g x
y f g x
dy dy du
dx du dx
xdx
du
udx
dydu
dyy
Extended Power Rule
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
Important Concepts
Left-hand limits and right-hand limits exists if f is continuous at a point if f is differentiable at a point if exists
lim ( )x a
f x
lim ( ) lim ( )x a x a
f x f x
lim ( ) ( )x a
f x f a
a ( )f a
Important Skills
Evaluate limits by using algebra. Finding derivatives using limits and
formula. Understand and able to perform implicit
differentiation. Solve word problems.
Remarks
Portion of points are designated for simplifying the answers.
Units are required for some answers.
Remarks
Review quiz solutions.
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