Download - MAT 1221 Survey of Calculus Exam 1 Info
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MAT 1221Survey of Calculus
Exam 1 Info
http://myhome.spu.edu/lauw
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Expectations
(Time = 15 min.) Use equal signs correctly Use and notations correctly Pay attention to the independent
variables: Is it or ?
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Tutoring Bonus Points
Turn in your paper today! Get the new paper for the next exam!
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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
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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.
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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.
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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
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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.
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Major Themes:
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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.)
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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
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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
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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
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Limit Laws Summary
lim ( ) ( )x ah x h a
( )h x
( ) continuous at h x a ( ) not continuous at h x a
Other methods
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Simplify2
1
1lim
1x
x
x
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Multiply by conjugate
0
2 2limh
h
h
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Differentiation Formulas
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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
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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
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Power Rule
If , then
(n can be any real number)
nxxfy )( 1)( nnxxf
1n ndx nx
dx
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Sum and Difference Rule
If , then)()( xvxuy )()( xvxuy
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Product Rule
If ,
then
)()( xgxfy
)()()()( xgxfxgxfy
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Quotient Rule
If
then
)(
)(
xg
xfy
2)(
)()()()(
xg
xgxfxgxfy
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Chain Rule
( ), ( )
Therefore, ( )( )
y f u u g x
y f g x
dy dy du
dx du dx
xdx
du
udx
dydu
dyy
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Extended Power Rule
dx
dunu
dx
dy
xguuy
xgy
n
n
n
1
)( ,
)(
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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
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Important Skills
Evaluate limits by using algebra. Finding derivatives using limits and
formula. Understand and able to perform implicit
differentiation. Solve word problems.
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Remarks
Portion of points are designated for simplifying the answers.
Units are required for some answers.
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Remarks
Review quiz solutions.