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Review of elements of Calculus(functions in one variable)
Mainly adapted from the lectures of prof Greg Kelly
Hanford High School, Richland Washington
http://online.math.uh.edu/HoustonACT/
https://sites.google.com/site/gkellymath/home/calculus-powerpoints
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Functions
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004
Golden Gate BridgeSan Francisco, CA
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A relation is a function if:
for each x there is one and only one y.
A relation is a one-to-one if also:
for each y there is one and only one
x.
In other words, a function is one-to-one
on domain D if:
f a f b whenever a b
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To be one-to-one, a function must pass the horizontal line test as well as the vertical line test.
-5
-4
-3
-2
-10
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-10
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-10
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
31
2y x 21
2y x 2x y
one-to-one not one-to-one not a function
(also not one-to-one)
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Inverse functions:
1
12
f x x Given an x value, we can find a y value.
-5
-4
-3
-2
-10
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
11
2y x
11
2y x
2 2y x
2 2x y
Switch x and y: 2 2y x 1 2 2f x x
Inverse functions are reflections
about y = x.
Solve for x:
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Inverse of function xf x a
This is a one-to-one function, therefore it has an inverse.
The inverse is called a logarithm function.
Example:416 2 24 log 16 Two raised to what power
is 16?
The most commonly used bases for logs are 10: 10log logx x
and e: log lne x x
lny x is called the natural log function.
logy x is called the common log function.
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Properties of Logarithms
loga xa x log x
a a x 0 , 1 , 0a a x
Since logs and exponentiation are inverse functions, they “un-do” each other.
Product rule: log log loga a axy x y
Quotient rule: log log loga a a
xx y
y
Power rule: log logy
a ax y x
Change of base formula:ln
logln
a
xx
a
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Trigonometric Functions
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008
Black Canyon of the GunnisonNational Park, Colorado
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Even and Odd Trig Functions:
“Even” functions behave like polynomials with even
exponents, in that when you change the sign of x, the yvalue doesn’t change.
Cosine is an even function because: cos cos
Secant is also an even function, because it is the reciprocal of cosine.
Even functions are symmetric about the y - axis.
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Even and Odd Trig Functions:
“Odd” functions behave like polynomials with odd
exponents, in that when you change the sign of x, the
sign of the y value also changes.
Sine is an odd function because: sin sin
Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function.
Odd functions have origin symmetry.
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Shifting, stretching, shrinking the graph of a function
y a f b x c d
Vertical stretch or shrink;
reflection about x-axis
Horizontal stretch or shrink;
reflection about y-axis
Horizontal shift
Vertical shift
Positive c moves left.
Positive d moves up.
The horizontal changes happen in the opposite direction to what you might expect.
is a stretch.1a
is a shrink.1b
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-1
0
1
2
3
4
-1 1 2 3 4 5x
Amplitude and period in trigonometric functions
2
sinf x A x C DB
Horizontal shift
Vertical shiftis the amplitude.A
is the period.B
A
B
C
D 2
1.5sin 1 24
y x
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23
2
2
2
3
2
2
Trig functions are not one-to-one.
However, the domain can be restricted for trig functions to make them one-to-one.
These restricted trig functions have inverses.
siny x
Invertibility of trigonometric functions
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Continuity
Grand Canyon, ArizonaGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
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Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same as the value of the function.
This function has discontinuitiesat x=1 and x=2.
It is continuous at x=0 and x=4, because the one-sided limits match the value of the function
1 2 3 4
1
2
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jump infinite oscillating
Essential Discontinuities:
Removable Discontinuities:
(You can fill the hole.)
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Removing a discontinuity:
3
2
1
1
xf x
x
has a discontinuity at .1x
Write an extended function that is continuous at .1x
3
21
1lim
1x
x
x
2
1
1 1lim
1 1x
x x x
x x
1 1 1
2
3
2
3
2
1, 1
1
3, 1
2
xx
xf x
x
Note: There is another discontinuity at that can not be removed.
1x
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Removing a discontinuity:
3
2
1, 1
1
3, 1
2
xx
xf x
x
Note: There is another discontinuity at that can not be removed.
1x
-5
-4
-3
-2
-10
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
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Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.
Also: Composites of continuous functions are continuous.
examples: 2siny x cosy x
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Intermediate Value Theorem
If a function is continuous between a and b, then it takes
on every value between and . f a f b
a b
f a
f b
Because the function is continuous, it must take on
every y value between
and .
f a
f b
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Rates of Change and Tangent Lines
Devil’s Tower, WyomingGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993
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The slope of a line is given by:y
mx
x
y
The slope at (1,1) can be approximated by the slope of the secant through (4,16).
y
x
16 1
4 1
15
3 5
We could get a better approximation if we move the point closer to (1,1). ie: (3,9)
y
x
9 1
3 1
8
2 4
Even better would be the point (2,4).
y
x
4 1
2 1
3
1 3
2f x x
0
123456789
10111213141516
1 2 3 4
Slope of a line
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The slope of a line is given by:y
mx
x
y
2f x x
0
123456789
10111213141516
1 2 3 4
Slope of a line
If we got really close to (1,1), say (1.1,1.21), the approximation would get better still
y
x
1.21 1
1.1 1
.21
.1 2.1
How far can we go?
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1f
1 1 h
1f h
h
slopey
x
1 1f h f
h
slope at 1,1
2
0
1 1limh
h
h
2
0
1 2 1limh
h h
h
0
2limh
h h
h
2
The slope of the curve at the point is: y f x ,P a f a
0
lim h
f a h f am
h
2f x x
Slope of a line
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In the previous example, the tangent line could be found
using . 1 1y y m x x
The slope of a curve at a point is the same as the slope of
the tangent line at that point.
If you want the normal line, use the negative reciprocal of
the slope. (in this case, )1
2
(The normal line is perpendicular.)
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Derivatives
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0
limh
f a h f a
h
is called the derivative of at .f a
We write:
0limh
f x h f xf x
h
“The derivative of f with respect to x is …”
There are many ways to write the derivative of
y f x
Derivatives
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f x “f prime x” or “the derivative of f with respect to x”
y “y prime”
dy
dx
“the derivative of y with respect to x”
df
dx
“the derivative of f with respect to x”
d
f xdx
“the derivative of f of x”
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0
1
2
3
4
1 2 3 4 5 6 7 8 9
y f x
-2
-1
0
1
2
3
1 2 3 4 5 6 7 8 9
y f x
The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.
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-3
-2
-1
0
1
2
3
4
5
6
-3 -2 -1 1 2 3x
2 3y x
2 2
0
3 3limh
x h xy
h
2 2 2
0
2limh
x xh h xy
h
2y x -6
-5
-4
-3
-2
-10
1
2
3
4
5
6
-3 -2 -1 1 2 3x
0lim2h
y x h
0
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A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.
Differentiability
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To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner cusp
vertical tangent discontinuity
f x x 2
3f x x
3f x x
1, 0
1, 0
xf x
x
Differentiability
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Rules for Differentiation
Colorado National MonumentGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003
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If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
0d
cdx
example: 3y
0y
The derivative of a constant is zero.
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Derivatives of monomials
2 2
2
0limh
x h xdx
dx h
2 2 2
0
2limh
x xh h x
h
2x
3 3
3
0limh
x h xdx
dx h
3 2 2 3 3
0
3 3limh
x x h xh h x
h
23x
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
(Pascal’s Triangle)
2
4dx
dx
4 3 2 2 3 4 4
0
4 6 4limh
x x h x h xh h x
h
34x
2 3
We observe a pattern: 2x 23x 34x 45x 56x …
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1n ndx nx
dx
examples:
4f x x
34f x x
8y x
78y x
power rule
We observe a pattern: 2x 23x 34x 45x 56x …
Derivatives of monomials
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d du
cu cdx dx
examples:
1n ndcx cnx
dx
Constant multiple rule:
5 4 47 7 5 35d
x x xdx
When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.
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(Each term is treated separately)
Sum and difference rules:
d du dv
u vdx dx dx
d du dv
u vdx dx dx
4 12y x x
34 12y x
4 22 2y x x
34 4dy
x xdx
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Product rule:
d dv du
uv u vdx dx dx
Notice that this is not just the product of two derivatives.
2 33 2 5d
x x xdx
5 3 32 5 6 15d
x x x xdx
5 32 11 15d
x x xdx
4 210 33 15x x
2 3x 26 5x 32 5x x 2x
4 2 2 4 26 5 18 15 4 10x x x x x
4 210 33 15x x
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Quotient rule:
2
du dvv u
d u dx dx
dx v v
3
2
2 5
3
d x x
dx x
2 2 3
22
3 6 5 2 5 2
3
x x x x x
x
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Derivatives of trigonometric functions
sin cosd
x xdx
cos sind
x xdx
2tan secd
x xdx
2cot cscd
x xdx
sec sec tand
x x xdx
csc csc cotd
x x xdx
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Derivatives of Exponential and Logarithmic Functions
Mt. Rushmore, South Dakota
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007
xx eedx
d aaa
dx
d xx ln
xx
dx
d 1ln
axx
dx
da
ln
1log
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dy dy du
dx du dx
Chain rule
If is the composite of and , then:
f g y f u u g x
at at xu g xf g f g
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Example
2sin 4y x
2 2cos 4 4d
y x xdx
2cos 4 2y x x
Differentiate the outside function...
…then the inside function
At 2, 4x y
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2cos 3d
xdx
2
cos 3d
xdx
2 cos 3 cos 3d
x xdx
2cos 3 sin 3 3d
x x xdx
2cos 3 sin 3 3x x
6cos 3 sin 3x x
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)
It looks like we need to use the chain rule again!
Example
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Higher Order Derivatives:
dyy
dx is the first derivative of y with respect to x.
2
2
dy d dy d yy
dx dx dx dx
is the second derivative.
(y double prime)
dyy
dx
is the third derivative.
4 dy y
dx is the fourth derivative.
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Extreme Values of Functions
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Absolute extreme values are either maximum or minimum points on a curve.
They are sometimes called global extremes.
They are also sometimes called absolute extrema.(Extrema is the plural of the Latin extremum.)
A local maximum is the maximum value within some open interval.
A local minimum is the minimum value within some open interval.
Global and Local extrema
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Local maximum
Local minimum
Notice that local extremes in the interior of the function
occur where is zero or is undefined.f f
Absolute maximum
(also local maximum)
Global and Local extrema
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Local Extreme Values:
If a function f has a local maximum value or a local
minimum value at an interior point c of its domain,
and if exists at c, then
0f c
f
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-2
-1
0
1
2
3
4
-2 -1 1 2
4 22 2y x x
2y
1y
34 4dy
x xdx
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-2
-1
0
1
2
3
4
-2 -1 1 2
4 22 2y x x
First derivative (slope) is zero at:
0, 1, 1x
34 4dy
x xdx
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Critical Point:
A point in the domain of a function f at which
or does not exist is a critical point of f .
0f f
Note:Maximum and minimum points in the interior of a differentiable function always occur at critical points,
BUTcritical points are not always maximum or minimum values.
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Critical points are not always extremes!
-2
-1
0
1
2
-2 -1 1 2
3y x
0f
(not an extreme)
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-2
-1
0
1
2
-2 -1 1 2
1/3y x
is undefined.f
(not an extreme)
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Find the absolute maximum and minimum values ofon the interval . 2/3f x x 2,3
2/3f x x
1
32
3f x x
3
2
3f x
x
There are no values of x that will makethe first derivative equal to zero.
The first derivative is undefined at x=0,so (0,0) is a critical point.
Because the function is defined over aclosed interval, we also must check theendpoints.
Finding absolute extrema
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0 0f
To determine if this critical point isactually a maximum or minimum, wetry points on either side, withoutpassing other critical points.
2/3f x x
1 1f 1 1f
Since 0<1, this must be at least a local minimum, and possibly a global minimum.
2,3D
At: 0x
At: 2x
2
32 2 1.5874f
At: 3x
2
33 3 2.08008f
Finding absolute extrema
Absoluteminimum:
Absolutemaximum:
0,0
3,2.08
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Finding Maxima and Minima Analytically:
1 Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points.
2 Find the value of the function at each critical point.
3 Find values or slopes for points between the critical points to determine if the critical points are maximums or minimums.
4 For closed intervals, check the end points as well.
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Using Derivatives for Curve Sketching
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First derivative:
y is positive Curve is rising.
y is negative Curve is falling.
y is zero Possible local maximum or minimum.
Second derivative:
y is positive Curve is concave up.
y is negative Curve is concave down.
y is zero Possible inflection point(where concavity changes).
Rules
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Example:Graph
23 23 4 1 2y x x x x
There are roots at and .1x 2x
23 6y x x
0y Set
20 3 6x x
20 2x x
0 2x x
0, 2x
First derivative test:
y
0 2
0 0
21 3 1 6 1 3y negative
2
1 3 1 6 1 9y positive
23 3 3 6 3 9y positive
Possible extreme at .0, 2x
We can use a chart to organize our thoughts.
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Example:Graph
23 23 4 1 2y x x x x
There are roots at and .1x 2x
23 6y x x
0y Set
20 3 6x x
20 2x x
0 2x x
0, 2x
First derivative test:
y
0 2
0 0
maximum at 0x
minimum at 2x
Possible extreme at .0, 2x
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Because the second derivative at
x = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum.
Example:Graph
23 23 4 1 2y x x x x
There are roots at and .1x 2x
23 6y x x Possible extreme at .0, 2x
Or you could use the second derivative test:
6 6y x
0 6 0 6 6y
2 6 2 6 6y Because the second derivative at
x = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum.
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inflection point at 1x
There is an inflection point at x = 1 because the second derivative changes from negative to positive.
Example:Graph
23 23 4 1 2y x x x x
6 6y x
We then look for inflection points by setting the second derivative equal to zero.
0 6 6x
6 6x
1 x
Possible inflection point at .1x
y
1
0
0 6 0 6 6y negative
2 6 2 6 6y positive
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43210-1-2
5
4
3
2
1
0
-1
43210-1-2
5
4
3
2
1
0
-1
Make a summary table:
x y y y
1 0 9 12 rising, concave down
0 4 0 6 local max
1 2 3 0 falling, inflection point
2 0 0 6 local min
3 4 9 12 rising, concave up
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Definite Integrals
Greg Kelly, Hanford High School, Richland, Washington
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When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.
0
1
2
3
1 2 3 4
211
8V t
subinterval
partition
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
Subintervals do not all have to be the same size.
Riemann sum
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0
1
2
3
1 2 3 4
211
8V t
subinterval
partition
If the partition is denoted by P,
then the length of the longest
subinterval is called the norm of Pand is denoted by .P
As gets smaller, the approximation for the area gets better.
P
0
1
Area limn
k kP
k
f c x
if P is a partition
of the interval ,a b
Riemann sum
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0
1
limn
k kP
k
f c x
is called the definite integral of
over .f ,a b
If we use subintervals of equal length, then the length of
a subinterval is: b ax
n
The definite integral is then given by:
Definite integrals
1
limn b
kan
k
f c x f x dx
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b
af x dx
IntegrationSymbol
lower limit of integration
upper limit of integration
integrand
variable of integration(dummy variable)
It is called a dummy variable because the answer does not depend on the variable chosen.
Definite integrals
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0
1
limn
k kP
k
f c x
b
af x dx
Area
Where F is a function :
F is called indefinite integral
)(xfdx
dF
)()( aFbF
Definite integral
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The Fundamental Theorem of Calculus
If f is continuous on , then the function ,a b
x
aF x f t dt
has a derivative at every point in , and ,a b
x
a
dF df t dt f x
dx dx
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0
1
2
3
4
1 2
Example2y x Find the area under the curve from
x=1 to x=2.
2
1
2dxxA
cx
F 3
3 The indefinite integral is defined up to a constant c
Proof:)(0
33 2
2
xfxx
dx
dF
3
7
3
1
3
8)1()2(
2
1
2 FFdxxA
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-1
0
1
Example
Find the area between the
x-axis and the curve
from to .
cosy x
0x 3
2x
2
3
2
3
2 2
02
cos cos x dx x dx
/ 2 3 / 2
0 / 2sin sinx x
3sin sin 0 sin sin
2 2 2
3
pos.
neg.
= =
= =
This because
cxF sin Proof: )(cos xfxdx
dF
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1.
0a
af x dx
If the upper and lower limits are equal, then the integral is zero.
2.
b a
a bf x dx f x dx
Reversing the limits changes the sign.
b b
a ak f x dx k f x dx 3. Constant multiples can be
moved outside.
b b b
a a af x g x dx f x dx g x dx 4.
Integrals can be added and subtracted.
5. b c c
a b af x dx f x dx f x dx
Intervals can be added(or subtracted.)
Rules for integrals
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The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.
Integration by Substitution
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Example 1:
5
2x dx Let 2u x
du dx5u du
61
6u C
6
2
6
xC
The variable of integration must match the variable in the expression.
Don’t forget to substitute the
value for u back into the problem!
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Example 2
21 2 x x dx The derivative of is .
21 x 2 x dx1
2 u du3
22
3u C
3
2 22
13
x C
2Let 1u x
2 du x dx
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Example 3:
4 1 x dx Let 4 1u x
4 du dx1
4du dx
1
21
4
u du3
22 1
3 4u C
3
21
6u C
3
21
4 16
x C
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Example 4
cos 7 5 x dx7 du dx
1
7du dx
1cos
7u du
1sin
7u C
1
sin 7 57
x C
Let 7 5u x
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Example 5
2 3sin x x dx3Let u x
23 du x dx21
3du x dx
We solve for because we can find it in the integrand.
2 x dx
1sin
3u du
1cos
3u C
31cos
3x C
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Example 6
4sin cos x x dx
Let sinu x
cos du x dx
4
sin cos x x dx
4 u du
51
5u C
51sin
5x C
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Integration By Parts
Start with the product rule:
d dv du
uv u vdx dx dx
d uv u dv v du
d uv v du u dv
u dv d uv v du
u dv d uv v du
u dv d uv v du
u dv uv v du
This is the Integration by Parts formula.
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Integration By Parts
Start with the product rule:
dxxvdx
xduxvxudx
dx
xdvxu )(
)()()(
)()(
dxxvdx
xdudxxvxu
dx
ddx
dx
xdvxu
xvdx
xduxvxu
dx
d
dx
xdvxu
dx
xdvxuxv
dx
xduxvxu
dx
d
)()(
)()()(
)(
)()(
)()()(
)(
)()()(
)()()(
u dv uv v du
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The Integration by Parts formula is a “product rule” for integration.
u can be always
differentiated
v is easy to
integrate.
dxxvdx
xduxvxudx
dx
xdvxu )(
)()()(
)()(
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Example 1
cos x x dx
Easy to integrate
u x
sinv x
dxxvdx
xduxvxudx
dx
xdvxu )(
)()()(
)()(
xdx
dvcos
1dx
du
Cxxx
dxxxxdxxx
cossin
sin1sincos
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Example 2
ln x dx lnu x
v x
1dx
dv
xdx
du 1
dxxvdx
xduxvxudx
dx
xdvxu )(
)()()(
)()(
Cxxx
dxxx
xxdxx
ln
1ln1ln
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Example 3
2u x
xv e
Ceexex
dxeexex
dxexex
dxex
xxx
xxx
xx
x
22
12
2
2
2
2
2
xdx
du2
xedx
dv
xu * xedx
dv
*
1*
dx
duxev *
Easy to integrate
Easy to integrate
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Taylor Series
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Suppose we wanted to find a fourth degree polynomial of the form:
2 3 4
0 1 2 3 4P x a a x a x a x a x
ln 1f x x at 0x that approximates the behavior of
If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.
0 0P f
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2 3 4
0 1 2 3 4P x a a x a x a x a x ln 1f x x
ln 1f x x
0 ln 1 0f
2 3 4
0 1 2 3 4P x a a x a x a x a x
00P a0 0a
1
1f x
x
1
0 11
f
2 3
1 2 3 42 3 4P x a a x a x a x
10P a 1 1a
2
1
1f x
x
1
0 11
f
2
2 3 42 6 12P x a a x a x
20 2P a 2
1
2a
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2 3 4
0 1 2 3 4P x a a x a x a x a x ln 1f x x
3
12
1f x
x
0 2f
3 46 24P x a a x
30 6P a 3
2
6a
4
4
16
1f x
x
40 6f
4
424P x a
4
40 24P a 4
6
24a
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2 3 4
0 1 2 3 4P x a a x a x a x a x ln 1f x x
2 3 41 2 60 1
2 6 24P x x x x x
2 3 4
02 3 4
x x xP x x ln 1f x x
-1
-0.5
0
0.5
1
-1 -0.5 0.5 1
-5
-4
-3
-2
-10
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
P x
f x
If we plot both functions, we see that near zero the functions match very well!
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Maclaurin Series:
(generated by f at )0x
2 30 0
0 0 2! 3!
f fP x f f x x x
If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:
Taylor Series:
(generated by f at )x a
2 3
2! 3!
f a f aP x f a f a x a x a x a