Antiderivatives and Indefinite IntegralsThe function F(x) is the antiderivative of f(x) if: Notation:

How many antiderivatives of are there? Why?

Verify the antiderivative:

What is the power rule for an antiderivative?

Find: Finding a particular solution:

if F(1) = 3

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Whiteboard warm up - Anti Derivatives:

Find the particular solution to if F(2) = 5

Test Yourself: Find the antiderivative

x2−2x+5− 2x sin x−e x

√ x ( x+1 )x ( x−5 )3√x

Definite Integrals

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Estimate the area under the curve on the interval [–2,1], using 3 rectangles:

Upper limit: Lower Limit:

Theorem 4.3: Limit of Upper and Lower Sums

https://www.desmos.com/calculator/tgyr42ezjqΔx=

Right: x i=a+iΔx

Left: x i=a+(i−1 ) Δx

Using limits, find the area under the curve on the interval [–2,1]

A=∑i=1

n

[ f ( xi )] Δx=∑i=1

n

[2(−2+i 3n )]( 3n )=∑i=1

n

[2(−6n+ 9 in2 )]=∑i=1

n

(−12n +18 in2 )=−12

n ∑i=1

n

1+18n2

∑i=1

n

i

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Find the definite integral using geometry

Properties of definite integrals:

Evaluate:

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Sometimes the symmetry of a graph can also help:

Whiteboards: 1997 – 24 1998 – 2, 20 2003 – 77 2008 – 9, 79 2012 – 8, 13, 86Test yourself: 2013 – 21, 76, 78 2014 – 83, 88 2015 – 84 2016 – 86

The Fundamental Theorem of Calculus

Evaluate:

Suppose I ran ½ mile to warm up and the graph above is of my treadmill screen when I’m done.How far did I run during my entire workout?

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MPH in 5 minute intervals0

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Other import interpretations of the fundamental theorem:

f (b )= f (a )+∫abf ' ( x )dx

As a sentence, it means…

Suppose the graph of is shown below, and :4

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-2

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-5 5 10

Let g ( x )=∫0

xf ' (t )dt

, find:g(4), g(–4), g’(1), g’’(1) f(–3)

Where are the minimum and maximum value(s) of f located and what are their values?

Although

ddx [∫ f ( x )dx]=∫ [ ddx f ( x )]dx=f ( x )

you must be more careful how you interpret the relationship with definite integrals:

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ddx [∫−5

xcos (t )dt ]= d

dx [∫3√x 1t dt ]=Mean value theorem for integrals: Average value of a function on an integral:

Shade in ∫062 xdx

. Can you find the average value without using the theorem?

Verify your answer using the theorem:

Find the average value of the function,

Multiple Choice Practice1997 - 1,3,9,20,78,87,88,90 1998 - 3,4,5,7,9,11,15,82,88,92 2003 – 5,22,23,82,83,84,88,91,922008 – 2,7,17,81,85,87,91,92 2012 – 3,6,15,17,26,79,81,83,89 2013 – 10,11,13,20, 26,27,79,80,812014 – 1,11,14,26,76,77,81,84,85,87 2015 – 6,12,23,26,78,80,87,90,91,92 2016 – 10,15,17,19,20,77,79,80,84,89

Written practice:2001 #3,5 2002 #4,6 2003 #4,6 2003B #5 2004 #5 2004B #4 2005 #4,52005 #4,5 2005B #4 2006 3 2006B #4 2007 3 2007 #3 2007B #42008 #3,4 2008B #4,5 2009 #6 2009B #3 2010 #1,3,5 2010B #2,4,6 2011 #1,4,62011B #1,2,4,6 2012 #3,6 2013 #1,2,4 2014 #1,3,5 2015 #1,5 2016 #1,2,3,5 2017 #2,3,5Assignment 4.4 – 2 blocks complete 10MC and 2WR each block. HW – 10 additional MC and 2 WR

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Integration by SubstitutionRecognizing the chain rule with antidifferentiation:

There a special technique we can use to help recognize this pattern more easily called integration by substitution:

Whiteboards:

Test yourself:

∫sin2 xcos xdx

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Sometimes the only difference between what we see and what we need is to multiply by a constant,

∫2 x (1−x2)2 dx

Whiteboards:

∫12 x2√2 x3+1dx

Test Yourself:

∫−3x sin x2 dx

Find the antiderivative of the given integral

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How would you find ? *hint:

Integrals of the 6 basic trigonometric functions:

Substitution to simplify the integrand:Ex.

Whiteboards:We may have to make several substitutions to find our antiderivative,

∫ 2x+1

√ x+4dx

Substitution for definite integrals:

∫15 4√2 x−1

dx

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Whiteboards:

1997 – 6,18 2003 – 2,8,11 2008 – 4,15 2012 – 12,90 2013 – 3,6,91 2014 – 4,8 Test yourself:2015 – 1,3 2016 – 2,8

Written Practice2002 – 3 2003 B – 4 2004 – 1 2004B – 6 2005 – 2 2009 – 1,2,3 2012 – 4 2015 – 3 2016 – 6

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Numerical Integration

Find the antiderivative of:So far the examples we’ve used have all been carefully chosen to have a nice antiderivative. In practice there are lots of functions whose antiderivative is not so simple.Riemann Sums:

Let’s consider Approximate the definite integral using the left and right point rule with 4 equally spaced intervals

Whiteboards:Approximate the definite integral using the midpoint point rule with 4 equally spaced intervals

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Area of a trapezoid

Use the trapezoid rule to approximate, for n = 4 (use calculator)

Whiteboards:

f is a continuous function for which and on the interval [3,10]. Use the values in the table below

to estimate the value of with the trapezoid, left, and right point rules. Are the values an over/under estimate? Justify your answer.

x 3 4 6 9 10f(x) 1 5 8 10 11

Multiple Choice1997 - #89 1998 - #85 2003 - #85 2008 - #10 2013 - #83 2014 - #12 2015 - #4 2016 #5

Written2001 - #2 2002 - #2 2002B - #2,3,4 2003 - #2,3 2003B - #2 2004 - #3 2004B - #2,32005 - #3 2005B - #2,3 2006 - #2,4 2006B - #2,6 2007 - #2,5 2007B - #2 2008 - #22008B - #2,3 2009 - #5 2009B - # 1,2,6 2010 - #2 2010B - #3 2011 - #2 2011B - #5

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2012 - #1 2013 - #3 2014 - #4Test Yourself:

Suppose a function has the following properties on the interval [1,5]: f ' ( x )>0 and f '' ( x )>0Form an inequality comparing an estimate made by the midpoint (M), trapezoid (T), right (R), and left (L) with

A=∫15f (x )dx

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