Is the Alternating Series Remainder Theorem Not Working?

 

This appears to be a counter example that would disprove the Alternating Series Remainder Theorem.... 

A u-Substitution Using the Same U Formula Two Ways (4.5 p. 306 #57)

 A u-Substitution Using the Same U Formula Two Ways (4.5 p. 306 #57)

Using the TI-84 to Explore Some Accumulator Functions (4-5 p 307 #87)

 Using the TI-84 to Explore Some Accumulator Functions (4-5 p 307 #87)

The Calculator can compute the area between the x-axis and the blue function using brute force.

We notice some relationships between them and spot points of inflection and extrema. 

A Definite Integral with u-Substitution (4-5 p 305 #61)

A Definite Integral with u-Substitution (4-5 p 305 #61).  

This one has a surprising result!

Some Complicated Integrals That Are Easier than they Appear! (4-5 p 307 #81)

 4-5 p 307 #81  Some Complicated Integrals That Are Easier than they Appear!

4-5 p 307 #81  Some Complicated Integrals That Are Easier than they Appear!  U-substitution to the rescue!

Using the Chain Rule with an Accumulator Function (4-4 p294 #83, 85)

Using the Chain Rule with an Accumulator Function (4-4 p294 #83, 85)

 If the derivative of the upper limit has a derivative more complicated than 1, then your better use the chain rule!

An Accumulaotr function without a constant 4-4 (p. 294 #81)

 

An Accumulaotr function without a constant 4-4 (p. 294 #81)

Without one of the limits being a constant, you can't use the 2nd FTC... what to do?  Invent one!

An Accumulation function (4.4 p. 293# 67)

An Accumulation function (4.4 p. 293# 67)

 An Accumulation function is a function that is a function of the accumulation of area under a curve.  A nice feature is that the derivative is based on the integrand itself

Estimating a Definite Integral (4.3 p278 #53)

 

Estimating a Definite Integral (4.3 p278 #53)

Estimating a definite integral with the area of rectangles. 

Integrals as Area Above and Below the x-axis (4.3 p. 278 #47)

 Integrals as Area Above and Below the x-axis (4.3 p. 278 #47)

Area can be negative now.... either by being below the x-axis, or by going backwards with area above the x-axis.  

Properties of Integrals (4.3 p 279 #49)

 Properties of Integrals (4.3 p 279 #49)

Here we demonstrate breaking apart integrals, changing the limits and capitalizing the properties of odd and even functions.

Review of Sigma Sums (4.2 p 267 #9, 22)

 Review of Sigma Sums (4.2 p 267 #9, 22)

We explain some useful sum formulas, work some examples, then show how to use the TI-84 to confirm our result.

A Particle Moves along the x-Axis (4.1 p. 256 #65)

 A Particle Moves along the x-Axis (4.1 p. 256 #65)

We often have this one-dimensional particle that moves along the x-axis as an AP exam question, and a number of things can be surmised fromthe first and second derivatives....

How High Will It Go (in Europe) (4.1 p. 256) # 60

 

How High Will It Go (in Europe) (4.1 p. 256) # 60

This time we find the maximum height of a ball thrown into the air in meters.  All we need to know is the velocity and height when it is thrown! In this one we use the constant acceleration of 9.8 meter per second per second.

How High will It Go? (4.1 (p 256 #57)

 How High will It Go? (4.1 (p 256 #57)

We find the maximum height of a ball thrown into the air.  All we need to know is the velocity and height when it is thrown! In this one we use the constant acceleration of 32 feet per second per second.

A Particular Antiderivative (4.1 (p.255) #44

 A Particular Antiderivative (4.1 (p.255) #44

Antiderivatives usually have a " + C" but given some particular points, we can find a particular solution.

Some Basic Antiderivatives (4.1 p. 255 27, 31, 33)

 Some Basic Antiderivatives (4.1 p. 255 27, 31, 33)

Some antiderivatives for beginners. Don't forget there is usually a nice table of anti derivatives on the inside cover of your textbook.  These are General solutions we add a C (for an arbitrary constant) to describe all the functions that have the integrand as a derivative.

Best Length Around a Corner (3.R (p 244) #85)

 Best Length Around a Corner (3.R (p 244) #85)

We find the best length of pipe to get around a corner where one hallway is narrower than the other.  

Optimal Area of 2 Shapes with a Perimeter of 10 (3.7 page 226 #35)

 Optimal Area of 2 Shapes with a Perimeter of 10 (3.7 page 226 #35) 

Another word problem where you use the relationship of perimeters to make a Area function base upon only one variable x.  Once we have the derivative at zero we are at an optimal place.

Minimizing Length (3.7 (p 225) #23)

Minimizing Length (3.7 (p 225) #23)

Here we use a calculator to find the minimum length of a line that goes through a point.  This is actually a great method for figuring out the length of something going around a corner, like we will demonstrate in a later video