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

 

Given the extrema and a POI, can we find the cubic equation (3.4 #61 page 196)?

 Given the extrema and a POI, can we find the cubic equation (3.4 #61 page 196)?

Using a Table of Slopes to Sketch a Graph(3.3 #73)

 Using a Table of Slopes to Sketch a Graph(3.3 #73)

Using what we know about the first derivative we can sketch a graph from a table of slopes.  I also show how to put the table into your calculator to see a graph of f'.

Graphing the Derivative from the graph of a f (3.3 # 59, p.188)

 Graphing the Derivative from the graph of a f (3.3 # 59, p.188)

The first derivative to a piecewise function (3.3 p.187 #39)

 The first derivative to a piecewise function (3.3 p.187 #39)

First Derivative, Increasing, Decreasing, and extrema (3.3 (p187) #11, 15, 17, 27, 35

 First Derivative, Increasing, Decreasing, and extrema (3.3 (p187) #11, 15, 17, 27, 35

The first derivative can find relative extremes when it changes from negative to positive (min) or positive to negative (max).  Here are 5 quick examples:

#11 -  0:00

#15 - 2:11

#17 - 3:28 

#27 - 6:00

#35 - 8:08

Another MVT example (3.2 #41)

Another MVT example (3.2 #41)

Another example of using the MVT and finding the place where the instantaneous slope matches the average (mean) slope.

Mean Value Theorem with a Calculator (3-2 #37)

 Mean Value Theorem with a Calculator (3-2 #37)

A basic polynomial to show what the MVT says and how to check your work with a TI-84 style calculator.

A Thinking Question About Slope Over Time (2.6 # 34)

 A Thinking Question About Slope Over Time (2.6 # 34)

Here we reflect on the how a positive derivative means something is increasing, and if it is negative, something is decreasing. We can then see an overall trend that is negative is indicating an inverse relationship between the variables (one goes up and the other goes down), and a positive trend in slope has a direct relationship (they go up and down together).  Looking at this should also get your brain in thinking about parametric relationships where both the x and y are are functions of time.  

Related Rates: Filling a Pool (2.6 # 19)

 Related Rates: Filling a Pool (2.6 # 19)

I encourage beginners to organize related rates word problems with these 5 steps:

1: Find what? 

2: Given what?

3: What is the connection between these?

4: Implicitly differentiate something.

5: Plug in what you know, solve for what you what to find (remember the units!)

Related Rates with The Volume of a Cone (2.6 #17)

 Related Rates with The Volume of a Cone (2.6 #17)

A related rates word problem has a lot of pieces, so try organizing your strategy:

1) Find what?

2) Given what?

3) Connect the given with what you need to find with an equation

4) Differentiate something

5) Substitute what you know, solve for what you what to find.

Related Rates (2.5 #5)

 Related Rates (2.5 #5)

We differentiate both sides of the equation with respect to time.  (After all, a "rate" is something over time)

TI-84 Tips to Show Curves are Orthogonal To Each Other (2.5 #63)

 TI-84 Tips to Show Curves are Orthogonal To Each Other (2.5 #63)

Here we use a TI-84 to show how two curves are orthogonal to each other.  If they are, then the tangent lines's slope at the points of intersection are perpendicular, and be negative reciprocals of each other:

If the slope is a/b, then the perpendicular slope is -b/a.  The calculator has a (CALC)5. intersect feature and a (CALC)6. dy/dx feature, so this is how we can tell if the curves are orthogonal.  

Tangents to a Circle (2.5 #57)

 Tangents to a Circle (2.5 #57)

While this is easily done with basic algebra, let's confirm our result with implicit differentiation!

A Second Derivative using Implicit Differentiation (2.5 #49)

 A Second Derivative using Implicit Differentiation (2.5 #49)

The second derivative is just the derivative of the first derivative.  When we differentiate implicitly two times, we can sometimes take advantage of substitution of a previous result to simplify our result with ease!

Basic Related Rates Example Quiz Solutions

 Basic Related Rates Example Quiz Solutions

These examples use the formulas of the area and alums of circles, spheres, and cones.  These techniques are applicable to cubes, cylinders, rectangles, triangles, prisms and pyramids using their respective formulas.  I encourage beginners to organize the given information by these 5 steps:

1: Find what?

2: Given what?

3:What is the connection between these?

4:Implicity differentiate something

5:Plug in what you know, solve for what you don't know

Here is a pdf of these 5 problems and their solutions

Implicit Differentiation (2.5 # 35, 37, 39)

 Implicit Differentiation (2.5 # 35, 37, 39)

We don't need to "solve for y" before we take a derivative.  This allows us to handle relationships that are not functions like these three examples.  The only thing to be mindful of it to use the chain rule! 

Do a derivative of y^2 is not merely 2y, it it 2y*(dy/dx) or if you like 2y*y'.

Finding Derivatives Based on a few facts (2.4 #98)

 Finding Derivatives Based on a few facts

Using a table of values to find a derivatives at a particular value of x. 

 

More Chain Rule Practice (2.4 #63)

 More Chain Rule Practice (2.4 #63)

The Chain rule to rescue, once again!

The Chain Rule in Action and Confirmed with our TI-84

 We attack a sec^3 function with the Chain Rule and confirm our slope with a TI-84

 We attack a sec^3 function with the Chain Rule and confirm our slope with a TI-84. The show the derivative of sec x is tan x * sec x.  Next we use the chain rule to twice to find the slope at x=0.

Tangent Line to A Circle with the Chain Rule (2.4 #79)

 Tangent Line to A Circle with the Chain Rule (2.4 #79)

He we find the slope of a semi-circle using the chain rule, and check our tangent line equation with a TI-84 style calculator.

Higher Order Derivatives (2.3 #93, 103, 115)

 Higher Order Derivatives (2.3 #93, 103, 115)

The rate of change of the rate of change is the derivative of the derivative, aka the "Second derivative".  We we write it " f''(x)" or sometimes "d^2x/dy^2".  But there is reason to stop there.  The derivative of the second derivative is the third derivative, and so on.  We will even find an eighth derivative in this video!

Rate of change of Area per second (2.3 #83)

 Rate of change of Area per second (2.3 #83)

Here we look at the rate of change of area per second which requires us to make a function (the Area of a rectangle) and compute the rate of change of that function.  This gives us a glimpse of what Calculus "word problems" look like!

Can Different Functions Have the Same Derivative? (2.3 # 79)

 Can Different Functions Have the Same Derivative? (2.3 # 79)

Here we investigate a claim that two function have the same derivative.  We use the quotient rule to confirm it, then looking at the graph, we see why these functions might actually have the same slope.

Quotient Rule to Find a Tangent Line with a Certain Slope (2.3 #77)

 Quotient Rule to Find a Tangent Line with a Certain Slope (2.3 #77)

After using the quotient rule to find a formula for slope, we set it to a certain slope to find when the function has that slope.  To get the equation of the line, we use the original function to find the y coordinate, and place it in the "Point-Slope" form of a linear equation: y-Y = m (x-X) where the point is (X,Y) and m is the slope. 

Finally we use the calculator to check our work both graphically as well as numerically.

Check your Derivative at a Point with a Calculator (2.3 # 59)

Check your Derivative at a Point with a Calculator (2.3 # 59)

After finding a derivative at a point with the quotient rule, we check our work with a TI-84 style calculator. 

Thinking Questions with Derivatives (2.2 #67,79)

 Thinking Questions with Derivatives (2.2 #67,79)

With the Power (or exponent)  Rule, it's easy to handle some questions that have to do with slope! We also use the TI-84 to confirm our result.

More Quotient and Product Rule (2.3 #35, 51, 55)

 More Quotient and Product Rule (2.3 #35, 51, 55)

Here the calculus is easy, but sometimes you have to use algebra to match your answer with the answer in the back of the book.  A great skill to have so you can select a multiple choice item that expresses the answer in a slightly different form than the way you have it.... 

Can a piece-wise function be differentiable? (2.1 #89)

 

Can a piece-wise function be differentiable? If it is continuous, and the slopes from the left and the right are the same, yes!