Definite Integrals / Area Under a Curve

Larson 7 Ch 4 Section 3 13-31 odds.

Anti-Derivatives, Indefinite Integrals, Differential Equations

Below are the problems we worked on today. They are in Larson 7.0 Chapter 4 Section 1. Please be work through all of the odd problems tonight. Thank you!

Solution videos for the Applications of Derivatives problem set are below (note that you do not need to complete 1991 AB1). These problems review all of the ideas we have been working with except for Rolle's Theorem and the Mean Value Theorem. I will post examples of those as well before the test. Study these!!! :)

I have fixed the sound issues on 1993 AB1 and 1992 AB1.

Applications of Derivatives Problem Set (printable)
Applications of Derivatives Problem Set Solutions (printable)

1996 AB1 Video #1
1996 AB1 Video #2
1996 AB1 Video #3

1994 AB1 Video #1
1994 AB1 Video #2
1994 AB1 Video #3

1993 AB4 Video #1
1993 AB4 Video #2
1993 AB4 Video #3

1993 AB1 Video #1
1993 AB1 Video #2

1992 AB1 Video #1
1992 AB1 Video #2

1991 AB5 Video #1
1991 AB5 Video #2
1991 AB5 Video #3

First, I have included the posters from the classroom that are relevant to our current topics. These posters cover all of the major concepts that we are working with on this upcoming test. In the next blog post, I will provide some examples that utilize these ideas.

Summary of Curve Sketching

Larson 7.0 Chapter 3 Section 6 #7-37 odds

Concavity / Points of Inflection

For Monday night Larson 7 Ch 3 Section 4 #11-25 odds
For Tuesday night Larson 7 Ch 3 Section 4 #27-39

Intervals of Increasing/Decreasing, Relative Min/Max

The first graphic shows the relationship between a function's behavior and it's first derivative and also the steps to identify the intervals of increasing/decreasing.



The second graphic states the First Derivative Test which tells us whether the function has a relative minimum, relative maximum or neither at each critical number.



The third graphic shows an example of finding intervals of increasing and decreasing in a function and makes use of the First Derivative Test to draw conclusions about relative minimums and maximums. Our notation is handwritten below. You can feel free to use any notation that works for you. In the end though, you must state specifically any conclusions you draw from the sign chart.



Please complete 11-35 odds for Monday. Thanks and enjoy your blizzard!

Mean Value Theorem

Remember the steps:
1) Verify that the function is continuous on the closed interval and differentiable on the open interval. If not, stop here. You wish you could be so lucky :)
2) Find the derivative
3) Find the slope of the secant line connecting the end points
4) Set 1 and 2 equal to eachother
5) Solve for x (which is c in this case)
6) Use only the values of x that lie WITHIN the interval (don't include end points here)

Complete 31-41 odds

Rolle's Theorem

Below is the definition of Rolle's Theorem.



Remember the steps:
1) Verify that the function is continuous on the closed interval and differentiable on the open interval. If not, stop here. You wish you were so lucky :)
2) Find the derivative
3) Set the derivative equal to 0
4) Solve for x (in this case it is c)
5) Use only the values of x that lie WITHIN the interval (don't include end points here)

Complete the odd problems for homework. Solutions are in the online book.

Extreme Value Theorem, Absolute Min/Max

I apologize for not posting this earlier. Please complete the odd problems listed. They are from Larson 7.0 Chapter 3 Section 1. Solutions are online in the digital book.

Remember: The Extreme Value Theorem states that a continuous function on a closed interval always has an absolute minimum and maximum value. Below is the book definition.



Remember: A Critical Number is an x-value where the derivative is zero or does not exist. Maximum and minimum values may occur at these points, but not always. The book definition is below.



Below are guidelines for finding Absolute Minima and Maxima.

Midterm

In addition to limits, you should know the following:

1. Implicit differentiation (has horizontal tangents when the numerator is zero, vertical tangents when the denominator is zero)
2. Find equations of tangent lines and normal lines (normal line is perpendicular to the tangent line at the point of tangency)
3. Derivatives of exponential and logarithmic functions
4. Don't forget the chain rule, the product rule, the quotient rule!
5. Remember how to show that something is differentiable - limit must exist, must be continuous and slopes from the left and right must agree.
6. A function has a horizontal tangent when its derivative is 0. Set the derivative equal to zero and solve for x.
7. Remember limits as x approaches infinity are the same as horizontal asymptotes.

Good luck!

Midterm Review

Below is a beginning of the midterm review. The first is on limits.

Limits Review

There are 8 videos that go with the limit review:

Limits overview (part 1)
Limits overview (part 2)
Limits overview (part 3)
Limits algebraically
Infinite limits (the behavior of a function around a vertical asymptote)
Limits at infinity (part 1)
Limits at infinity (part 2)
Limits of trig functions

More to follow!

Derivatives of Exponential, Logarithmic and Inverse Trigonometric Functions

Video: The derivative of e^x
Video: The derivative of ln(x) and a^x
Video: The derivative of log (base a)(x)
Video: The derivative of arcsin(x)
Video: The derivative of arccos(x), arctan(x), arccot(x)
Video: The derivative of arcsec(x) and arccsc(x)
Video: The derivative of the inverse of any function

Chapter 5 Assignments (printable solutions):

Ch 5 Sect 1 45-69 odds (solutions in the e-book)

Video: Ch5 Sect 1 45-49 odd
Video: Ch5 Sect 1 49(continued)-51 odd
Video: Ch5 Sect 1 53-57 odd
Video: Ch5 Sect 1 57(continued)-59 odd

Ch 5 Sect 4 39-61 odds (solutions in the e-book)

Video: Ch5 Sect 4 39-45 odd
Video: Ch5 Sect 4 47-51 odd
Video: Ch5 Sect 4 53-57 odd

Ch 5 Sect 5 41-59 odds (solutions in the e-book)

Video: Ch5 Sect 5 41-47 odd
Video: Ch5 Sect 5 49-55 odd

Ch 5 Sect 8 41-59 odds (solutions in the e-book)

Video: Ch5 Sect 8 41-43 odd
Video: Ch5 Sect 8 45-47 odd
Video: Ch5 Sect 8 49

Happy New Year! Implicitly and Explicitly

I have put together a simple sequence of videos that gives an introduction to implicit differentiation using a circle graph.

Into to Implicit Differentiation Pt. 1
Into to Implicit Differentiation Pt. 2
Into to Implicit Differentiation Pt. 3
Into to Implicit Differentiation Pt. 4

I will post video solutions to homework problems as well.

The homework for this unit is Larson Chapter 2 Section 5 #1-15, 21-27, 35-39, 41-43 odds only. There are worked out solutions in the online textbook, but I will try to put at least some videos out as well. We haven't worked with trig functions yet, so I will include them. Finally I will post a few actual AP problems like the one we did in class today.

Larson 7.0 Ch 2 Section 5 #1,3
Larson 7.0 Ch 2 Section 5 #5,7
Larson 7.0 Ch 2 Section 5 #9,11
Larson 7.0 Ch 2 Section 5 #13,15
Larson 7.0 Ch 2 Section 5 #21,23
Larson 7.0 Ch 2 Section 5 #25,27
Larson 7.0 Ch 2 Section 5 #35
Larson 7.0 Ch 2 Section 5 #37,39
Larson 7.0 Ch 2 Section 5 #41,43

Solutions in PDF form

Problem from this morning:

Video for Problem 1

A second similar one:

Video for Problem 2

And one more:

Video for Problem 3

Printable Solutions

Please study and work through all these examples. If you do, you own Implicit Differentiation!