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!