Two Common Tangent Lines to Two Functions! (2.1 #77)

Finding a slope of a tangent line on one function, but can we find a common tangent line to two functions?  How about 2 Common tangent lines? It takes some algebra together with our calculus, but it can be done!

When is a function not differentiable?(2.1 #80)

 When is a function not differentiable? (p 109 (2.1) number 80

 A function may not have a derivative.  After all slope is rise over run, and if it is a vertical line, it will have no slope.  Here we have an example that has no slope at x=0, since the slopes from the left and the right do not agree (so the limit does not exist, so neither does the slope!

Integration by Parts: Tablular style (8.2 #17)

 

Integration by Parts: Tablular style (8.2 #17)

An example of a case where integration by parts has more then one step, so I demonstrate the tabular method of integration by parts.

Welcome back to School 2020

 Welcome back to school - remote style!

Intro to Derivatives (Ch 2.1)

 

Introducing the idea that a slope of a curve depends on the value of x... so we call the slope formula a derivative... The limit of the slope of a secant line is how we get this major topic started!

Some Ch P and Chapter 1 Review Questions (P2. 69, #79) (P.revw #39) (1.3# 79)(1.4 #84, 93)

 A few problems are explained, including tangent lines, and the Intermediate Value Theorem (IVT).

The arc length of a predator chasing prey (7.4 #65)

 Using the arc length integral we can show that a predator can travel twice the distance in the same amount of time!

Derivatives of Common Functions Review (Chapter 8 Basics)

 

A review of the derivatives of the basic functions, with a derivation of a few of them.

Answer it yourself before you watch.

One Sided, "Type 2" Limits (1.5 #37,49)

 

Type 2 limits are the type that when you substitute the number that x is going towards, you get a fraction with a number in the numerator and a 0 in the denominator (->L/0).  The division by zero is associated with a vertical asymptote, so the limit will go to either positive infinity or negative infinity.

Continuity and the IVT (1.4, #63, 83, 85)

 

Using the requirements of Continuity, we can find a way to make a piece wise function continuous.  We also look at using the Intermediate Value Theorem (IVT) to prove the existence of a zero of a function on a certain closed interval.

Arc Length (7.4 #11)

An exact length of arc can be computed with the arc length formula, together with a bit of algebra, so the integrand is a product of factors where one factor is the derivative of the other so we can use "U-substition"

Exact Arc Length (7.4 # 19)

 

An exact length of arc can be computed with the arc length formula, together with a bit of factoring, recalling a^2+2ab+b^2 = (a + b)^2

Surface Area of a Function Rotated around an Axis

 

Knowing how to use calculus to find arc length helps us find the surface area a function rotated around an axis.  I use u-substitution to find the surface area of y=x^2 two different ways.

Continuity (1.4 #51, #53, and 1.5 #31)

Use limits to test Continuity! We check if the function exits, the limit from left and right exist, and they are all match! 

Continuity and Limits (Section 1.4; 9, 13, 18, 20, 25)

 A function can be continuous at a point, and interval or everywhere.  We can tell if three things are required!

Using a Theorem to find a limit (1.3 #74)

 We can use Theorems together with algebra tricks to find limits that have the indeterminate 0/0 form.

The conjugate comes to the rescue again! (1.3 #75)

To find the limit, we turn to the conjugate once more!

Limits that go to 0/0 (1.3 #53)

Don't fear the "∆x" it works like a single variable, and refers to the change in x.

More Algebra Tricks: The Conjugate (1.3 #52)

 Her we take advantage of (a+b)(a-b) = a² - b ².

to resolve an indeterminate form.

Using Algebra Tricks to Find a Limit (1.3 #51)

 If you substitute 4 for x, we get a 0/0 situation we call an indeterminate form which can resolved with some algebra tricks.