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!

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!