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Exponential growth

Doubling something (after 1 year, say) is what we call 100% growth. Interest is generally paid at a much smaller amount. This video begins first with 100% growth, so the growth of the ball is easy to draw. This way the lecturer can get you to “e”. He then goes on to show a more reasonable growth rate r.


Compound interest, present and future value problems

This is a good presentation of the several types of problems seen in finding present and future value when interest of an investment or loan is compounded. The lecturer uses the variable A where we use F for future value.


Basic ideas of limits

Finding limits from a graph

These give a good overview and some examples of limit. Patrick is one of my favorite internet teachers.

Most important idea: A function f(x) has a limit at x = a only if its left-hand limit is equal to its right-hand limit as x approachesa.


Continuity and limits made easy

Discontinuities in a function (piecewise)

More inspecting for discontinuities of a piecewise function

Patrick discusses limits and the relationship to continuity. Watch for the important ideas, as we will discuss at length.

Most important idea: A function f(x) is continuous at x = a if f(a) exists and if limit of f(x) as x approaches a is equal to f(a).


Difference quotient (DQ): Definition of derivative

Finding derivative with the DQ, Example 1

Finding derivative with the DQ, Example 2

Discovering the instantaneous rate of change of a function at a point (as opposed to an average rate of change between two points) is the work of the derivative function.

The numerical slope of a tangent lanine at some point of a function is derived from the function itself by means of the difference quotient. The function that describes the behavior of the slope of the tangent line at any point along the graph of a function is called the “derivative function” (or simply, the “derivative”).

This function we will soon see is the marginal cost function.

We evaluate a derivative function at a given x (say, x = a), we find the instantaneous rate of change of the function at that point.


Finding equation of tangent line to a curve at given point using DQ by Prof Poyser

Finding equation of tangent line to a curve at given point using DQ by Patrick

Both videos are good, giving the method you need to demonstrate for this unit. You may not use the shortcut formula for derivative to find the slope yet. You have to use the difference quotient.

Summary: We're using point-slope form of the line, where the slope is found via the DQ:

y - y1 = (DQ at x1)(x - x1)


Proof of product rule

Proof of quotient rule

Proof of chain rule

These are short and understandable proofs of the derivative rules.

Product rule: (fg)' = f'g + g'f

Quotient rule: (f/g)' = [f'g - g'f]/ g squared

Chain rule: [f(g(x))]' = f'(g(x))g'(x)

By the way, it is handy to use u, v notation, for memorizing each of the above rules. For example, to express the product of two functions using u and v rather than f and g, we begin by letting f(x) = u(x)v(x), then further shorten the notation to f' = u'v + v'u.

There are plenty of videos with worked examples of product and quotient rules. The second example in each of Patrick's shows trig function derivatives, which you are NOT responsible for. It is hard to find examples that don't have trig, but the idea is the same:

Product rule examples

Quotient rule examples


While you are not responsible for the proofs, only the rules, you ought to see where the formulas come from. They rely on d/dx notation, implicit methods, and log properties, as well as limit process. None of these are difficult. They are short and very clear.

Proof of derivative of exponential function (base e)

Proof of derivative of natural log function

Proof of derivative of exponential function (base a)

Relationship between displacement, velocity and acceleration


Implicit differentiation method often lets us find rtes of change of one variable with respect to another even if there is no explicit function present. For example, the circle isn't a function, but its tangents are of interest to us, especially where they fail to exist. Differentiating the equation without solving for either of its branches (top or bottom semicircle) is handy using ID.

Implicit differentiation ID


Real-world related rate problems are best introduced with problems of basic geometry. Clearly, physical parameters of plane figures (rectangles, triangles, circles) and solid figures (boxes, cylinders, spheres) are related by formulas. As such, if any one of the parameters changes, then so do the others. More to the point, the rate of change of one parameter affects the rate of change of another.

Implicit differentiation comes into play because if we say (for example) that area of a circle is function of radius, and the rate at which the radius changes is a function of time, then the rate at which the area of the circle changes is also a function of time. We end up differentiating area (an explicit function of radius) implicitly with respect to time using a given rate of change of the radius.

Related rates 1: Area of circle and changing radius rate

Related rates 2: Area of triangle and changing side length rate

Related rates 3: Ladder sliding down the wall problem

And finally, a video on related rates that applies to a business application:

Cost and profit with respect to time


Finding critical numbers is the preliminary step to using first and second derivative tests, used to examine where functions are increasing, decreasing, topping or bottoming out, (local and global extremes), and onto curve sketching.

Finding critical numbers of a fcn

Critical numbers and First and Second Derivative Tests

First derivative test

Second derivative test and concavity


Curve sketching with calculus

Limits at infinity of rational functions (including "tricks" at 6:16)

Graphing a polynomial

Graphing a rational function


Optimization

Optimization problem (instructive)

The fence problem

More optimization: Wherein n = number of price reductions or increases. Notice I prefer n to the book's x for this variable, as we are used to x being quantity of sales.

Computer software sales

There are two examples done on this video. The first should be enough, and I like that it gives a graph as I did for the lamp sales problem today. The whole idea of graphing the upside down parabola and asking what its max and its y-intercept indicate is very important. (By the way, on the video's first problem, the y-intercept shown is incorrect, as it is definitely not the case that n = 0 price reductions would mean $0 revenue; but more about that in class).

The hot dog problem

Finally, the frequently promised hot dog problem. You decide on the merits!


Partial differentiation

Partial derivatives

Second order partial derivatives


Local max and min of a function of two variables f(x,y)

In three-space, we find critical points essentially same as we do in two-space, but to determine if they are max, min, or saddle points, we have no first derivative test. Rather, we go straight to a second derivative test after finding critical values.

As you watch this be aware of the following:

1. Calculation of partial derivatives

2. Solving for critical numbers, which generally entails solving a simply linear system, but sometimes a non-linear one, and even discarding some values (Theorem 28.1).

3. Employing the SDT for f(x,y) (Theorem 28.2).

Critical points and second derivative test for local max and min of multi-variable function f(x,y)


Lagrange multipliers for solving problems in optimization with a constraint

Optimization via Lagrange multiplier


Antiderivatives (indefinite integrals)

Antiderivatives and indefinite integration

Examples of basic indefinite integration

Antiderivative with initial conditions (finding a particular F(x))

u-substitution

Integration by parts


Definite integrals

Definition of definite integral

Examples of finding definite integrals


Improper integrals


Imroper integrals basic idea

Improper integrals continues


Applications of the definite integral

Finding area between two curves

Average value of a function

Position, velocity and acceleration

Present and future value of a continuous income stream

people/mckenzie/videos.txt · Last modified: 2016/12/02 15:09 by mckenzie