Cost, Revenue, Profit
For a little later in the unit: Marginal Revenue, Average Cost, Profit, Price & Demand Function
Compound interest, present and future value
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.
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 for F for future value. For F I used P(t) to show the it is a function of time, and Po for present value.
This gives a good overview and some examples of limit. Patrick is among my favorite Internet teachers for simplicity and clarity of message. Further techniques and examples:
Infinite limits in which a function goes to positive infinity or negative infinity as x approaches a.
Optional, good insight, might actually help you better understand the actual limits we have done:
Derivative (slope of a tangent line: the instantaneous rate of change of a function at point)
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 line 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.
This is an essential video: Difference quotient (DQ) and the definition of derivative
Here is the process: Finding derivative with DQ, Ex 1
Here are examples: Using derivative to find eqn of a tangent line, Ex 1 , Ex 2 , Ex 3 and Ex 4
Now here are the same idea using shortcuts:
A practical application from the laws of physics (motion): Relationship between displacement, velocity and acceleration
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).
The Derivative Rules and examples
Recall Shortcuts to the derivative
The following are short, understandable proofs of the derivative rules:
Now examples of all. But first a note: It is handy to use u, v notation for memorizing the above rules. For example, let f(x) = u(x)v(x), then (shortening further) 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:
Decent videos of chain rule that don't involve trig are hard to come by. Skip over the trig examples in the following (though, quite simply, sin x and cos x are derivatives of one another except for a negative coefficient).
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.
This proof requires a result from a proof that uses a trig graph (i.e., not in the 'scope' of this course, but not hard either.
Implicit differentiation method often lets us find rates 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.
(Go to 3:30 for example that involves non-trig equation)
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 when we encounter a situation in which dimension 'a' changes with respect to another dimension 'b', and 'b' is changing with respect to time 't.' Then it's clear 'a' changes with respect to 't' through 'b'.
Here is a video Process behind related rates
For example, the area of a circle is a function of its radius; if radius changes with time (i.e., radius is a function of time), then the rate at which the circle's area changes is also a function of time. We end up differentiating area (an explicit function of radius) implicitly with respect to time. The videos demonstrate this.
And finally, a video on related rates that applies to a business application:
Critical numbers, first and second derivative tests
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.
Curve sketching with calculus
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.
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).
Here's a similar one.
Finally, one that requires us to come up with the price function (called the demand function in this video), which we will use in the revenue function:
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).
Lagrange multipliers for solving problems in optimization with a constraint
This is one of the best mini-lectures I have seen.
The lecturer mentions the curves that have to be drawn 'many times in the course.' We won't have to draw many curves, except as we discuss the meaning.
Antiderivatives (indefinite integrals)
Applications of the definite integral
It's not the clearest video in terms of production quality, but it's really the best I've seen in terms of a spare explanation. And, the lecturer does both present and future in one video.
NOTE: Most video lecturers do not bring out the constant e^rT, as we do. But it is easier to deal with the constant term on the outside when you integrate e^(-rt) rather than in the integrand as e^(rT - rt).
Notation equivalence between our text and this video: Our time T = video M. Our f(t) = video S(t).
The easy way to remember which formula gets the multiplier is to note that, since, PV < FV, present value lacks the multiplier that future value has. No need to overthink that one.
What is the usefulness of Present Value? It's a tool for comparing what you would make if you were to hold on to your business vs if you sell it right now and invest the money you are given.