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. First, an easy one:
This last one 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.
Finding limits from a graph (This is one of his rougher videos, but well explained.)
More techniques for evaluating limits, Ex 3 (gives a little jump on continuity)
IMPORTANT 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:
Main 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).
First, check these helpful videos on graphing piecewise functions by Patrick, if you need them:
Patrick discusses limits and the relationship to continuity. Watch for the important ideas, as we will discuss at length.
Instantaneous rate of change of a function at point via the difference quotient (DQ)
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 for finding equation l of line tangent to f(x) at a given point:
A practical application from the laws of physics (motion): Relationship between displacement, velocity and acceleration
The Derivative Rules and examples
Recall Shortcuts to the derivative
The following are short, understandable proofs of the derivative rules:
Proof of chain rule The lecturer uses Leibniz notation dy/dx. There's a little fudging where on u(x) and g(x) being the same, but he says so.
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, as one parameter changes, then so do the other parameters. 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 both 'a' and 'b' change with respect to time 't.' 'a' changes with respect to 't' through 'b'.
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.
Here's a nice video by Krista, who reviews implicit differentiation before doing a related rate problem. You can't go wrong looking her up when you seek video explanation. Like Patrick, she is clear, no frills, and her conveyance of mathematics methods is great!
And finally, a video on related rates that applies to a business application:
Finding critical numbers is the preliminary step to using first and second derivative tests, used to examine where functions are attaining their local extremes, and the intervals on which they are increasing, decreasing. We use these skills to sketch the curves of functions and so examine what kind of behavior a function models.
A critical number f(x) is that value of x (call it 'c') in the domain of f where EITHER f'(x) = 0 OR where f'(x) does not exist.
There are many more videos on the HW page for this topic.
First and second derivative tests
Curve sketching with calculus
Optimization problems dealing with geometry:
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 where we have to come up with the price function (called the “demand function” here), to create the revenue function:
Multivariable functions up to 9:45
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)
Fundamental Theorem of Calculus (FTC) and Definite Integrals
The area under the curve f(x) represents the accumulation of that function on a stated interval.
And a more rigorous version:
Here are examples:
Riemann Sum and the Definite Integral
Further understanding of this construct of area under the curve and the definite integral is found in the notion of the Riemann sum:
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.