Cost, Revenue, Profit
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
All videos show 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.
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
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)
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:
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.
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.
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.
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).
Finally, the frequently promised hot dog problem. You decide on the merits!
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
Antiderivatives (indefinite integrals)
Applications of the definite integral