Problem of the Week
Hilton Memorial Lecture
Cost, Revenue, Profit
Cost/revenue/profit 1, Cost/revenue/profit 2, Cost/revenue/profit 3
Demand function and cost function
For a little later in the unit: Marginal Revenue, Average Cost, Profit, Price & Demand Function
Economics summary of cost, revenue and profit
Logarithms, Exponential and e
Graphing 2^x and log base 2 of x
Explaining Euler's number e, the natural log base
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, some background on e:
Understanding the number e (exponential growth).
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.
Compound interest, present and future value problems
Finding limits from a graph (This is one of his rougher videos, but well explained.)
Evaluate limits using properties, Ex 1
Evaluate limits using properties, Ex 2
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:
Ex 6 involving rational expressions
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:
Graph a piecewise fcn 1 and Graph a piecewise fcn 2
Patrick discusses limits and the relationship to continuity. Watch for the important ideas, as we will discuss at length.
Continuity and limits made easy
Discontinuities in a function (piecewise)
More inspecting for discontinuities of a piecewise function
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
and Finding derivative with DQ, Ex 2
Here are examples for finding equation l of line tangent to f(x) at a given point:
Longer example of finding equation of tangent line
A practical application from the laws of physics (motion): Relationship between displacement, velocity and acceleration
Derivative Rules, Properties and Examples
Derivative of an exponential fcn with base a
Derivative of a log fcn with base a
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.
Chain rule explained by Patrickmjt
Ex of chain rule for radical function
Ex of chain rule for natural log function
Many great examples of chain rule involving ln[u(x)]
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)
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.
Proof of derivative of natural log function
Proof of derivative of exponential function (base a)
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)
One dimension (x) changes with respect to time (t), causing a related dimension (y) also to change with respect to time.
For example, area A of a circle is a function of its radius r; if r changes with time, then area changes also with respect to radius and ultimately time. We note the basic equation for this: dA/dt = (dA/dr)(dr/dt). The videos demonstrate this.
[Note: Videos by Krista, like Patrick's, are clear and no frills. Both convey mathematics methods effectively!]
Implicit differentiation and related rates
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
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.
Finding the critical numbers of a function is the first step to applying the first and second derivative tests, by which we examine where functions attain local extremes, the intervals where the function is increasing or decreasing, intervals of concavity and points of inflection. We use these skills to sketch the curves of functions and so examine what kind of behavior a function models.
Finding critical numbers of a fcn
There are many more videos on the HW page for this topic.
First and second derivative tests
Second derivative test and concavity
Curve sketching with calculus
Examples first and second derivative test to graph functions
Graphing another rational function
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:
Optimizing revenue given two points of data
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).
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:
Prof. Kumar shows Lagrange multiplier method of solving optimization w/constraint
Lagrange Method with Krista Ex 1 and Lagrange Method with Krista Ex 2
First, watch the excellent presentation that explains the overall picture, NO CALCULUS, explained by an economics professor in a very accessible way.
Prof Tarrok's Elasticity of Demand
Second, the formulas explained and worked problems OHM Elasticity 1 and OHM Elasticity 2, and note that the demand function is labeled D(x), but it means the same as q(p), which we use in class.
Antiderivatives (indefinite integrals)
Antiderivatives and indefinite integration
Examples of basic indefinite integration
Antiderivative with initial conditions (finding a particular F(x))
Displacement, velocity, acceleration example
Examples of simple substitution
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.
Prof Dave on Integration as Area under a Curve
Prof Dave on Fundamental Theorem of Calculus
And a more rigorous version:
Here are examples:
Example 0 of finding definite integral
Example 1 of finding a definite integral (polynomial and power rule)
Example 2 of finding a definite integral (log and IBP)
Examples 3 and 4 (two more IBP)
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:
Riemann sum to approximate the area under curve on interval [a, b]
Definition of definite integral via Riemann Sum
Applications of the definite integral
Finding area between two curves
Position, velocity and acceleration
Finally, Present and future value of a continuous income stream
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.