Week #7 in MATH 1420

Here is a guide to events and assignments in the coming week.

Monday, February 27, 2017

1. A trivia question will be posted on Blackboard.
2. Take Mini-exam 5.
3. Worksheet 2.7 is due by 3 pm.
4. Read/watch Lecture 2.8 on higher derivatives.
5. Begin Worksheet 2.8

Tuesday, February 28, 2017

1. Review OpenStax chapter 3.
2. Finish Worksheet 2.8.
3. Open office hours (for ANY student in any math class) are offered by me, from 2 to 3:30 at the BSM (Baptist Student Ministry) building on campus.

Wednesday, March 1, 2017

1. Worksheet 2.8 is due by 3 pm.
2. Read/watch Lecture 2.9 on hyperbolic trig functions and their derivatives.
3. Begin Worksheet 2.9.

Thursday, March 2, 2017

1. A trivia question will be posted on Blackboard.
2. Continue to review OpenStax chapter 3.
3. Finish Worksheet 2.9.

Friday, March 3, 2017

1. Worksheet 2.9 is due by 3 pm.
2. Read/watch Lecture 2.10 on the mastery of the derivative.
3. Begin Worksheet 2.10.

Saturday-Sunday, March 4-5, 2017

1. Continue to review OpenStax chapter 3.
2. Finish Worksheet 2.10.
3. Review for Mini-exam 6, given on Monday.
4. Monday's mini-exam is over lectures 2.1-2.10 and OpenStax chapter 3, on computing the derivative.

2.6, Derivatives of Inverse Functions (W 2017 0222)

2.6 Derivatives of inverse functions

Give some examples of geometric effect.

2.5, The Chain Rule Applied to the Exponential Function

A generic exponential function has form
f(x) = b^x
where b is a positive real number.

2.5, The Chain Rule Applied to Some Trig Functions

We take an extra day on the chain rule to allow students time to assimilate this rule and build up their skills in manipulating it.

When we apply the chain rule to trig and exponential functions, we gain information and understanding.

Once we work out the derivatives of the basic trig functions, sine, tangent and secant, the chain rule gives us a nice way to keep up with the other functions.  Here is how:

If theta is an acute angle of a right triangle, the opposite acute angle is said to be complementary. (In the figure below, the complementary angle of the black angle theta is the blue angle pi/2 - theta.)

The sine of the complementary angle is call the co-sine; the tangent of the complementary angle is call the co-tangent; secant of the complementary angle is call the co-secant -- this explains the syllable "co" in cosine, cotangent and cosecant.

Since the derivative of

pi/2 - theta 

is -1, then by the chain rule, the derivative of

f(pi/2-theta)

is

-f'(pi/2-theta).

Here is a table of the six trig functions, each matched with its complementary function.

And here are the derivatives.

Note how the derivatives listed in the right column are the negatives of the complementary functions of the derivatives given in the left column.

Week #6 in MATH 1420

Week #6 in MATH 1420

Here is a guide to events and assignments in the coming week.

Monday, February 20, 2017

1. A trivia question will be posted on Blackboard.
2. Take Mini-exam 4, over lectures 2.1-2.3.
3. Worksheet 2.4 is due by 3 pm.
4. Read/watch Lecture 2.5 on the chain rule for trig and exponential functions
5. Begin Worksheet 2.5.

Tuesday, February 21, 2017

1. Read OpenStax section 3.9.
2. Finish Worksheet 2.5.
3. Open office hours (for ANY student in any math class) are offered by me, from 2 to 3:30 at the BSM (Baptist Student Ministry) building on campus.

Wednesday, February 22, 2017

1. Worksheet 2.5 is due by 3 pm.
2. Read/watch Lecture 2.6 on the derivatives of inverse functions
3. Begin Worksheet 2.6.

Thursday, February 23, 2017

1. A trivia question will be posted on Blackboard.
2. Read OpenStax section 3.7.
3. Finish Worksheet 2.6.

Friday, February 24, 2017

1. Worksheet 2.6 is due by 3 pm.
2. Read/watch Lecture 2.7 on implicit differentiation.
3. Begin Worksheet 2.7.

Saturday-Sunday, February 25-26, 2017

1. Read OpenStax section 3.8.
2. Finish Worksheet 2.7.
3. Review for Mini-exam 5, given on Monday.
4. Monday's mini-exam is over lectures 2.1-2.7.

2.4, The Chain Rule

The chain rule allows us to use our derivative rules to compute the derivative of compositions of functions.  It is the most powerful tool in our derivative toolkit and we will spend a week or more ferreting out its power.

The chain rule is motivated by a symbolic simplification,

This should be interpreted as a statement about the derivative of a composition of a function y and a function u. Imagine that y(x) = sin x and u(x) = x^3.  Then the composition of y and u is a new function,  sin(u), or, written in terms of x,

f(x) = sin(x^3). 

Since the derivative of sin x is cos x and the derivative of x^3 is 3x^2 then the derivative of

f(u) = sin(u) 

with respect to u is cos(u) and so the derivative of f(x) with respect to x is

cos(u)(du/dx) = cos(x^3)(3x^2).

This rule allows us to take a complicated composition of functions, such as f(x)=(sin(x^3+e^x))^5, and work out the derivative of the composition based on the individual derivatives of the links in the chain.

2.3, Derivatives of Trig Functions

We take an extra class day after demonstrating the product and quotient rules for students to practice those new rules and to demonstrate how the quotient rule leads to the derivatives of trig functions.

Once we understand the quotient rule, it is easy to take the derivatives of

1. tan x = (sin x)/(cos x)
2. sec x = 1/(cos x)
3. cot x = (cos x)/(sin x)
4. csc x = 1/(sin x)

In each case we can use the quotient rule and possibly the Pythagorean identity

cos^2 x + sin^2 x = 1.

For example, in taking the derivative of tan x = (sin x)/(cos x), by the quotient rule, we get the numerator

(cos x)(cos x)-(sin x)(-sin x) = (cos x)^2 + (sin x)^2 = 1

while the denominator is (cos x)^2.

So our answer is 1/(cos x)^2 which is the same as (sec x)^2.

It is customary in the notation for trig functions to put the exponent over the trig function, writing sec^2 x for (sec x)^2 and so the derivative of tan x is sec^2 x.

Using the quotient rule, the derivative of sec x = 1/(cos x) will be  sin x/(cos x)^2 (since the derivative of 1 is 0) and it is customary to rewrite
sin x/(cos x)^2 = (sin x/cos x)(1/cos x) = tan x sec x.

Similar steps give formulas for the derivatives of the cotangent and cosecant functions.

Here is a summary.

One might notice that the trig functions which have minus signs in the answer are all "co"-functions. There is a good reason for this!  We will look more at this when we explore the Chain Rule.

A 538 Minimization Problem

A riddle from FiveThirtyEight.

I assume that this problem means the drowning person is 100 meters offshore from a point 100 meters to the lifeguard's right.  This is a fairly common (but not especially easy) problem in calculus; see the "Minimizing Travel Time" island problem in the OpenStax calculus text or Do Dog's Know Calculus?

This can be solved using differential calculus. The answer is to run approximately 79.588 meters to the right and then dive in, swimming approximately 102.062 meters, taking 11.938 seconds along the beach and 76.547 seconds in the water, about 88.485 seconds in all, instead of the 90 seconds required if one were to run to the point opposite the swimmer.

The problem can be represented by right triangle with the lifeguard's position at one corner and the right angle at the point directly opposite the swimmer.  Let x be the number of yards that the lifeguard runs along the beach.  Then the time running is 

T_r = x(15/100) seconds 

and the time swimming is, by the Pythagorean theorem,

T_s = sqrt(100^2 + (100-x)^2)(75/100) seconds.

The total time is 

T_r + T_s = T_r = x(15/100) + sqrt(100^2 + (100-x)^2)(75/100) secs.

I will factor out (15/100) and rewrite this as

T_r + T_s = (15/100)[x +  5 sqrt(100^2 + (100-x)^2)] secs.

Our goal is to minimize this value by taking the derivative with respect to x and then setting our derivative equal to zero.

The derivative of the total time, with respect to x, is 

(15/100)[1  + (5/2)(100^2 + (100-x)^2)^(-1/2)(-2(100-x))].

We may rewrite this expression as a constant (15/100) multiplied by a fraction whose numerator is
sqrt(100^2 + (100-x)^2) – 500 + 5x.
The denominator is sqrt(100^2 + (100-x)^2).
We do not care about the constant term (15/100) or the denominator so we focus on setting the numerator sqrt(100^2 + (100-x)^2)  -500+5x equal to zero. This means

sqrt(20000 -200x +x^2) = 500-5x

and so, squaring both sides,

20000-200x+x^2 = 250000-5000x+25x^2

And so

24x^2-4800x+230000 = 0

Solving this quadratic formula gives solutions 

x = 100 +/- 25 sqrt(2/3)

Only the minus sign makes sense so 

x =100 – 25 sqrt(2/3)

which is about 79.588 meters.  Thus the lifeguard runs almost 80 meters (not 100!) and then swims about 102 meters.

The solution given by 538 is in this blog post.

Week #5 in MATH 1420

Here is a guide to events and assignments in the coming week.

Monday, February 13, 2017

1. A trivia question will be posted on Blackboard.
2. Take Exam 1.

Tuesday, February 14, 2017

1. Read through OpenStax section 3.3 on the product and quotient rules
2. Finish Worksheet 2.2.
3. Open office hours (for ANY math student in any class!) are offered by me, from 2 to 3:30 at the BSM (Baptist Student Ministry) building on campus.

Wednesday, February 15, 2017

1. Worksheet 2.2 is due by 3 pm.
2. Read/watch Lecture 2.3 on the derivatives of trig functions
3. Begin Worksheet 2.3.

Thursday, February 16, 2017

1. A trivia question will be posted on Blackboard.
2. Read OpenStax section 3.5
3. Finish Worksheet 2.3.

Friday, February 17, 2017

1. Worksheet 2.3 is due by 3 pm.
2. Read/watch Lecture 2.4 on the chain rule
3. Begin Worksheet 2.4.

Saturday-Sunday, February 18-19, 2017

1. Read OpenStax section 3.6.
2. Finish Worksheet 2.4.
3. Review for Mini-exam 4, given on Monday.
4. Monday's mini-exam is over lectures 2.1-2.4.

2.2, The Product and Quotient Rules

Noticeably absent from our small collection of algebra rules about computing derivatives is a rule for computing derivatives of products or quotients.  This is because those rules are a bit more complicated than one might first expect.

The product rule

The derivative of a function which can be written as a product, such as f(x) = x^3sin x  is not the product of the derivatives, but a sum involving derivatives of the factors.

If

f(x) = u(x)v(x)

then

f'(x) = u'(x)v(x)+u(x)v'(x)

So the derivative of  f(x) = x^3sin x  is  f'(x) = 3x^2 sin x + x^3 cos x.

The quotient rule

If

f(x) = u(x)/v(x)

then

f'(x) = [u'(x)v(x)-u(x)v'(x)]/v^2(x)

For example, the derivative of  f(x) = x^3/sin x  is  f'(x) = [3x^2 sin x - x^3 cos x]/sin^2 x.

2.1, The Derivative As a Function

Given a function f(x), the slope of the curve y = f(x) at the point (a, f(a)) is the derivative at a. Given f(x), we can work out formulas for this slope (derivative) and so create a new function,

f '(x)

where the slope of the curve y = f(x) at the point (a, f(a)) is f '(a).

Some basic formulas

We have already worked out a few derivative formulas.

1. The derivative of  x^n is nx^(n-1) (the power rule).
2. The derivative of sin x is cos x; the derivative of cos x is –sin x.
3. The derivative of e^x is e^x.

Here, from my lecture notes, is a summary:

We will combine these basic rules with some algebra rules that allow us to find the derivative of a large number of functions.

Some basic algebra rules

We have two algebra rules, both very natural, a "sum" rule and a "scalar multiplication" rule.

If we know the derivative of two functions f(x) and g(x), the definition of the derivative shows us that the derivative of

f(x) + g(x)

should just be

f '(x) + g '(x).

Thus the derivative of the sum is the sum of the derivatives.

For example, since the derivative of x^5 is 5x^4 and the derivative of sin x is cos x then the derivative of  x^5+sin x is  5x^4+cos x.

In a similar manner, if we multiply a function f(x) by a constant, c, the derivative of cf(x) is just cf '(x).  Thus the derivative of (10)(x^5) is (10)(5x^4) = 50x^4.

Multiplying a function by a constant is called "scalar multiplication" (we are "scaling" the function) and so the derivative preserves scalar multiplication.

One rule we do not yet have is the derivative of a product of two functions!  That turns out to be more complicated and will be saved for next time.

Other resources

There are some good online applets that show this off.  Let me see if I can find some....

Part 2, Computing the Derivative, Spring 2017

In Part 2 of our first calculus course, we use the concept of the IRC of function to create a new function, the derivative.  We then focus on computational aspects of the creation of this new function. Here is our schedule for this spring semester.

Wednesday, February 8,
2.1, The Derivative as a Function

Friday, February 10,
2.2, The Product and Quotient Rules

Monday, February 13, Exam 1

Wednesday, February 15,
2.3, Derivatives of Trig Functions

Friday, February 17,
2.4, The Chain Rule

Monday, February 20,
2.5, The Chain Rule for Trig and Exponential Functions

Wednesday, February 22,
2.6, The Derivative of Inverse Functions

Friday, February 24,
2.7, Implicit Differentiation

Monday, February 27,
2.8, Higher Derivatives

Wednesday, March 1,
2.9, Hyperbolic Trig Functions and Their Derivatives

Friday, March 3,
2.10, Mastery of Derivatives

Part 1, The Derivative Concept, Spring 2017

As we finish the first part of the course and prepare for the first exam, it is time to clean up the details on the concept of the derivative.

The concepts of ``calculus" were discovered in the 1600's towards the end of the Renaissance period.  Prior to that (indeed by 300 AD) most of the mathematics of our high school curriculum had been discovered.  But the new powerful mathematics discovered in the Seventeenth Century changed the world!  It initiated the modern age of technology (beginning with the Industrial Revolution) and ushered in an age of science and prosperity unrivaled in human history.

The new mathematics was so efficient and explained so many things, that people spoke of it as "the way to calculate" or, in short, "the calculus."  Once people understood the new way to calculate, there was no reason to return to the old ways of the Greeks or Babylonians!

This first part of our calculus class focuses on the main concept of Calculus 1, the concept of slope.  I will take this in an order that is not in the standard textbooks, since I want to emphasize concepts over calculation.  The concept of slope is really a simple one, but we will use it to great effect, by generalizing the slope of a line to the slope of an arbitrary function.

As we do this, we introduce the average rate of change (ARC) and the instantaneous rate of change (IRC) and then make these concepts precise by introducing the concept of limits.

We cover this material in nine lectures across the first three weeks of this semester.

The course material in available in Ken's Google Drive folder. Here is the schedule for Spring 2017, along with links to the relevant material in that folder.

Wednesday, January 18, 2017
1.1, The Slope of a Function

Friday, January 20, 2017
1.2, The ARC and the IRC

Monday, January 23, 2017
1.3, Limits and the IRC

Wednesday, January 25, 2017
1.4, Limit Laws and  Continuity

Friday, January 27, 2017
1.5, Limits and Infinity

Monday, January 30, 2017
1.6, The Formal Definition of Limit and Derivative

Wednesday, February 1, 2017
1.7, The IRC of Powers of x

Friday, February 3, 2017
1.8, The IRC of Transcendental Functions

Monday, February 6, 2017
1.9, Euler's Marvelous Formula

Also in that folder are some subfolders with quizzes and exams from Spring 2017, their solutions, and some older exams and solutions.

Tomorrow we move on to the second part of our semester, the computation of the derivative.

1.9, Euler's Marvelous Formula!

Long ago (about 1740) Euler discovered that if x is fixed then as n goes to infinity

(1+x/n)^n goes to e^x.

cos(x/n) goes to 1

and

sin(x/n)/(x/n) goes to 1.

He then rewrote DeMoivre's formula

(cos z + i sin z)^n = cos nz + i sin nz

as 

(cos x/n + i sin x/n)^n = cos x + i sin x

He noted that sending n to infinity does not effect the right side of this equation but does effect the left side and so, "letting n be infinite" (!) he said that the lefthand side of this equation was

(1+i x/n)^n

But then, as n goes to infinity, that is just e^ix. So
e^(ix) = cos x + i sin x!

This remarkable result can be proven in other ways  – I like the Taylor series proof, which Euler came up with later – but most important are the many applications of this identity.  This identity first explains that trigonometric functions and exponential functions are just different views of the same mathematical object.

Additional resources

My notes on Euler's Marvelous Formula are in my Google Drive folder.  A precalculus lecture on Euler's formula is available here.

On my Google Drive folder for precalculus are notes on complex numbers and the polar coordinate form of complex numbers.

Some other notes on the web about complex numbers are here and here.

Week #4 in MATH 1420

Here is a guide to events and assignments in the coming week.

Monday, February 6, 2017

• A trivia question will be posted on Blackboard. (Subscribe to that thread so you get an email as soon as it is posted!)
• Take Mini-exam 3.
• Worksheet 1.8 is due by 3 pm.
• Read/watch Lecture 1.9 on Euler's Marvelous Formula. (This lecture involves complex numbers; these can be reviewed by looking at the precalculus lecture notes 2.4 and precalculus lecture notes 5.7 and 5.8.) 
• Begin Worksheet 1.9.

Tuesday, February 7, 2017

• Read through my precalculus lecture notes 2.4 on complex numbers and polar coordinate form for these, lectures 5.7 and 5.8.
• Finish Worksheet 1.9.
• Open office hours (for ANY math student in any class!) are being offered by me, from 2 to 3:30 at the BSM (Baptist Student Ministry) building on campus, just west of the parking garage.  Drop by!

Wednesday, February 8, 2017

• Worksheet 1.9 is due by 3 pm.
• Read/watch Lecture 2.1 on the derivative as a function.
• Begin Worksheet 2.1.

Thursday, February 9, 2017

• A trivia question will be posted on Blackboard.
• Read OpenStax section 3.2.
• Finish Worksheet 2.1.

Friday, February 10, 2017

• Worksheet 2.1 is due by 3 pm.
• Read/watch Lecture 2.2 on the product and quotient rules.
• Begin Worksheet 2.2. (It is due the following Wednesday.)

Saturday-Sunday, February 11-12, 2017

• Review for Exam 1, given on Monday, February 13. 
• Monday's Exam 1 is over part 1 of the course, lectures 1.1-1.9. It will not include 2.1 or 2.2.

1.6, Precise Definitions of the Limit and the Derivative

In Lecture 1.6 we focus on making the limit concept precise and then using that concept to define the derivative.

The formal definition of the limit

The idea of a limit, that as x goes to a, y goes to L, is now interpreted to mean that given an arbitrary small tolerance around L, one can find a small interval around a so that if x is in the interval around a, y is within the small tolerance from L.  The small tolerance on the y-axis is traditionally denoted by the Greek letter epsilon.  The interval on the x-axis is traditionally balanced around a, so that is is (a-delta, a+delta) for some positive number delta.

The formal definitions of the limit appears in the OpenStax section 2.5. In that section we "quantify closeness" by using epsilon for the tolerance (or "closeness") on the y-axis and delta for the "closeness" on the x-axis.  That OpenStax section has a nice applet that let's one experiment with delta and epsilon.

In my Google Drive folder for lecture 1.6 I also have a short video exploring the Epsilon-Delta game at the Desmos webpage.

The formal definition of the limit is a precise but subtle concept. I have brief lecture notes on it in my Google Drive folder, but the Open Stax section 2.5 is probably a better reference.

The formal definition of the derivative

The IRC, or derivative at a point P, is formally defined as the limit of the ARCs as the second point Q approaches the point Q.  This is discussed in the OpenStax section 3.1.  I explored this concept back in the beginning of the semester, in Lecture 1.2.  I have a short video exploring a Math Insight applet in my Google Drive lecture 1.2 folder.

Additional resources

There are some nice applets driven by Geogebra at this Shippensburg University webpage.

1.8, The IRC of Transcendental Functions

In this lecture (of Friday, Feb 3), we develop the IRC (or derivative) of three fundamental transcendental functions.

In order to do this, we need the limits of

(sin h)/h, 

(1 - cos h)/h,

and

(e^h-1)/h 

as h goes to zero.

We use these limits to then show that the IRC of

f(x) = sin x

at a point x=x0 is

f '(x0) = cos x0;

the IRC of

f(x) = cos x

at a point x=x0 is

f '(x0) = –sin x0 

and the IRC of

f(x) = e^x

at x=x0 is

f'(x0) =  e^x0.

We will use these results throughout the course.  This lecture demonstrates how we found these answers using the definition of the derivative.

1.7, The IRC of Power Functions

The main idea of this lecture is that we can compute the IRC of f(x) = x^n at an arbitrary point.

We do this by first assuming that the exponent n is a positive integer and writing out the formal definition of the derivative. According to that definition, we need to take the limit, as h goes to zero, of

f(x+h)-f(x)         (x+h)^n - x^n

-------------  =   -------------------

h                           h

We expand (x+h)^n by the binomial theorem (a.k.a Pascal's Triangle) and discover that x^n will cancel out, leaving us with an expression whose numerator begins

nx^(n-1)h +  (... higher powers of h...)

However, we must divide by h and then take the limit.  Dividing by h will give us

nx^(n-1) + h( ... more terms ...)

and when we evaluate the limit by setting h to zero, we get nx^(n-1).

It turns out that this formula works for any power of x, even if n is not an integer. (We won't prove that.)  Thus if

 f(x) = x^n

then the derivative is 

 f ' (x) = nx^(n-1).

This is our first derivative rule, often called the "power rule".

This lecture also gives us a change to teach (or review) Pascal's triangle.

The binomial theorem can be visualized by a triangle that begins with a ones at the top and has ones descending down the sides. We begin with row 0, which just has a 1 in it, followed by row 1 which is 1 1

As the triangle grows, the interior positions are filled by numbers created by adding the digits just above the number, on its left and right shoulders. Thus we row 2 with entries 1 2 1.

and after that, row 3 with entries  1 3 3 1

and so on.

 Here are rows 0 to 16!

(This version of Pascal's Triangle, created by Paul Gaborit (2009) under the Creative Commons attribution license, was found here.)