MAT 136, Calculus I

Prof. Swift, Spring 2024

Course information

Here is a link to NAU policy statements, and the math department policies including links to Career Ready resources that are part of the syllabus.

Office hours are usually in my office, AMB 110. Occasionally I will be down the hall in the MAP room (AMB 137). If I am not at my office during office hours, look for me there.
My office hours are
Tuesday 10:30-12:15
Thursday 11:30-12:30
Friday 11:30-1:30
You can always send me e-mail, drop in, or make an appointment if these times aren't convenient. Here is my weekly schedule.

The official textbook, Whitman, is freely available online.
Another free textbook Herman and Strang, is on OpenStax.
If you want to get a physical textbook, we recommend any edition of Calculus, Early Transcendentals, by Rogawski et al.
There many freely available resources on the internet, for example the Paul's Notes, by Paul Dawkins of Lamar University, and wonderful videos at the Khan Academy
There are videos and other resources at the MOOCulus site, https://mooculus.osu.edu/.

An excellent web-based calculator is www.desmos.com/calculator

Most homework is assigned and graded using WeBWorK. Your username and password is the same as for LOUIE. (A typical username is abc123.) Use any WeBWorK link at this web site, or type in https://webwork.math.nau.edu/webwork2/JSwift_136/.

This link has detailed information on WeBWorK (in pdf format). Your username and password are the same as for Louie.

Chart of letters of the Greek alphabet.

I hope you will come to my office hours to introduce yourself. Aside from my office hours, we have lots of free help available.

The Math Achievenment Program (MAP) room is at the end of the hall, just north of our classroom, in AMB 137. Starting January 18, it will be open M-Th 10-6 and F 10-2. This is staffed by Peer Math Assistants (PMAs), and I will also have some of my office hours there.
Free drop in tutoring and one-to-one tutoring is available at the North and South Academic Success Centers.

Help, Notes, and Worksheets in reverse Chronological Order

Remember that we need to know the differentiation shortcuts forward and backward.

Monday, April 29: Set 25, Substitution
Section 5.3: Substitution Rule for Indefinite Integrals
Section 5.4: More Substitution Rule
Chain Rule says \(\displaystyle \frac d{dx} F(u) = F'(u) \frac{du}{dx}\). This says the same thing as \(\displaystyle \int F'(u) \frac{du}{dx} dx = F(u) + C\).
We can re-write this as the substitution rule \(\displaystyle \int f(g(x)) g'(x) dx = \int f(u) \, du\), where \(u = g(x)\). Note that \( du = \frac{du}{dx} dx = g'(x) dx\).
In-class worksheet on the method of \(u\)-substitution, with the scanned solutions.

Monday, April 22: Set 25, Net Change as an Integral
This set is about applications of the FTC I.
\(\int_a^b f'(x)\, dx = f(b) -f(a)\) says that the integral of the rate of change is the total change.
The worksheet today is from the sample Midterm 4, which will be handed out in class.

Friday, April 19: More on set 24, The Fundamental Theorem of Calculus, Part II
Here is the Big Picture of Calculus.

The FTC II is useful because many indefinite integrals are non-elementary, meaning that the general antiderivative is not elementary. An elementary function can be written in terms of powers, roots, trig functions, inverse trig functions, logs, and exponentials.
The indefinite integrals \( \int e^{-x^2} dx\) and \( \int \sqrt{x + \sin(x)} \, dx\) are not elementary. Click on the integrals to see what Wolfram Alpha does. (It either cheats, or punts.)
As a consequence, \(f(x) = \int_0^x e^{-t^2} dt\) is a non-elementary function that satisfies \(f(0) = 0\) and \(f'(x) = e^{-x^2}\). This function is featured in the wikipedia article on elementary functions.

Thursday, April 18: Set 24, The Fundamental Theorem of Calculus, Part II
Note that Paul's notes and our WeBWorK switch the names of Part I and Part II. Paul does what he calls the Funamental Theorem of Calculus, Part I at the end of Section 5.6: Definition of the Definite Integral

The Fundamental Theorem as stated in the previous section implies that
FTC II: \(\displaystyle \frac{d}{dx} \int_a^x F'(t)\, dt = \frac{d}{dx}[F(x) - F(a)] = F'(x) \). This is usually written with \(f\) in place of \(F'\):

FTC II: \(\displaystyle \frac{d}{dx} \int_a^x f(t)\, dt = f(x) \). Note that \(f(t)\) is inside the integral and \(f(x)\) is on the right-hand side. But \(f\) is the same function.

In-class worksheet on The Fundamental Theorem of Calculus, Part II, with the scanned solutions.

Wednesday, April 17: Set 23, The Fundamental Theorem of Calculus, Part I
Section 5.7: Computing Definite Integrals
FTC I: \(\displaystyle \int_a^b F'(x)\, dx = F(b) - F(a) \). This can also be written as
FTC I: \(\displaystyle \int_a^b f(x)\, dx = F(b) - F(a) \), where \(F'(x) = f(x)\).
That right-hand side is so commonly used that we have special notation: \(F(x) |_a^b = [F(x)]_a^b = F(b)-F(a)\).

Proof of the Fundamental Theorem of Calculus, Part I.

In-class worksheet on The Fundamental Theorem of Calculus, with the scanned solutions.

Monday, April 15: Set 22, The Indefinite Integral
In-class worksheet on The Indefinite Integral, Part 2, with the scanned solutions.

Friday, April 12: Set 22, The Indefinite Integral
Section 5.1: Indefinite Integrals
Section 5.2: Computing Indefinite Integrals

Definition: A function \(F(x)\) is an anti-derivative of the function \(f(x)\) provided that \(F'(x) = f(x)\).

Fact: If \(F(x)\) is an anti-derivative of \(f(x)\) then the most general anti-derivative of \(f(x)\) is \(F(x) + C\), where \(C\) is an arbitrary constant.
Definition: The most general anti-derivative of \(f(x)\) is called the indefinite integral of \(f(x)\), and is denoted \(\displaystyle \int f(x) \, dx = F(x) + C\), where \(F\) is any antiderivative of \(f\) and \(C\) is an arbitrary constant.

We find anti-derivatives, which we call “doing integrals”, by knowing the derivative shortcuts forward and backward. We use the power rule so often that it is useful to re-phrase it with a shifted exponent:
\(\displaystyle \int x^a \, dx = \frac{x^{a+1}}{a+1} + C\), provided \(a \neq -1\). Note that if \(a = -1\) then \(a + 1 = 0\) and we cannot divide by zero. Instead, use the known fact that \(\displaystyle \frac{d}{dx}\ln|x| = \frac 1 x\) to get the integral formula
\(\displaystyle \int x^{-1}\, dx = \int \frac{1}{x} \, dx = \ln|x| + C\).

Thursday, April 11: Set 21, Area and the Definite Integral, continued
On Wednesday we gave the Geometric Definition:
\(\displaystyle \int_a^b f(x) dx \) is the signed area between \(y = 0 \) and \(y = f(x)\) with \(a \leq x \leq b\). (The integral is the “area under the curve.”)
On Thursday we give the Limit Definition:

\(\displaystyle \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i = 1}^{n}f(x_i^*) \Delta x \), where \(\displaystyle \Delta x = \frac{b-a}{n}\), \(x_i = a + i \Delta x\), and for each \(i\), the sample point \(x_i^*\) is any point in the sub-interval\([x_{i-1}, x_i]\). We consider three cases for the Riemann sum:
The Left sum \(\displaystyle L_n = \sum_{i = 0}^{n-1}f(x_i) \Delta x \), the Right sum \(\displaystyle R_n = \sum_{i = 1}^{n}f(x_i) \Delta x \), and the Midpoint sum \(\displaystyle M_n = \sum_{i = 1}^{n}f(\overline{x_i}) \Delta x \), where \(\displaystyle \overline{x_i} = \frac{x_{i-1}+ x_i}{2}\) is the midpoint of the sub-interval.

In-class worksheet on The Definite Integral, Part 2, with the scanned solutions.
Here is a video showing how Gauss discovered \(\displaystyle \sum_{i=1}^n i = \frac{n(n+1)}{2}.\) For your last problem on the WeBWorK set you will need the fact that \(\displaystyle \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}.\)

Wednesday, April 10: Set 21, Area and the Definite Integral
Section 5.4: The Area Problem
Section 5.6: Definition of the Definite Integral.
In-class worksheet on The Definite Integral, with the scanned solutions.

Thursday, April 4: Set 20, Optimization and Newton's Method
Section 4.13: Newton's Method.
Newton's Method: To find an approximate solution to \(f(x) = 0\), start with a guess \(x_0\) and compute \(x_1\), \(x_2\), \(x_3\), etc. using \(\displaystyle x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\). Stop when \(x_{n+1} \approx x_n\) to the desired accuracy, since it follows that \(f(x_n) \approx 0\).
Here is a YouTube video of me showing how to do Mewton's Method with a Spreadsheet.

Wednesday, April 3: Set 20, Optimization
Section 4.8: Optimization, and Section 4.9: More Optimization.
The key is setting up a function of one variable, \(Q = f(x)\), whose output \(Q\) is the thing you want to optimize. Here are some steps:
(1) Understand the problem (Read it carefully, and frequently)
(2) Draw a diagram. Identify the given fixed quantities, and the variable quantities.
(3) Introduce notation. The word "Let" is super-important.
(4) Write the quantity you want to optimize (we will call it \(Q\)) in terms of other quantites.
(5) Usually \(Q\) will depend on more than one quantity. Use the constraints to eliminate all but one variable quantity to get \(Q = f(x)\). Write down the domain of \(f\).
(6) Use calculus to find the global max (or min) value of \(f\), or the input \(c\) at which the global extremum occurs. Read the question again and answer it.

Monday, April 1: More on set 19, L'Hospital's Rule
The indeterminate forms are
\( \frac 00,~ \frac{\infty}{\infty},~ 0\cdot\infty,~ \infty - \infty,~ 0^0,~ 1^\infty, \text{ and } \infty^0 .\)
L'Hospital's rule can only be used for limits of type \(\frac 00\) and of type \(\frac{\infty}{\infty}\).
In-class worksheet on Indeterminate Forms, with the scanned solutions.

Friday, March 29: Set 19, Section 4.10: L'Hospital's Rule
L'Hospital's Rule, also written L'Hôpital's Rule, says that if \(\displaystyle \lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\), or \(\displaystyle \lim_{x \to a} f(x) = \pm \infty\) and \(\displaystyle \lim_{x \to a} g(x) = \pm \infty\), then
\(\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\)
L'Hospital's Rule can be applied to \(\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)}\) and \(\displaystyle \lim_{x \to -\infty} \frac{f(x)}{g(x)}\), provided they are indeterminate forms of type \(\frac 00\) or type \(\frac{\infty}{\infty}\).
In-class worksheet on L'Hospital's rule, with the scanned solutions. Here is a Desmos Graph.

Wednesday and Thursday, March 27, 28: Set 18, Section 4.6 : The Shape Of A Graph, Part II
Let \(I\) be an interval in what follows.
If \(f' > 0\) on \(I\), then \(f\) is increasing on \(I\).
If \(f' < 0\) on \(I\), then \(f\) is decreasing on \(I\).
If \(f'' > 0\) on \(I\), then \(f'\) is increasing on \(I\), and \(f\) is concave up on \(I\).
If \(f'' < 0\) on \(I\), then \(f'\) is decreasing on \(I\), and \(f\) is concave down on \(I\).

There are two more cases:
If \(f' = 0\) on \(I\), then \(f\) is constant on \(I\).
If \(f'' = 0\) on \(I\), then \(f'\) is constant on \(I\), and \(f\) is straight on \(I\).

Definition: A turning point of \(f\) is a point on the graph of \(f\), namely \((c, f(c))\) , where \(f\) has a local extremum at \(c\).
Definition: An inflection point of \(f\) is a point on the graph of \(f\), namely \((c, f(c))\), where the concavity of \(f\) changes.

Theorem: If \(f''(c) = 0\), and \(f''\) changes sign at \(c\), then \((c, f(c))\) is an inflection point of \(f\).

Theorem: If \(f'\) has a local extremum at \(c\), then \((c, f(c))\) is an inflection point on the graph of \(f\).

Theorem: (The Second Derivative Test) If \(f'(c) = 0\) and \(f''(c)>0\), then \(f\) has a local minimum at \(c\).
Similarly, (\(f'(c) = 0\) and \(f''(c)< 0 ) \implies \) \(f\) has a local maximum at \(c\).
Note that if \(f'(c) =0 \) and \( f''(c) = 0\), then then the second derivative test says absolutely nothing.

In-class worksheet on the Shape of Graphs, with the scanned solutions. Here is a desmos graph of \(y = e^{-x^2/2}\)

Here are the scanned solutions to group work on the Shape of Grpahs.

Definition: A function \(f\) is concave up on an interval \(I\) provided that every secant line with x-values in I is above the graph of \(f\).
Similarly, \(f\) is concave down if every secant line is below the graph.

Theorem: A function \(f\) is concave up on an interval \(I\) provided if \(f'\) is increasing on \(I\).

Theorem: A function \(f\) is concave up on an interval \(I\) provided if \(f''\) is positive on \(I\).
Similar theorems holds for concave down.

For in-class worksheet do your version of Set 17, problem 9, on the MVT.

Thursday, Friday, and Monday, March 21, 22, and 25: Set 17, Critical Points and Local Extrema, and the Mean Value Theorem.
Section 4.3: Minimum and Maximum Values,
Section 4.5: The Shape of a Graph, Part I.
Section 4.7: Mean Value Theorem.

Definition: A function \(f\) is increasing on an interval \(I\) provided that \(f(x_1) < f(x_2)\) for all \(x_1 < x_2\) in \(I\).
A similar definition holds for decreasing.
Examples: \(f(x) = x^2\) is increasing on \([0, \infty)\) and decreasing on \( (-\infty, 0] \).
Similarly, \(g(x) = x^3\) is increasing on \( (-\infty, \infty)\).

Theorem: If \(f'(x) > 0\) for all \(x\) in an interval \(I\), then \(f\) is increasing on \(I\).
A similar theorem holds: \(f' < 0\) on \(I\) implies that \(f\) is decreasing on \(I\).

We can get a stronger result:
Theorem: If \(f'(x) \geq 0\) for all \(x\) in an interval \(I\), and \(f'(x) = 0\) only at a finite number of points (possibly no points), then \(f\) is increasing on \(I\).
A similar theorem holds for \(f'(x) \leq 0\) and \(f\) decreasing.

Examples: We can see from the definition that \(f(x) = x^2\) is increasing on \( [0, \infty) \), and the stronger theorem proves this, since \(f'(x) = 2x \geq 0\) on \( [0, \infty) \), and only \(x = 0\) satisfies \(f'(x) = 0\).
Similarly, the stronger theorem says that \(g(x) = x^3\) is increasing on \( (-\infty, \infty)\), since \(g'(x) = 3 x^2 \geq 0\) for all \(x\), and the only solution to \(g'(x) = 0\) is \(x = 0\).

In-class worksheet: Find the largest interval(s) on which the following functions are increasing.
\(f(x) = \frac 1 3 x^3 - \frac 1 2 x^2 -2 x\) and \(g(x) = \frac 1 3 x^3 - x^2 + x\). Here are the scanned solutions.
Here are desmos graphs for the functions f and g.

Definition: We say a function \(f\) has a local maximum at \(c\) if \(f(c) > f(x)\) for all \(x\) sufficiently close to, but not equal to, \(c\)
A similar definition holds for a local minimum of \(f\) at \(c\).

Theorem: (The First Derivative Test) Suppose that \(f\) is continuous at \(c\).
1. If \(f'(x) > 0\) for \(x < c \) and \(f'(x) < 0\) for \(c < x\), then \(f\) has a local maximum at \(c\).
2. If \(f'(x) < 0\) for \(x < c\) and \(f'(x) > 0\) for \(c < x \), then \(f\) has a local minimum at \(c\).
3. If \(f'(x) \) has the same sign for \(x < c\) and \(c < x\), then \(f\) does not have a local extremum at \(c\).
Note: Those inequalities about the sign of \(f'(x)\) only need to hold for \(x\) close to \(c\).

Theorem: If \(f\) has a local extremum at \(c\), then \(c\) is a critical point of \(f\).
The converse is not true. If \(c\) is a critical point of \(f\), then \(f\) might or might not have a local extremum at \(c\).

In-class worksheet on The First Derivative Test to Classify Critical Points. Here are the scanned solutions.
Here is a desmos graph of f and f-prime.
Here is a desmos graph of g and g-prime. The hardest thing about graphing cubic functions is finding the \(x\)-intercepts!

Wednesday and Thursday, March 20 and 21: Section 4.2: Critical Points and Section 4.4: Finding Global Extrema (set 16).
Definition: A critical point of a function \(f\) is a number \(c\) such
(1) \(f'(c)=0\) or \(f'(c)\) is undefined, and
(2) \(f(c)\) is defined. (That is, \(c\) is a point in the domain of \(f\).)
Definition of Global Extrema, and a theorem about global extrema of a continuous function on a closed interval.
In-class worksheet on Critical Points and Global Extrema, and the scanned solutions.

Monday, March 18: Section 4.11: Local Linearization and the Tangent Line Approximation. WeBWorK set 15.
The local linearization of \(f\) at \(a\) is \(\ell_a(x) = f(a) + f'(a)(x-a)\). It is OK to write \(\ell_a(x) = f'(a)(x-a) + f(a)\).
(1) \(y = \ell_a(x)\) is an equation of the tangent line to the graph of \(f\) at \(a\).
(2) \(f(x) \approx \ell_a(x)\) for \(x \approx a\) is the tangent line approximation.
For a linear function, the tangent line approximation is exact.
Here is one of Richard Feynman's many amusing stories: Feynman vs. the Abacus.
In-class worksheet on Local Linearization. Here are the scanned solutions.

Monday, Wednesday, Thursday, March 4-7: Logarithmic Differentiation, and Review for Friday's Midterm 2
Here's another link to Differentiation Shortcuts.
Here are the scanned solutions to Friday's quiz
Differentiation review Thursday is an in-class gateway exam. This was not collected for a grade. (The gateway exam was used in the old days. Students had multiple opportunities to get 7 or 8 out of 8 correct. They had to pass to continue with the course.)
Two versions of the scanned solutions are available here.
The first solutions show the work, and uses the rules as Swift taught them.
The second solutions skip some steps
If you do it correctly, you can skip steps on the exam. But to maximize partial credit, I suggest you follow the pattern of the solutions that show the work.

Friday, March 1: more on WeBWorK set 14
Today's facts.
\( \frac d {dx} \arcsin(x)= \frac {1}{\sqrt{1-x^2}}\), and \( \frac d {dx} \arctan(x)= \frac {1}{1+x^2}\).
Note: WeBWorK frequently uses \(\sin^{-1}(x)\) as a notation for \(\arcsin(x)\), and \(\tan^{-1}(x)\) as a notation for \( \arctan(x)\).
Desmos graphs for arcsine and arctangent
Worksheet on derivatives of inverse trig functions, and the tangent line to a curve, with the scanned solutions. Here is a desmos graph of the tangent line to the curve.

Wednesday and Thursday, February 28, 29: WeBWorK set 14
Wednesday: Section 3.6 : Derivatives of Exponential and Logarithmic Functions,
Thursday: Section 3.10 : Implicit Differentiation.
Friday:Section 3.7 : Derivatives Of Inverse Trig Functions,
Yet another desmos graph with numerical derivative. This one starts without graphing the numerical derivative.
New derivative facts obtained using the chain rule.
\(\frac d {dx} a^x = \ln(a) a^x\), where \(a\) is any positive constant,
\(\frac d {dx} \ln(x) = \frac 1 x \), with domain all \(x\) such that \(x > 0\), and
\( \frac d {dx} \ln|x| = \frac 1 x \), with domain all \(x\) such that \(x \neq 0\).

Implicit differentiation: Start with an equation involving \(x\) and \(y\). (So \(y\) is an implicit function of \(x\).) Use the chain rule to differentiate both sides of the equation with respect to \(x\), treating \(y\) as an unknown function of \(x\). Then, solve for \(\frac{dy}{dx}\).
Worksheet on derivative of logs, and implicit differentiation, with the scanned solutions. Here is a check of the calculation of \(h'(x)\) in problem 1 with a desmos graph: Curve 2 obscures curve 1 (the original \(h(x)\)), and curve 4 is a check of the derivative formula plotted in curve 3.

Monday, February 26: More on WeBWorK set 13
Here is the verbal Chain Rule.
Here is a page with a derivative table from a book published in 1933. Note how the chain rule is used.

Friday, February 23: WeBWorK set 13 Section 3.9 : Chain Rule
The chain rule can be stated in two equivalent ways:
If \(y = f(u)\) and \(u = g(x)\), then \(\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}\)
\(\frac{d}{dx} f(g(x)) = f'(g(x))\cdot g'(x) \)
Desmos graph for the Numerical derivative of h(x) = sin(x)
Here are scanned solutions of today's in-class worksheet.

Wednesday, February 21 and Thursday, February 22: WeBWorK set 12
Note: We are skipping set 11, and will return to this material after learning how to differentiate any formula.
Set 12 covers Section 3.5 : Derivatives Of Trig Functions, and Section 3.12 : Higher Order Derivatives.

Today's facts:
\(\frac d {dx} \sin(x) = \cos(x)\), \(\frac d {dx} \cos(x) = -\sin(x)\), \(\frac d {dx} \tan(x) = \frac 1 {\cos^2(x)}\).

Everything you need to know about computing derivatives in this class are summarized in the page Differentiation Shortcuts.
Note that you can use trig identities to avoid memorizing the derivatives of \(\csc(x)\), \(\sec(x)\) and \(\cot(x)\).
Derivative Shortcuts 3, with the scanned solutions

Thursday: Here is a desmos graph showing the numerical derivative.
Here is today's worksheet on Derivative Shortcuts 4, with the scanned solutions

Monday, January 19: More on set 10
Here are the verbal Product and Quotient Rules.
Don't make this mistake.
Worksheet on Product and Quotient Rules, with the scanned solutions

Friday, February 16: More on WeBWork set 10
Rule:
\( \displaystyle \frac{d}{dx} \left [ \frac {f(x)}{ g(x)} \right ] = \frac{f'(x)\cdot g(x) - f(x) \cdot g'(x) }{g(x)^2} \quad\) The quotient rule
Here are proofs of the Product Rule and the Quotient Rule.
Here is Friday's Quiz 4.

Thursday, February 15: More on WeBWork set 9, start set 10
Set 10 covers Section 3.4 : Product And Quotient Rule

Rule:
\(\frac{d}{dx} [f(x)\cdot g(x)] =f'(x)\cdot g(x) + f(x) \cdot g'(x) \quad\) The product rule
Fact:
\(\frac{d}{dx} e^x = e^x \quad \) The derivative of the natural exponential function is itself!
Worksheet on Derivative Shortcuts 2, with the scanned solutions. Note: We did not do this worksheet in class.

Wednesday, February 14: The Derivative as a Function, WeBWork set 9
We start to learn how to differentiate any nice function, using Section 3.3: Differentiation Formulas.
Rules:
\(\frac{d}{dx} [c f(x)] = c f'(x) \quad\) The constant multiple rule
\(\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \quad \) The sum and difference rules
Facts:
\(\frac{d}{dx} [c] = 0 \quad \) The derivative of a constant function is the 0 function
\(\frac{d}{dx} [x^c] = c x^{c-1} \quad \) The Power Function Fact (PFF)
Worksheet on Derivative Shortcuts 1, with the scanned solutions

Monday, February 5 and Wednesday, February 7 (class cancelled): The definition of the derivative, WeBWork set 8
Here are the solutions to Friday's quiz.
Note: A sample exam for Friday's Exam 1 is on Canvas in the "Modules" section.

Section 3.1: The definition of the Derivative.
Section 3.2: Interperetation of the Derivative.

Definition: The derivative of the function \(f\) at \(a\) is the slope of the tangent line to \(y = f(x)\) at the point (\(a, f(a))\).
The notation for the derivative is \(f'(a)\), pronounced "\(f\) prime of \(a\)", and it can be computed as
\(\displaystyle f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}\).

Replace \(x\) with \(a + h\), so \(\Delta x = x -a = (a+h) - a = h\), and this becomes
\(\displaystyle f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\). This is illustrated in this Desmos graph.

Since \(a\) can be any point in the domain of \(f\), we can replace \(a\) by \(x\) and get our most useful way to compute the derivative.
\(\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\).
Given a function \(f\), this defines a new function, \(f'\).

Today's in-class ``worksheet'': Find an equation of the tangent line to the graph of \(f(x) = x^2 - 4x + 7\) at the point \( (3, f(3)) \).
Here is a graph of \(f\) and the point (\(3, f(3))\).
Solution: Since \(f(3) = 4\), the point-slope form of the tangent line is \(y = m(x-3) + 4\). The slope of the tangent line is
\(m = f'(3) = \displaystyle \lim_{h \to 0} \frac{f(3+h)-f(3)}h = \lim_{h \to 0} \frac{(3+h)^2 - 4(3+h) + 7 \ -4}h = \lim_{h \to 0} \frac{9 + 2\cdot 3 h + h^2 - 12 - 4h + 7 \ -4}h = \lim_{h \to 0} \frac{6 h + h^2 - 4h}h = \lim_{h \to 0} \frac{h(2 + h)}h = \lim_{h \to 0} (2 + h) = 2 + 0 = 2. \)
To summarize, the slope of the tangent line is \(f'(3) = 2\). Thus an equation of the tangent line is \(y = 2(x-3)+4\). Here is a graph of both \(f\) and the tangent line at \(x = 3\).
This point-slope form of the tangent line is perferred, in calculus, to \(y = 2x - 2\).

Thursday and Friday, February 1 and 2: More Limits, WeBWorK set 7
These sections in Paul's notes will be helpful
Section 2.6 : Infinite Limits
Section 2.7 : Limits At Infinity, Part I
Section 2.8 : Limits At Infinity, Part II
Worksheet 9, Limits at infinity, with the scanned solutions

Set 7 also talks about these two theorems which are part of sections we already worked on
The Squeeze Theorem: From Section 2.5: Computing Limits
Assume \(a < b < c\) and \(f_\ell(x) \leq f(x) \leq f_u(x)\) for all \(x \in (a,b)\cup( b,c)\), and \(\displaystyle \lim_{x \to b} f_\ell(x) = \lim_{x \to b} f_u(x) = L\). Then \(\displaystyle \lim_{x \to b} f(x) = L\).
The famous example I did in class is also done in this Khan Academy video:
limit of sin(x) over x as x approaches 0.

The Intermediate Value Theorem (IVT): From Section 2.9: Continuity
Suppose \(f(x)\) is continuous on \([a,b]\), and \(M\) is a number strictly between \(f(a)\) and \(f(b)\). Then there exists a number \(c\) such that
\(a < c < b\), and \(f(c) = M\).

Wednesday, January 31: Continuity and Algebraic Limits, WeBWorK set 6
Here are some useful theorems that allow you to compute most of the limits you will encounter:
If \(f(x) = \tilde f(x)\) for all \(x \neq a\), and \(\tilde f\) is continuous at \(a\), then \(\displaystyle \lim_{x \to a} f(x) = \displaystyle \lim_{x \to a} \tilde f(x) = \tilde f(a) \).
If \(f(x) = \tilde f(x)\) for all \(x > a\), and \(\tilde f\) is continuous at \(a\), then \(\displaystyle \lim_{x \to a^+} f(x) = \lim_{x \to a^+} \tilde f(x) = \tilde f(a) \).
If \(f(x) = \tilde f(x)\) for all \(x < a\), and \(\tilde f\) is continuous at \(a\), then \(\displaystyle \lim_{x \to a^-} f(x) = \lim_{x \to a^-} \tilde f(x) = \tilde f(a) \).
Worksheet 8, True/False questions about limits and continuity, with the scanned solutions.

Here is the desmos graph of the my version of set 6, problem 12. You can play around with the sliders for \(a\) and \(b\) and try to make the function \(f\) continuous at both 2 and 3, but that is too hard!
The key is to get 2 equations in the 2 unknowns \(a\) and \(b\) by imposing the condition that \(f\) is continuous at 2 and 3. Here is the solution to my version of this problem. The solution to the system of 2 equations is \(a = -1\), \(b = -4\). You can go to that desmos graph and use the sliders to get a continuous function.

Monday, January 28: Continuity and Algebraic Limits, WeBWorK set 6
These sections in Paul's notes will be helpful:
Section 2.4: Limit Properties
Section 2.5: Computing Limits
Section 2.9: Continuity
Worksheet 7, with the scanned solutions

Friday, January 26: Graphical Limits, WeBWorK set 5
These sections of Paul's notes will be helpful:
Section 2.2: The Limit
Section 2.3: One-Sided Limits
Section 2.6: Infinite Limits
Rules for sketching graphs: An open circle stands for a point on the graph. A closed circle stands for a point not on the graph.
\(\displaystyle \lim_{x \to 2} f(x) = 3\) means that \(f(x) \approx 3\) for all \(x \approx 2\), except possibly \(x = 2\).
An alternative notation is \(f(x) \to 3\) as \(x \to 2\).
Note: The value of \(f(2)\), has nothing to do with the limit as \(x\) approaches 2. Frequently, we are asked to determine \(\displaystyle \lim_{x \to 2} f(x) \) in the cases where \(f(2)\) is not defined.
Here is the in-class Quiz 2. Here are the scanned solutions to the quiz.

Thursday, January 25: Rates of Change, WebWorK set 4
(1) How to numerically estimating the slope of the tangent line to a graph, which is the average rate of change of the fuction.
(2) How to graphically estimate the slope of the tangent line to a graph.
Today we finish problem 6 and 11 in set 4, using the two Desmos Graphs from yesterday.
We also work on Problem 4 in set 4.

Wednesday, January 24: Rates of Change, WebWorK set 4:
We finally start Calculus!
Here is the introduction to Paul's Notes Chapter 2. Set 4 is loosely based on Paul's Section 2.1 : Tangent Lines And Rates Of Change
\(m_{PQ}\) is the slope of the secant line through points \(P\) and \(Q\) on the graph of a function. Furthermore, \(m_{PQ}\) is the average rate of change of the function. The slope of the tangent line at \(P\) is the limit of \(m_{PQ}\) as \(Q\) approaches \(P\).
Here is a Desmos Graphing Calculator Link the plots the secant lines for f(x) = 2/x with the point \(P = (2, 1)\).
Today's assignment is to modify this for the function \(f(x) = \ln(x)\) with the x-value from your problem 11 on the webwork. Here is the secant line for f(x) = ln(x) with point \(P = (5 \ln(5))\).

Monday, January 22: Population Modeling, Composition of Functions, and Inverse Functions, WeBWorK set 3:
Definition of open and closed intervals: (Needed in webwork set 3, problem 1)
\( (a, b) = \{x \in \mathbb{R} \mid a < x < b\}\). The open interval, written with parentheses, does not include endpoints.
\( [a, b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}\). The closed interval, written with square brackets, does include endpoints.
\( (a, b] = \{x \in \mathbb{R} \mid a < x \leq b\}\). This interval does not include \(a\), but it does include \(b\). This interval is neither open nor closed.
Desmos graphs sent over the weekend:
Cubic function, intercept form: (Problem 15),
Linear function, point-slope form
Exponential Function, point-base form (problem 16) Exponential Function, point-base form (problem 16)

Worksheet 3. Here is the desmos graph referred to in the worksheet:
Population Models start
Population Models solutions

Friday, January 19: Trig functions and function notation, WeBWorK set 2:
Ten minute Quiz 1, with scanned solutions. on linear and exponential functions, at the end of class.

Thursday, January 18: Linear and Exponential functions, WeBWorK set 1:
In class worksheet 2 on the Point-Slope Form for Linear Functions. scanned solutions.

Wednesday, January 17: Introduction:
Today's ice-breaker worksheet is on paper.

Required knowledge from Precalculus Here is the Precalculus Chapter in Paul's Notes.