# Lecture Notes

MATH 124

## Navigation

- Week 1 Friday
- Week 2 Monday
- Week 2 Wednesday
- Week 2 Friday
- Week 3 Monday
- Week 3 Wednesday
- Week 3 Friday
- Week 4 Monday
- Week 4 Wednesday
- Week 4 Friday
- Week 5 Monday
- Week 5 Wednesday
- Week 6 Monday
- Week 6 Wednesday
- Week 6 Friday
- Week 7 Monday
- Week 7 Wednesday
- Week 8 Wednesday
- Week 8 Friday
- Week 9 Monday
- Week 9 Wednesday
- Week 10 Monday
- Week 10 Wednesday

## Week 1 Friday

Questions to Collect:

- What is it?
- How to compute it?
- Applications?
- Special cases?
- Properties

Tangent Lines to Circles

- Tangent Line intesects with circle exactly one time.
- Tool - make use of the fact that radial line through point of tangency and tangent line are perpendicular (\(m_t = -\frac{1}{m_r}\)).
- One problem in which point of tangency is given, another type in which the point of tangency is not given.
- When the point of tangency is given, we need to obtain three equations to solve for the three variables (\((a,b)\) - point of tangency and \(m_t\) - tangent line slope).
- Circle equation - \(a^2 + b^2 = r^2\)
- Slope of the tangent line - \(m_t = -\frac{a}{b}\)
- Slope of the tangent line with intersecting point \((h_x, h_y)\) - \(m_t = \frac{b-h_y}{a-h_x}\)m

- When the point of tangency is given, we need to obtain three equations to solve for the three variables (\((a,b)\) - point of tangency and \(m_t\) - tangent line slope).

Tangent Lines to Graphs of Functions

- Zoom in until graph does not show a curve.
- Instantaneous rate of change.
- Algebraically, begin with a secant line; note the slope; move the other point closer until you obtain a closer estimate.
- Take secant slope - \(\lim_{x\rightarrow}\frac{f(x) - f(a)}{x-a}\).

Do in-class problems for practice on quiz.

## Week 2 Monday

- Rise over run - \(\lim_{a \to 0} \frac{f(x) - f(a)}{x-a}\); the limit of the difference quotient.
- \(x\) is the point \(a\) gets closer too.

- On the Thursday quiz, you must be able to describe the secant line approach.
- Secant line - average rate of changes. Tangent line - instantaneous rate of change.
- Currently, our tools do not allow us to evaluate limits nicely.
- Today’s section: take away difference quotience and discuss limits in full generality.
- Limit of a function - look closer and closer to a certain point.
- Never look at the point itself.
- One-sided limits: \(\lim_{x\to a^+}\) vs \(]lim_{x \to a^-}\)
- Methods: tables, ommon sense, graphing.
- WHen does a limit not exist? When the positive and negative values are not equal, or when the limit is infinite. Consider \(\lim_{x\to a} \sin\left(\frac{1}{x}\right)\); gets ‘infinitely squished’ near \(x=0\) and we cannot discern any direction at all.
- \(x=a\) is the vertical asymptote in cases of infinite limits.

## Week 2 Wednesday

- The tangent line slope can only be found if we know how to calculate limits.
- First, identify the limit type and observe the simplified outcome - this is \(\frac{0}{0}\) type, this is \(c\) type, etc.
- If plugging in is defined, use the limit laws to solve; a large limit problem can be split into smaller chunks as long as everything is defined.
*Direct Substitution Property*: if \(f\) is an algebraic function and \(a\) is in the domain, then \(\lim_{x\to a} f(x) = f(a)\).

- If plugging in obtains something weird - infinity, dividing by zero, etc.
- If \(\frac{c}{0}\) case, check vertical asymptotes.
- If is \(0\times\infty\), \(\infty-\infty\), \(\frac{\pm\infty}{\pm\infty}\), or \(\frac{0}{0}\) type, use algebra to transofrm the limit into a different type.

- Algebraic tools: factorization, rationalization, simplification, common denominator, squeeze theorem.
- Sandwich/Squeeze Theorem: if two squeezing functions have the same limit at some point, the squeezed function must also have the same limit at that point.

## Week 2 Friday

- Continuity of functions at a place \(x=a\).
- \(f\) is continuous at \(a\) if the graph does not break at \(a\).
- \(f\) is continuous at \(a\) if \(\lim_{x\to a} f(x) = f(a)\).
- \(f\) is continuous if it is continuous at all places onthe domain.

- New interpretation of DSP: algebraic functions are continuous on their domain.
Example: $$\frac{ x+3 }{x+3}\(is continuous for all\)x\neq -3$$ because it is a well-defined combination of well defined continuous functions.

## Week 3 Monday

*Asymptotos*- never falling together.- We already have vertical asymptotes in our toolbox; we can similar discuss horizontal asymptotes.
- The only space for the graph to get closer and closer but never touch is to \(-\infty\) and \(\infty\).
- Horizontal asymptotes must come with \(x\to\pm\infty\).
- Understand the behavior of a function far to the right and far to the left; a function can have at most two horizontal asymptotes.
- If \(f(x)\) becomes constant to one side, the outcome of the one-sided limits to infinity are the respective horizontal asymptotes of the function \(f(x)\).
- You can have an asymptote on one side and none on the other.
- Algebraic tools to find limits to infinity:
- In a polynomial, the highest power dominates the long term behavior.
- Rational functions - the highest power dominates.
- \(e^x\) dominaates over every \(x^r\).
- Limit types \([\infty - \infty]\) and \(\left[\pm\frac{\infty}{\infty}\right]\) need more algebra toget a different type.

- Special limits to know:
- \(\frac{1}{x^r}\) - horizontal and vertical asymptote at \(y=0\) and \(x=0\) for \(r>0\).
- \(\lim_{x\to\infty} x^r = \infty\), \(\lim_{x\to-\infty} x^r = -\infty\) for odd \(r\).
- \(\lim_{x\to\infty} x^r = \lim_{x\to-\infty} x^r = \infty\) for even \(r\).

## Week 3 Wednesday

- Quiz grades are out; mistakes may have been made by graders.
- Limits you cannot solve yet - when there is something like \(\lim_{x\to 0} \frac{\sin(x)}{x}\) or \(\lim_{x\to 0} \frac{x}{\sin(x)}\).
- Begin sharing solutions on Ed.
- Originally, we were interested in finding the tangent line to a graph.

The tangent line to a curve \(y = f(x)\) at the point \(P(a, f(a))\) is the line through \(P\) with slope \(f'(a) = \lim_{x\to a} \frac{f(x) - f(a)}{x-a} = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}\). \(f'(a)\) is the

derivativeof \(f\) at \(a\).

- Rephrasing of original slope definition using \(\lim_{h\to 0}\).
- Will always be a \(\left[\frac{0}{0}\right]\) type; transform it into a different type of canceling the \(h\).
*It is easier to cancel out \(h\) than \(x-a\).* - Tangent line equation of \(f\) at \((a, f(a))\): \(y - f(a) = f'(a)(x-a)\).
- Average rate of change - secant. Instantaneous rate of change - derivative.
- Units of a derivative are \(\frac{y \text{ unit}}{x \text{ unit}}\).
- The “tangent” line to a line is equal to itself.

## Week 3 Friday

- We are interested in the instantaneous rate of change.
- We can find tangent line slopes if the function allows it so far.
- The derivative is the slope of the tangent line at that point.
- Reminder: \(f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}\).

- Repeat process with \(f'(x)\) and get the second derivative \(f''(x)\), etc. - higher order derivatives.
- You will stil look to pull out an \(h\) somewhere when finding the derivative using the limit definition.
- \(f'(x)\) is now a function.
- Generalizaiton: find the limit once and use it to find the slope of a tangent line at any point.
*Differentiability*- \(f\) is differentiable at \(a\) if \(f'(a)\) exists.- \(f\) is differentiable on the interval \((a,b)\) if it is differentiable at every position in \((a,b)\).
- How can a function fail to be differentiable? If it is not continuous, if there is a corner, vertical tangent line. vertical tangents like in \(\sqrt{x}\) and \(x^\frac{1}{3}\).
- Asymptotic or break discontinuities.
- Identifying the derivative for graphs:
- Top of mountain/bottom of valley: \(f'(x) = 0\)
- \(f\) increasing: \(f'(x) > 0\)
- \(f\) decreasing: \(f'(x) < 0\)

## Week 4 Monday

- Midterm will cover everything including Section 3.4 (not including chain rule).
- Sum of two functions - \((f(x) \pm g(x))' = f'(x) \pm g'(x)\).
- \[(cf(x)' = cf'(x)\]
- \[(x^n)' = nx^{n-1}\]
- Product Rule and Quotient Rule
- \[c' = 0\]
- Finding \(e\): \(\lim_{h\to 0}\frac{a^{x+h} - a^x}{h} = \lim_{h\to 0}\frac{a^x(a^h - 1)}{h} = a^x \lim_{h\to 0} \frac{a^h - 1}{h} \implies \lim_{h\to 0} \frac{a^h - 1}{h}\). We can observe that between \(a=2\) and \(a=3\), there is a number \(e\) such that \(\lim_{h\to 0} \frac{e^h - 1}{h} = 1\). Thus, \(\frac{d}{dx} e^x = e^x \cdot \lim_{h\to 0} \frac{e^h - 1}{h} = e^x \cdot 1 = e^x\).

## Week 4 Wednesday

### Administrative

- Midterm will include all content until 3.2.
- 80 minutes long
- 8 problems
- There will be a large range of problems, easy and difficult.
- Only get stuck on a problem for a few minutes; don’t spend too much time on just one problem. Be smart about your time.

### Notes

- Product rule and quotient rule.
- We can understand \(\left(\frac{f}{g}\right)' = \left(f \cdot \frac{1}{g}\right)'\).
- You don’t need to simplify it if the problem asks only for the derivative; if you need to use the derivative it is probably wise to further simplify it.

## Week 4 Friday

- Look at Midterm 1 revieew
- Whenever a ‘co-‘ is involved (cotangent, cosine, cosecant), the derivative has a negative sign. Otherwise, there is no negative sign (tangent, sine).
- \[\lim_{h\to 0} \frac{\sin h}{h} = 1; \lim_{h\to 0} \frac{\cos h}{h} = 1\]
- e.g.: \(\lim_{x\to 0}{\frac{\sin(7x)}{11x}\) - multiply numerator and denominator by \(\frac{7}{11}\) and apply above limit rule.

## Week 5 Monday

\(\lim_{x\to 2} \frac{(x-2)(x+1)}{x^2 - 4} = \frac{3}{4}\)

\(\lim_{x\to 2} \frac{(x-2)(x+1)}{x^2 - 4x + 4} = \lim_{x\to 2} \frac{x+1}{x-2}\) - we know a \(\left[\frac{c}{0}\right]\) results in \(\infty\), \(-\infty\), or \(DNE\). In this case, one-sided limit analysis concludes that \(\lim_{x\to2^+} \to \infty\) and \(\lim_{x\to2^- \to -\infty\), yielding \(DNE\).

*Be careful when pulling terms into an even root, which does not preserve input sign.*

When you have a \(\frac{c}{x^n}\) situation, rewrite as \(cx^{-n}\) and use power rule to find derivative.

## Week 5 Wednesday

\(f\) concave up \(\implies\) \(f'(x)\) increasing \(\implies\) \(f''(x) > 0\)

Chain rule: \(f(g(x))' = f'(g(x)) g'(x)\), or \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}, f(u) = f(g(x))\).

## Week 6 Monday

- Quiz on Thursday - mainly about derivatives.

Parametric equations: see the parameter \(t\) as time and \(x\) and \(y\) as positions.

- \(x\) and \(y\) are presented in terms of \(t\); they represent the position in the coordinate system.
- To obtain a curve for the path in the \(xy\) coordinate system, eliminate \(t\).
- You can use \(\sin^2 t + \cos^2 t = 1\) trigonometric identity to eliminate \(t\).
- If the time is restricted \(a\le t\le b4\), \((x(a), y(a))\) is the initial point; \((x(b), y(b))\) is the terminal point.
- You need to memorize the generic form of uniform linear motion - pair of equations. Task is typically to determine \(a, b, c, d\).
- Uniform circular motion - much hated.
- \(\omega = \frac{\Delta \theta}{\Delta t}\) - angular speed.

- Use Pythagorean theorem to find distance; use Pythagorean theorem on horizontal and vertical instantaneous velocities to calculate speed at a point.

## Week 6 Wednesday

- Parametric equation - \(x\) and \(y\) are separate equations, written as a function of a third variable \(t\). Together, \(x\) and \(y\) represent a point in the coordinate system.
- We can find te derivative \(\frac{dy}{dx}\) aas \(\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
- The tangent line equation is \(y = \frac{dy}{dx} (x - x_1) + y_1\).
- Horizontal tangent line - when \(\frac{dy}{dt} = 0\). Vertical tangent line - when \(\frac{dx}{dt} = 0\).

## Week 6 Friday

- Logarithmic differentiation: useful when many factors/quotients and ane xponent are involved.
- If you have an expression with an \(x\) in the base and in the exponent.
- \(\ln(x^b) = b\ln(x)\) - useful property of logarithms.
- In other cases, it is convenient rather than required to use logarithmic differentiation by using properties like \(\ln(xy) = \ln(x) + \ln(y)\).

## Week 7 Monday

- Relating desired quantities to one another.
- Use geometric tools to establish the mathematical relationship - areas, volume, surface of common solid, Pythagorean Theorem, similar triangles, trigonometric laws, uniform linear and circular motions, etc.
- In midterms, you will have to make a sketch and properly label it.
- When implicitly differentiating, we are chasing \(t\); everything else is a function of \(t\).

## Week 7 Wednesday

- Linear approximation, linearization.
- We can approximate a value by walking on a tangent line rather than one the function itself.

## Week 8 Wednesday

Collect information about

- horizontal asymptotes
- critical numbers
- \(f''(x) = 0\) or DNE
- sign of \(f'(x)\) (monotonicity4$ and \(f''(x)\) (concavity) in between critical numbers and \(f''(x) = 0\) or DNE

## Week 8 Friday

- Works only for 0/0 or infty/infty cases. However, not save-the-world universal approach.
- When working with roots, can be difficult to make simpler with L’Hopital.
- Indeterminate limit types: \(0 \cdot \pm \onifty, \infty - \infty, 1^\infty, 0^0, \infty^0\). Require more work, “forbidden types”.
- \([0\cdot \pm\infty] \cdot fg = \frac{f}{1/g}\) or \(fg \cdot \frac{g}{1/f}\)
- \([\infty - \infty]\): common denominator or factorization. \(f - g = f(1 - \frac{g}{f}) = g(\frac{f}{g} - 1)\).
- Exponentials involving \(1, 0, \infty\): use logarithmic approach.

## Week 9 Monday

- Symmetry is helpful. However, remember that most functions are not symmetric. Evaluate symmetry with respect to origin, too.
- If you don’t get too many points - mainly concavity and an asymptote, etc.; feel free to compute more points and add them to the coordinate system.

## Week 9 Wednesday

Preparing for final exam:

- Go to the test archive and begin doing all the finals. Our final will be very similar.
- Related rates and optimization problems can vary quite a bit. Other questions are quite predictable for the majority of the problems.
- For the first practice midterm exam, consider it to be a review. Go back to respective chapters and read.
- Derivative problems should be done in ~ 10 minutes, very quickly.

Tips:

- Don’t worry about simplifying your derivative. If you simplify and make a mistake, points will be deducted. Find the derivative and you’re good.

## Week 10 Monday

- Quiz on Thursday will be about graphing.
- There will be a quiz next Tuesday - cover broad problems.
- Goal of finding derivative using definition of a derivative - cancel out an \(h\).
- Asymptotic behavior, evening out.
- Monotonicity and increasing/decreasing attributes.
- Verical asymptotes - denominator zero, log argument is 0.
**Critical numbers must be in the domain of the function.**

## Week 10 Wednesday

- Optimization problems - application of derivative skills. Find minimm or maximum of a function (e.g. cost, surface area, distance).
- Reading task - identify the quantity we need to minimize or maximize.
- Find a formula for what we wnat ot minimize or maximize.
- Solve the formula for one of the variables and plug into the optimized function by taking the derivative and setting to zero.