Coverage: Lessons 1–10

Exam Date and Time: Wednesday, September 23, 8:00–9:00 pm

Locations: LILY 1105, MA 175, WALC 1055. Refer to the posted seating chart for your location.

Review of Functions [Lesson 1]

Video Notes Practice

Functions, Domain, and Range

A function \(f\) assigns each input \(x\) a unique output \(f(x)\). The domain is the set of admissible inputs; the range is the set of outputs.

\[\begin{gathered} \text{Dom}(f\pm g)=\text{Dom}(fg)=\text{Dom}(f)\cap\text{Dom}(g)\\[4pt] \text{Dom}(f/g)=\text{Dom}(f)\cap\text{Dom}(g)\cap\{g(x)\neq 0\} \end{gathered}\]

Two families are common enough to know by sight: the domain of a polynomial is all of \((-\infty,\infty)\), and the domain of a rational function \(f(x)=p(x)/q(x)\) is every \(x\) with \(q(x)\neq0\).

Vertical Line Test

A curve is the graph of a function if and only if every vertical line crosses it at most once.

Composition and Symmetry

\[(f\circ g)(x) = f(g(x))\]

In general, \(f(g(x)) \neq g(f(x))\). The composition \((f\circ g)(x)\) is defined precisely when the range of \(g\) is contained in the domain of \(f\) — always check this before composing.

Even function: \(f(-x)=f(x)\) — graph is symmetric about the \(y\)-axis.

Odd function: \(f(-x)=-f(x)\) — graph is symmetric about the origin.

Transformations of \(y=f(x)\)

Transformation Rules
Transformation Effect
\(y=f(x)+k\)Vertical shift \(k\) units: up if \(k>0\), down if \(k<0\)
\(y=f(x-k)\)Horizontal shift \(k\) units: right if \(k>0\), left if \(k<0\)
\(y=kf(x)\)Vertical stretch if \(|k|>1\), compression if \(0<|k|<1\)
\(y=f(kx)\)Horizontal compression if \(|k|>1\), stretch if \(0<|k|<1\)
\(y=f(-x)\)Reflect across the \(y\)-axis
\(y=-f(x)\)Reflect across the \(x\)-axis

Practice Problems

Inverse Functions and Logarithms [Lesson 2]

Video Notes Practice

Horizontal Line Test

A function is one-to-one, and therefore invertible, if and only if every horizontal line crosses its graph at most once. If a function is not 1–1 on its whole domain, restrict the domain to a piece where it is.

Finding \(f^{-1}\)

By definition, \(y=f^{-1}(x) \iff f(y)=x\), which gives the round-trip identities \(f^{-1}(f(x))=x\) and \(f(f^{-1}(y))=y\). To find \(f^{-1}\) explicitly: (i) solve \(y=f(x)\) for \(x\) to get \(x=f^{-1}(y)\); (ii) interchange \(x\) and \(y\). The graph of \(f^{-1}\) is the reflection of the graph of \(f\) across \(y=x\).

Exponential and Logarithmic Functions

\(y=b^x\) (\(b>0,\ b\neq1\)): domain \((-\infty,\infty)\), range \((0,\infty)\)

\(y=\log_b x\): domain \((0,\infty)\), range \((-\infty,\infty)\). The natural log is \(\ln x=\log_e x\).

Law of Exponents: \(b^{x+y}=b^xb^y, \quad b^{x-y}=\dfrac{b^x}{b^y}, \quad (b^x)^r=b^{xr}\)

Log Rules

  • \(\log_b(xy)=\log_b x+\log_b y\)
  • \(\log_b(x/y)=\log_b x-\log_b y\)
  • \(\log_b(x^r)=r\log_b x\)
  • \(\log_b b^x=x, \quad b^{\log_b x}=x\)
  • Change of base: \(\log_b x = \dfrac{\ln x}{\ln b}\)

Practice Problems

Trigonometry Review and Inverse Trig Functions [Lesson 3]

Video Notes Practice

Right-Triangle Ratios and the Unit Circle

\[\sin\theta=\frac{\text{opp}}{\text{hyp}}, \quad \cos\theta=\frac{\text{adj}}{\text{hyp}}, \quad \tan\theta=\frac{\sin\theta}{\cos\theta}\]

Unit circle: \((\cos\theta,\sin\theta)\) is the point at angle \(\theta\). Radians: \(2\pi\) rad \(=360^\circ\); arc length \(s=r\theta\).

Key Trig Identities

\[\begin{gathered} \sin(\theta+2\pi)=\sin\theta,\quad \cos(\theta+2\pi)=\cos\theta\\ \tan(\theta+\pi)=\tan\theta\\[6pt] \cos(-\theta)=\cos\theta\ \text{(even)}\\ \sin(-\theta)=-\sin\theta\ \text{(odd)}\\[6pt] \sin^2\theta+\cos^2\theta=1\\ \tan^2\theta+1=\sec^2\theta,\quad 1+\cot^2\theta=\csc^2\theta \end{gathered}\]

Double-angle / power-reduction: \(\cos2\theta=\cos^2\theta-\sin^2\theta\), \(\sin2\theta=2\sin\theta\cos\theta\), \(\cos^2\theta=\dfrac{1+\cos2\theta}{2}\), \(\sin^2\theta=\dfrac{1-\cos2\theta}{2}\)

General Sine Wave

For \(f(\theta)=a\sin(b(\theta-c))+d\): amplitude \(|a|\), period \(2\pi/|b|\), horizontal shift \(c\), vertical shift \(d\).

Inverse Trig Functions

Domain and Range (Restricted for Invertibility)
FunctionDomainRange
\(\sin^{-1}x\)\([-1,1]\)\([-\pi/2,\pi/2]\)
\(\cos^{-1}x\)\([-1,1]\)\([0,\pi]\)
\(\tan^{-1}x\)\((-\infty,\infty)\)\((-\pi/2,\pi/2)\)

Reference-triangle technique: to simplify expressions like \(\sin(\cos^{-1}x)\), let \(\theta=\cos^{-1}x\), so \(\cos\theta=x/1\); draw a right triangle with adjacent \(x\) and hypotenuse \(1\), so the opposite side is \(\sqrt{1-x^2}\) by the Pythagorean theorem. Then \(\sin(\cos^{-1}x)=\sin\theta=\sqrt{1-x^2}\). The same idea — label the triangle from the inner inverse-trig definition, then read off the requested ratio — handles any composition like \(\cos(\tan^{-1}x)\).

Practice Problems

Velocity, Definition of Limit, One-Sided Limits [Lesson 4]

Video Notes Practice

Average and Instantaneous Velocity

\[\text{Average velocity on } [a,b]: \quad \frac{f(b)-f(a)}{b-a}\]

Instantaneous velocity at \(a\) is the limit of average velocity as the interval shrinks to \(a\).

The same quotient is also the slope of the secant line through \((a,f(a))\) and \((b,f(b))\): \(m_{\text{sec}}=\dfrac{f(b)-f(a)}{b-a}\). Letting \(b\to a\) turns the secant into the tangent line, with slope \(m_{\text{tan}}=\displaystyle\lim_{b\to a}\dfrac{f(b)-f(a)}{b-a}\) — the same limiting process as instantaneous velocity, just viewed geometrically.

Definition of a Limit

\(\lim_{x\to a}f(x)=L\) means \(f(x)\) gets arbitrarily close to \(L\) as \(x\) gets close to \(a\) (from either side), regardless of \(f(a)\) itself.

One-Sided Limits and Existence

\[\lim_{x\to a}f(x)=L \iff \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\]

The value \(f(a)\) is irrelevant to whether \(\lim_{x\to a}f(x)\) exists — only the behavior of \(f\) near \(a\) matters. If the one-sided limits disagree, the two-sided limit does not exist (DNE).

Practice Problems

Computing Limits, Limit Laws, Squeeze Theorem [Lesson 5]

Video Notes Practice

Limit Laws

For \(\lim f=L\), \(\lim g=M\): sum/difference \(L\pm M\); product \(LM\); quotient \(L/M\) (\(M\neq0\)); power \(L^n\); constant multiple \(cL\). For a polynomial, \(\lim_{x\to a}P(x)=P(a)\); for a rational function with \(Q(a)\neq0\), \(\lim_{x\to a}\frac{P(x)}{Q(x)}=\frac{P(a)}{Q(a)}\).

0/0 Indeterminate Forms

Factor and cancel the common factor, or rationalize (multiply by the conjugate) when a square root is involved, then substitute.

Squeeze Theorem

If \(f(x)\le g(x)\le h(x)\) for all \(x\) near \(a\) (except possibly at \(a\) itself) and \[\lim_{x\to a}f(x)=\lim_{x\to a}h(x)=L,\] then \(\displaystyle\lim_{x\to a}g(x)=L\) as well.

Practice Problems

Infinite Limits & Vertical Asymptotes [Lesson 6]

Video Notes Practice

Vertical Asymptotes

\(x=a\) is a vertical asymptote (VA) of \(f\) if any of \(\lim_{x\to a}f(x)\), \(\lim_{x\to a^+}f(x)\), or \(\lim_{x\to a^-}f(x)\) is infinite (\(+\infty\) or \(-\infty\)).

For a rational function \(f=p(x)/q(x)\) in lowest terms, vertical asymptotes occur exactly at the zeros of \(q(x)\). If a factor cancels between numerator and denominator, that point is a hole (removable), not a VA.

Common mistake: setting only the denominator to zero without checking whether the same factor cancels with the numerator.

Practice Problems

Limits at Infinity & Horizontal/Slant Asymptotes [Lesson 7]

Video Notes Practice

Limits at Infinity of Powers and Common Functions

\[\begin{gathered} \lim_{x\to\pm\infty}x^n=\infty\ \ (n\text{ even})\\[4pt] \lim_{x\to\infty}x^n=\infty,\quad \lim_{x\to-\infty}x^n=-\infty\ \ (n\text{ odd})\\[4pt] \lim_{x\to\pm\infty}\frac{1}{x^n}=0\ \ (n\ge1)\\[8pt] \lim_{x\to\infty}e^x=\infty,\quad \lim_{x\to-\infty}e^x=0\\[4pt] \lim_{x\to\infty}e^{-x}=0,\quad \lim_{x\to-\infty}e^{-x}=\infty\\[4pt] \lim_{x\to0^+}\ln x=-\infty,\quad \lim_{x\to\infty}\ln x=\infty \end{gathered}\]

A polynomial's end behavior is determined entirely by its leading term (the highest power dominates as \(x\to\pm\infty\)). Also useful: \(\lim_{x\to\infty}\tan^{-1}x=\pi/2\) and \(\lim_{x\to-\infty}\tan^{-1}x=-\pi/2\), so \(y=\pm\pi/2\) are horizontal asymptotes of \(\tan^{-1}x\).

End Behavior of Rational Functions

Compare degrees of \(p(x)\) and \(q(x)\) in \(f=p(x)/q(x)\):

If \(f=p(x)/q(x)\) is in reduced form (no common factors), its vertical asymptotes are precisely the zeros of \(q(x)\) — see Lesson 6.

Square roots: for \(x>0\), \(\sqrt{x^2}=x\); for \(x<0\), \(\sqrt{x^2}=-x\) (i.e., \(\sqrt{x^2}=|x|\)). This sign flip matters when evaluating limits as \(x\to-\infty\) with a square root in the expression.

Practice Problems

Continuity and Discontinuities [Lesson 8]

Practice

Definition of Continuity

\(f\) is continuous at \(x=a\) if and only if all the following three conditions are satisfied:
(1) \(f(a)\) is defined;
(1) (2) \(\lim_{x\to a}f(x)\) exists (and is finite);
(1) (3) \(\lim_{x\to a}f(x)=f(a)\)

Polynomials, rational functions (away from zeros of the denominator), compositions of continuous functions, inverse functions, and inverse trig functions are all continuous on their domains; a piecewise function is continuous wherever each piece is continuous and the pieces agree at the break points.

Types of Discontinuity

Limits of Composite Functions

(a) If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a)\), then \[\lim_{x\to a}f(g(x)) = f\!\left(\lim_{x\to a}g(x)\right) = f(g(a)).\]

(b) More generally, if \(\lim_{x\to a}g(x)=L\) and \(f\) is continuous at \(L\), then \[\lim_{x\to a}f(g(x)) = f\!\left(\lim_{x\to a}g(x)\right) = f(L).\]

Intermediate Value Theorem (IVT)

If \(f\) is continuous on \([a,b]\) and \(L\) is any value between \(f(a)\) and \(f(b)\), then there is at least one \(c\in(a,b)\) with \(f(c)=L\).

Practice Problems

Introduction to Derivatives [Lesson 9]

Practice

Derivative at a Point

\[\begin{gathered} f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\\[6pt] =\lim_{h\to 0}\frac{f(a+h)-f(a)}{h} \end{gathered}\]

Three interpretations of \(f'(a)\): the slope of the tangent line to \(y=f(x)\) at \(x=a\); the (instantaneous) velocity at time \(a\), if \(f\) is a position function; and, in general, the instantaneous rate of change of \(f(x)\) at \(a\).

Tangent Line and Normal Line

Tangent line at \(a\) (point-slope form, slope \(m=f'(a)\)): \[y-f(a)=f'(a)(x-a)\]

Normal line at \(a\) (perpendicular to the tangent, when \(f'(a)\neq0\)): \[y-f(a)=-\frac{1}{f'(a)}(x-a)\]

The normal line is perpendicular to the tangent line at the same point; if the tangent has slope \(m\), the normal has slope \(-1/m\).

Practice Problems

The Derivative as a Function [Lesson 10]

Practice

Definition

\[f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]

This is a new function giving the tangent slope at every \(x\). A function is differentiable at \(x\) if this limit exists there.

Key implication: if \(f\) is differentiable at \(x=a\), then \(f\) is continuous at \(a\). The converse is false — a function can be continuous at a point without being differentiable there (e.g. \(f(x)=|x|\) at \(x=0\)).

When \(f'(a)\) Fails to Exist

\(f'(a)\) fails to exist when there is a vertical tangent at \(a\); \(f\) is discontinuous at \(a\); or there is a corner/cusp at \(a\) (left- and right-hand derivatives disagree, as with \(f(x)=|x|\) at \(x=0\)).

Graphical Relationship

Where \(f\) is increasing, \(f'>0\); where \(f\) is decreasing, \(f'<0\); where \(f\) has a horizontal tangent, \(f'=0\).

Practice Problems