Untitled

Average Velocity

The change in position divided by the change in time: $v_{\text{avg}}=\frac{f(b)-f(a)}{b-a}$ It represents the slope of the secant line between two points.

Instantaneous Velocity

The velocity at one specific instant, which is the derivative of position: $v(t)=s'(t)$ It represents the slope of the tangent line at a point.

Difference between Average and Instantaneous Velocity

Average velocity is over an interval (secant line); instantaneous velocity is at one moment (tangent line).

Instantaneous Velocity using a Limit

$v(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$ This is the definition of the derivative.

\sin^{-1}(x) Meaning

Inverse sine (arcsin), not $\frac{1}{\sin(x)}$. It answers: 'What angle has sine equal to x?'

\cos^{-1}(x) Meaning

Inverse cosine: $\cos^{-1}(x)=\arccos(x)$ It asks: 'What angle has cosine equal to x?'

\tan^{-1}(x) Meaning

Inverse tangent: $\tan^{-1}(x)=\arctan(x)$ It asks: 'What angle has tangent equal to x?'

Domains of Inverse Trig Functions

$\arcsin(x): [-1,1]$, $\arccos(x): [-1,1]$, $\arctan(x): (-\infty,\infty)$

SOH-CAH-TOA

SOH: $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$; CAH: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$; TOA: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$.

Relationship between Tangent, Sine, and Cosine

$\tan(x)=\frac{\sin(x)}{\cos(x)}$; tangent is undefined whenever $\cos(x)=0$.

Reciprocal Trig Functions

$\csc x=\frac{1}{\sin x}, \quad \sec x=\frac{1}{\cos x}, \quad \cot x=\frac{1}{\tan x}$.

Sine Positive Quadrants

Quadrants I and II.

Cosine Positive Quadrants

Quadrants I and IV.

Tangent Positive Quadrants

Quadrants I and III.

ASTC Rule

Moving counterclockwise: I: All positive; II: Sine positive; III: Tangent positive; IV: Cosine positive. Mnemonic: All Students Take Calculus.

Pythagorean Trig Identity

$\sin^2x+\cos^2x=1$.

Identities from Pythagorean Identity

Divide by $\cos^2x$: $\tan^2x+1=\sec^2x$; Divide by $\sin^2x$: $1+\cot^2x=\csc^2x$.

Reciprocal Identities

$\sin x=\frac{1}{\csc x}, \quad \cos x=\frac{1}{\sec x}, \quad \tan x=\frac{1}{\cot x}$.

Quotient Identity for Tangent

$\tan x=\frac{\sin x}{\cos x}$.

Limit Case When Denominator Higher Degree

The limit is: $0$ as $x \to \infty$.

Limit Case Same Degree

The limit is the ratio of the leading coefficients.

Limit Case Numerator Higher Degree

The function approaches: $\pm \infty$ or may have no finite horizontal asymptote.

Degree Rule for Rational Functions

Top < Bottom: 0; Top = Bottom: Ratio of leading coefficients; Top > Bottom: Usually $\pm \infty$ or polynomial-type growth.

Basic Limits of Sine and Cosine

$\lim_{x \to \infty}\sin x$ and $\lim_{x \to \infty}\cos x$ do not exist because they continually oscillate.

Limit of Tangent at Infinity

$\lim_{x \to \infty}\tan x$ does not exist because tangent repeatedly becomes undefined.

Important Trig Limit in Calc 1

$\lim_{x \to 0}\frac{\sin x}{x}=1$; measure x in radians.

Another Important Trig Limit

$\lim_{x \to 0}\frac{\tan x}{x}=1$; angles must be in radians.

Evaluating Limit with Sine

Substitute: $\sin(a)$; sine and cosine are continuous everywhere.

Evaluating Limit with Cosine

Direct substitution works: $\lim_{x \to a}\cos x=\cos(a)$.

Evaluating Limit with Tangent

Directly evaluate when $\tan(a)$ is defined.

Constant Limit Law

$\lim_{x\to a}c=c$.

Sum Law

$\lim[f(x)+g(x)]=\lim f(x)+\lim g(x)$.

Difference Law

$\lim[f(x)-g(x)]=\lim f(x)-\lim g(x)$.

Product Law

$\lim[f(x)g(x)]=\left(\lim f(x)\right)\left(\lim g(x)\right)$.

Quotient Law

$\lim\frac{f(x)}{g(x)}=\frac{\lim f(x)}{\lim g(x)}$ provided $\lim g(x)\neq0$.

Power Law

$\lim[f(x)]^n=\left[\lim f(x)\right]^n$.

Absolute Value Meaning

Represents distance from zero; always nonnegative: $|x|\ge0$.

Absolute Value as Piecewise Function

$|x|=\begin{cases} x,&x\ge0 \\ -x,&x<0 \end{cases}$.

Evaluating Limit with Absolute Value

$\lim_{x \to a}|f(x)|=|L|$ if $\lim_{x \to a}f(x)=L$.

Sign Change with Absolute Value

Check whether the expression changes sign around $a$, for instance in $|x-a|$.

Piecewise Function Definition

A function defined using different formulas over different intervals.

Evaluating a Piecewise Function

Determine the condition that contains your x-value, then use only that formula.

Common Mistake in Evaluating Piecewise Functions

Using the wrong piece; always check the inequality first.

Evaluating Limit at Boundary of Piecewise Function

Calculate left-hand limit: $\lim_{x\to a^-}f(x)$ and right-hand limit: $\lim_{x\to a^+}f(x)$; compare them.

Existence of a Two-Sided Limit

$\lim_{x\to a}f(x)\text{ exists}$ only if $\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)$.

Does f(a) Have to Equal the Limit?

No, the function’s actual value at $a$ can be different from the limit.

When to Use L'Hôpital's Rule

When direct substitution produces indeterminate forms: $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$.

What L'Hôpital's Rule States

If the limit produces $0/0$ or $\infty/\infty$: $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$.

First Step Before Using L'Hôpital's Rule

Try direct substitution first; do not automatically differentiate.

Applying L'Hôpital's Rule Again

You can apply L'Hôpital's Rule again if the resulting limit still satisfies the conditions.

Using L'Hôpital's Rule on 0\cdot\infty

Not directly; rewrite as a quotient first.

Using L'Hôpital's Rule on \infty - \infty

No; rewrite the expression into a quotient first.

When to Consider Using a Conjugate

When a limit contains radicals and direct substitution produces $\frac{0}{0}$.

Conjugate of \sqrt{x}+3

$\sqrt{x}-3$.

Useful Identity for Conjugates

$(a+b)(a-b)=a^2-b^2$.

Basic Steps for Radical Limit with Conjugate

Substitute, identify radical, multiply by conjugate, simplify, cancel common factor, substitute again.

Vertical Asymptote Definition

A vertical line: $x=a$ where the function approaches $+\infty$, $-\infty$, or becomes unbounded.

Finding Vertical Asymptotes

Factor numerator and denominator, cancel common factors only if appropriate, set remaining denominator to zero.

Difference Between a Hole and Vertical Asymptote

If a factor cancels, it creates a hole; if denominator remains zero after simplifying, it creates a vertical asymptote.

Horizontal Asymptote Definition

A horizontal line: $y=L$ that the function approaches as $x\to\infty$ or $x\to-\infty$.

Finding Horizontal Asymptotes

Compare degrees of numerator and denominator.

Horizontal Asymptote when Num Degree < Den Degree

$y=0$.

Horizontal Asymptote when Num Degree = Den Degree

Ratio of leading coefficients.

Horizontal Asymptote when Num Degree > Den Degree

Generally no horizontal asymptote may exist.

Six Basic Trig Functions

$\sin x, \quad \cos x, \quad \tan x, \quad \csc x, \quad \sec x, \quad \cot x$.

When Sine Equals Zero

$\sin x=0$ at $x=n\pi$ where $n$ is any integer.

When Cosine Equals Zero

$\cos x=0$ at $x=\frac{\pi}{2}+n\pi$.

When Tangent is Undefined

$\tan x$ is undefined when $\cos x=0$, hence $x=\frac{\pi}{2}+n\pi$.

Ranges of Sine and Cosine

$-1\le\sin x\le1$ and $-1\le\cos x\le1$.

Is Tangent Bounded?

No, $-\infty<\tan x<\infty$ where it is defined.

Period of Sine and Cosine

$2\pi$.

Period of Tangent

$\pi$.

Amplitude of y=A\sin x or y=A\cos x

$|A|$.

General Form of Sine/Cosine Function

$y=A\sin(Bx-C)+D$, where: $|A| = \text{amplitude}, \frac{2\pi}{|B|} = \text{period}, \frac{C}{B} = \text{phase shift}, D = \text{vertical shift}$.

General Form of Tangent Function

$y=A\tan(Bx-C)+D$ with period $\frac{\pi}{|B|}$.

Limit Problem First Steps

Step 1 — Substitute. If normal number, DONE. If 0/0, try factoring/simplifying/conjugate/trig identities/L'Hôpital's Rule. If $\frac{\infty}{\infty}$, consider highest power/$L'Hôpital's Rule$.

Important Infinity Rules for Rational Functions

Compare powers: Bottom bigger → 0; Same power → leading coefficient ratio; Top bigger → no finite horizontal asymptote.

What to Memorize First for Calc 1 Quiz

Limit laws, direct substitution, 0/0 scenarios, limits at infinity, piecewise left/right limits, basic trig functions + unit circle, identities, vertical/horizontal asymptotes.