Study Guide: Trigonometry and Limits (Calculus I, Sections 2.1–2.3)

Trigonometric Functions and Their Graphs

Graphing Sine, Cosine, and Tangent Functions

Understanding the graphs of the basic trigonometric functions is essential for calculus. Each function has unique characteristics, including periodicity, amplitude, and key points such as intercepts and extrema.

• Sine and Cosine: Both functions are periodic with period $2\pi$.

• Tangent: The tangent function has period $\pi$ and vertical asymptotes where the function is undefined.

• Key Points: Identify x-intercepts, maximum and minimum values, and the behavior within one period.

Example: The graph of $y = \sin(x)$ passes through the origin, reaches a maximum of 1 at $x = \frac{\pi}{2}$, and a minimum of -1 at $x = \frac{3\pi}{2}$ within one period.

Trigonometric Values and Identities

Exact Values of Trigonometric Functions

It is important to compute the exact values of all six trigonometric functions for primary terminal points in quadrant I and use symmetry to find values in other quadrants.

• Primary Angles: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$

• Symmetry: Use reference angles and the unit circle to determine values in all quadrants.

Example: $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$, $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

Simplifying Trigonometric Expressions

Trigonometric identities are used to simplify expressions and solve equations.

• Reciprocal Identities: $\sin(x) = \frac{1}{\csc(x)}$, $\cos(x) = \frac{1}{\sec(x)}$, $\tan(x) = \frac{1}{\cot(x)}$

• Quotient Identities: $\tan(x) = \frac{\sin(x)}{\cos(x)}$, $\cot(x) = \frac{\cos(x)}{\sin(x)}$

• Pythagorean Identities: $\sin^2(x) + \cos^2(x) = 1$, $1 + \tan^2(x) = \sec^2(x)$, $1 + \cot^2(x) = \csc^2(x)$

Example: Simplify $\frac{1 - \cos^2(x)}{\sin(x)} = \frac{\sin^2(x)}{\sin(x)} = \sin(x)$

Average Velocity and Tangent Lines

Calculating Average Velocity

Average velocity is the change in position over the change in time, and can be computed from a table, function, or graph.

• Formula: $\text{Average velocity} = \frac{f(b) - f(a)}{b - a}$

Example: If $f(2) = 5$ and $f(5) = 20$, average velocity from $t=2$ to $t=5$ is $\frac{20-5}{5-2} = 5$ units/time.

Tangent Lines and Slope Estimation

The tangent line to a curve at a point represents the instantaneous rate of change (the derivative) at that point.

• Drawing: Sketch the tangent so it just touches the curve at the point of interest.

• Estimating Slope: Use nearby points to estimate the slope numerically.

Example: For $y = x^2$ at $x = 1$, the tangent line has slope $2$(since$f'(1) = 2$).

Functions: Characteristics and Graphs

Identifying Function Characteristics

Given a graph, identify properties such as average rate of change, limits, and vertical asymptotes.

• Average Rate of Change: Slope between two points on the graph.

• Limits: The value the function approaches as $x$ approaches a point.

• Vertical Asymptotes: Lines $x = a$ where the function grows without bound as $x$ approaches $a$.

Sketching Functions with Specified Properties

Be able to draw a function that meets given criteria, such as certain limits or asymptotes.

• Example: Sketch a function with a vertical asymptote at $x=2$ and a removable discontinuity at $x=1$.

Vertical Asymptotes

Solving for Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is nonzero.

• Find zeros of the denominator: Set denominator equal to zero and solve for $x$.

• Check numerator: Ensure numerator is not also zero at those points (if so, it may be a hole/removable discontinuity).

Example: $f(x) = \frac{1}{x-3}$ has a vertical asymptote at $x=3$.

Limits

Types of Limits Covered

Limits are foundational in calculus, describing the behavior of functions as inputs approach a value. The following types are emphasized:

• 0/0 Indeterminate Forms (Algebraic): Use algebraic manipulation (factoring, rationalizing) to resolve.

• Piecewise Functions: Evaluate limits from the left and right; check for continuity at the joining point.

• Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

• Limit Laws and Direct Substitution: Apply limit laws and substitute directly when possible.

• Oscillating Case: Some functions (e.g., $\sin(1/x)$ as $x \to 0$) do not have a limit due to oscillation.

• Non-zero over Zero (Vertical Asymptotes): If the numerator is nonzero and the denominator approaches zero, the limit may be $\infty$ or $-\infty$.

• Graphical Limits: Estimate limits by observing the graph near the point of interest.

Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ is a $0/0$ form; factor numerator: $\frac{(x-2)(x+2)}{x-2} = x+2$, so the limit is $4$.

Study and Exam Preparation Strategies

Effective Study Habits

• Begin reviewing at least a week before the exam.

• Work through all assigned homework, quizzes, and worksheets without notes to test understanding.

• Practice writing out full solutions, not just looking at answers.

• Use resources such as the Math Place, office hours, and study centers for additional help.

• Review exercises from the textbook and WebAssign, focusing on problems similar to those assigned.

Exam Expectations

• Show all work and reasoning for full credit.

• Be prepared to write answers in exact form (e.g., $\frac{\sqrt{3}}{2}$ instead of decimals).

• If unsure about the level of detail required, consult your instructor before the exam.

Summary Table: Types of Limits

Type of Limit	Method/Key Idea	Example
0/0 Indeterminate	Algebraic manipulation (factor, rationalize)	$\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
Piecewise Function	Check left and right limits	$f(x) = \begin{cases} x^2, & x < 0 \\ x, & x \geq 0 \end{cases}$
Squeeze Theorem	Bound function between two others	$\lim_{x \to 0} x^2 \sin(1/x) = 0$
Limit Laws/Direct Substitution	Apply limit laws, substitute directly	$\lim_{x \to 2} (3x + 1) = 7$
Oscillating Case	No limit due to oscillation	$\lim_{x \to 0} \sin(1/x)$ does not exist
Non-zero/Zero (Vertical Asymptote)	Limit approaches $\infty$ or $-\infty$	$\lim_{x \to 0^+} \frac{1}{x} = \infty$
Graphical	Estimate from graph	See graph of $f(x)$ near $x = a$

Additional info: This guide covers foundational calculus concepts from the beginning of a Calculus I course, focusing on trigonometric functions, limits, and introductory function analysis. Mastery of these topics is essential for success in subsequent calculus topics such as derivatives and integrals.