The Big Picture: From Foundations to the Derivative

This exam covers the foundational concepts of calculus, building from essential precalculus ideas to the formal definition of the derivative. The first part of the course is about building a solid base: reviewing functions, trigonometry, and logarithms that are the building blocks of more complex calculus problems.

The heart of this unit is the concept of the limit. We explore limits intuitively, graphically, and algebraically to understand how a function behaves near a point. This leads to the crucial idea of continuity. Finally, we use the limit to answer two fundamental geometric questions: "What is the slope of a curve at a single point?" (the tangent line problem) and "What is the instantaneous rate of change of a function?" The answer to both is the derivative, which we define formally using the limit of a difference quotient.

The best way to study is to practice problems. Questions on this exam will be similar in style and difficulty to those found in the homework, in-class worksheets, and the "Check Your Understanding" problems from the lecture notes.

Topic-by-Topic Breakdown

Week 1: Precalculus and Function Review

Key Concepts

  • A function describes a rule that assigns each input to exactly one output. Key properties include domain, range, and vertical line test.
  • New functions can be created from old ones through transformations (shifting, stretching, reflecting) and composition ($f \circ g$).
  • An inverse function, $f^{-1}(x)$, "undoes" the original function. A function must be one-to-one (pass the horizontal line test) to have an inverse.
  • Exponential functions ($b^x$) and logarithmic functions ($\log_b x$) are inverses of each other.
  • The unit circle provides the fundamental definitions for trigonometric functions.

Key Skills

  • Find the domain and range of a function.
  • Given the graph of $f(x)$, sketch transformations like $f(x+c)$, $f(x)+c$, $cf(x)$, and $-f(x)$.
  • Find the composition of two functions, $(f \circ g)(x)$.
  • Find the inverse of a one-to-one function algebraically.
  • Use the properties of logarithms to expand or condense logarithmic expressions.
  • Solve exponential and logarithmic equations.
  • Evaluate trigonometric and inverse trigonometric functions for key angles.

Section 2.1: The Tangent and Velocity Problems

Key Concepts

  • The tangent line problem asks for the slope of a curve at a single point.
  • The velocity problem asks for the instantaneous velocity of an object at a single moment in time.
  • Both problems are solved by the same underlying idea: finding the limit of average rates of change over progressively smaller intervals.

Key Skills

  • Estimate the slope of a tangent line by calculating the slope of secant lines over small intervals.
  • Estimate instantaneous velocity by calculating average velocities over small time intervals.

Section 2.2: The Limit of a Function

Key Concepts

  • The limit, $\lim_{x \to a} f(x) = L$, describes the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$, without necessarily reaching $a$.
  • The value of the limit at $a$ does not depend on the value of the function at $a$.
  • A limit exists if and only if both the left-hand limit and the right-hand limit exist and are equal.
  • An infinite limit occurs when function values increase or decrease without bound near a point, which corresponds to a vertical asymptote on the graph.

Key Skills

  • Estimate the value of a limit by creating a table of values for $x$ approaching $a$.
  • Determine the value of a limit by looking at a graph.
  • Determine when a limit does not exist (DNE) from a graph (due to a jump, oscillation, or infinite behavior).
  • Find and describe vertical asymptotes using infinite limits.

Section 2.3: Calculating Limits Using the Limit Laws

Key Concepts

  • The Limit Laws provide a systematic way to evaluate limits of functions built from simpler functions.
  • Direct Substitution Property: If $f$ is a polynomial or rational function and $a$ is in its domain, then $\lim_{x \to a} f(x) = f(a)$.
  • Recognizing the two special trigonometric limits: $\lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1$ and $\lim_{\theta \to 0} \frac{\cos\theta - 1}{\theta} = 0$.

Key Skills

  • Use the Limit Laws (sum, difference, product, quotient, power, etc.) to evaluate limits.
  • Evaluate limits using direct substitution when applicable.
  • Use algebraic techniques to evaluate limits that result in the indeterminate form $\frac{0}{0}$, such as:
    • Factoring and canceling common factors.
    • Multiplying by the conjugate to rationalize a numerator or denominator.
    • Simplifying complex fractions (by finding a common denominator).
  • Use the Squeeze Theorem to find the limit of a function trapped between two other functions.

Section 2.5: Continuity

Key Concepts

  • A function is continuous at a point $a$ if three conditions are met: (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists, and (3) the limit equals the function value.
  • Geometrically, a continuous function is one whose graph can be drawn without lifting your pencil.
  • Types of discontinuities: removable (hole), jump, and infinite.
  • The Intermediate Value Theorem (IVT) states that for a continuous function on $[a, b]$, the function must take on every y-value between $f(a)$ and $f(b)$.
  • Polynomials, rational functions, root functions, and trigonometric functions are all continuous on their domains.

Key Skills

  • Determine if a function is continuous at a point from a graph or a formula.
  • Identify the type and location of discontinuities.
  • Find the value of a parameter that makes a piecewise function continuous.
  • Use the Intermediate Value Theorem to show that a root exists within an interval.

Section 2.6: Limits at Infinity; Horizontal Asymptotes

Key Concepts

  • The limit at infinity, $\lim_{x \to \infty} f(x) = L$, describes the long-term behavior of a function.
  • If this limit exists, the line $y=L$ is a horizontal asymptote of the graph of $f(x)$.

Key Skills

  • Evaluate limits at infinity, particularly for rational functions, by dividing the numerator and denominator by the highest power of $x$ in the denominator.
  • Find the horizontal asymptotes of a function.

Sections 2.7 & 2.8: The Derivative

Key Concepts

  • The derivative $f'(a)$ is the instantaneous rate of change of $f$ at $x=a$. Geometrically, it is the slope of the tangent line to the curve at that point. Physically, if $f$ is a position function, $f'(a)$ is the instantaneous velocity.
  • The derivative function, $f'(x)$, is a new function whose output at any $x$ is the slope of the original function $f$ at that $x$.
  • Graphical Relationship: The value of $f'(x)$ is the slope of $f(x)$. This means if $f'(x) > 0$, then $f(x)$ is increasing. If $f'(x) < 0$, then $f(x)$ is decreasing. If $f'(x)=0$, then $f(x)$ has a horizontal tangent.
  • A function is differentiable at a point if the derivative exists there. This requires the graph to be smooth and without corners, cusps, discontinuities, or vertical tangents.
  • Differentiability implies continuity, but continuity does not imply differentiability.

Key Skills

  • Use the limit definition to find the derivative at a point $a$: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.
  • Use the alternative limit definition to find the derivative at a point $a$: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$.
  • Use the derivative $f'(a)$ to find the equation of the tangent line at a point.
  • Use the limit definition to find the derivative function $f'(x)$: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
  • Given the graph of $f(x)$, sketch a plausible graph of its derivative, $f'(x)$.
  • Given the graph of $f'(x)$, determine where the original function $f(x)$ is increasing or decreasing.

Key Formulas Reference Sheet

Note: This reference sheet is provided as a study aid. It will not be provided on the exam.

Limits and Continuity

The Limit of a Function

$$\lim_{x \to a} f(x) = L$$ The value $f(x)$ approaches as $x$ approaches $a$.

Condition for Limit to Exist

$$\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = L \text{ and } \lim_{x \to a^+} f(x) = L$$

Definition of Continuity at a Point $a$

A function $f$ is continuous at $a$ if:
  1. $f(a)$ is defined.
  2. $\lim_{x \to a} f(x)$ exists.
  3. $\lim_{x \to a} f(x) = f(a)$.

Horizontal Asymptote

The line $y=L$ is a horizontal asymptote if either $$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$$

Derivatives

Slope of the Secant Line (Average Rate)

$$m_{\text{sec}} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$

The Derivative at a Point $a$

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Alternative Definition of Derivative at a Point

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$$

The Derivative as a Function

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Equation of the Tangent Line

At the point $(a, f(a))$, the equation is: $$y - f(a) = f'(a)(x-a)$$