Calculus I Exam 1 Study Guide: Functions, Limits, Derivatives, Exponential & Logarithmic Functions

Functions

Definition and Domain

In calculus, a function is a rule that assigns each input value (from the domain) to exactly one output value. The domain of a function is the set of all possible input values for which the function is defined.

• Finding the domain: Identify all values of x for which the function produces a real output.

• Function operations: Functions can be added, subtracted, multiplied, divided, and composed (using composition).

Basic Functions and Their Graphs

• Linear functions: $y = mx + b$; slope $m$ is the rate of change.

• Parabolas: $y = ax^2 + bx + c$; vertex at $x = -\frac{b}{2a}$; intercepts and min/max values occur at the vertex.

• Vertical line test: A graph is a function if any vertical line crosses it at most once.

Mathematical Modeling

• Revenue: $\text{Revenue} = \text{Price} \times \text{Quantity}$

• Cost: Total cost of producing a quantity of goods.

• Profit: $\text{Profit} = \text{Revenue} - \text{Cost}$

• Break-even point: Occurs when $\text{Revenue} = \text{Cost}$, so $\text{Profit} = 0$

• Market equilibrium: Supply equals demand.

Limits and Continuity

Limits

The limit of a function describes the behavior of the function as the input approaches a particular value. Limits are foundational for defining derivatives and continuity.

• Notation: $\lim_{x \to a} f(x)$

• Finding limits algebraically: Pay special attention to cases where $x$ approaches $0$,$\infty$, or where the function is piecewise or polynomial/rational.

Continuity

A function is continuous at a point if the limit exists and equals the function value at that point.

• Continuity at $x = a$: $\lim_{x \to a} f(x) = f(a)$

• Types of discontinuity: Jump, infinite, and removable discontinuities.

The Derivative

Definition of Derivative

The derivative of a function at a point measures the instantaneous rate of change of the function at that point.

• Formal definition:

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

• Prime notation: $f'(x)$

• Meaning: The slope of the tangent line to the graph at $x$.

• Interpretation:

  • $f'(x) > 0$: $f$ is increasing at $x$

  • $f'(x) < 0$: $f$ is decreasing at $x$

Derivative Rules

These rules allow you to compute derivatives efficiently.

Derivative Rule	Prime Notation
The Constant Rule	$c' = 0$
The Power Rule	$x^r' = r x^{r-1}$
The Constant Multiple Rule	$(c f)' = c f'$
The Sum Rule	$(f + g)' = f' + g'$

Applications of the Derivative

• Marginal analysis: Marginal cost is $C'(q)$, marginal revenue is $R'(q)$.

• Approximate change: $f(a + h) \approx f(a) + f'(a) h$

The Exponential and Logarithmic Functions

Exponential Functions

An exponential function has the form $y = b^x$, $b > 0$, $b \neq 1$. Exponential functions model growth and decay processes.

• Properties:

  • If $0 < b < 1$, the function is decreasing.

  • If $b > 1$, the function is increasing.

  • As $x \to \infty$, $b^x \to \infty$ (if $b > 1$).

  • As $x \to -\infty$, $b^x \to 0$ (if $b > 1$).

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. The logarithm base $b$ is written $\log_b x$.

• Properties:

  • $\log_b (xy) = \log_b x + \log_b y$

  • $\log_b (x/y) = \log_b x - \log_b y$

  • $\log_b (x^r) = r \log_b x$

  • $\log_b 1 = 0$

  • $\log_b b = 1$

Relationships Between Exponential and Logarithmic Functions

• If $b^x = y$, then $x = \log_b y$

• If $x = \log_b y$, then $b^x = y$

Applications

• Compound interest:

  • Interest compounded $n$ times per year: $A(t) = P (1 + \frac{r}{n})^{nt}$

  • Interest compounded continuously: $A(t) = P e^{rt}$

  • Present value of an investment: $P = \frac{A}{(1 + r)^t}$

• Exponential growth: $Q(t) = Q_0 e^{rt}$, $r > 0$

• Exponential decay: $Q(t) = Q_0 e^{rt}$, $r < 0$

How to Study

• Make a list of all definitions and formulas you need to know.

• Organize material by topic for easier recall.

• Practice solving problems without notes or textbooks to reinforce understanding.