# MATH 150A - Calculus I: Comprehensive Study Guide
---
## Table of Contents
1. [**Chapter 1: Functions, Limits, and Continuity**](#chapter1-en)
* 1.1 Review of Functions
* 1.2 The Intuitive Concept of a Limit
* 1.3 Computing Limits
* 1.4 One-Sided Limits and Infinite Limits
* 1.5 The Precise Definition of a Limit ($\epsilon-\delta$ Definition)
* 1.6 Continuity
* 1.7 The Intermediate Value Theorem
2. [**Chapter 2: The Derivative**](#chapter2-en)
* 2.1 The Derivative and the Tangent Line Problem
* 2.2 The Derivative as a Function
* 2.3 Basic Differentiation Rules
* 2.4 The Product and Quotient Rules
* 2.5 The Chain Rule
* 2.6 Implicit Differentiation
* 2.7 Higher-Order Derivatives
3. [**Chapter 3: Applications of Differentiation**](#chapter3-en)
* 3.1 Related Rates
* 3.2 Extreme Values of Functions
* 3.3 The Mean Value Theorem
* 3.4 Derivatives and the Shape of a Graph
* 3.5 L'Hôpital's Rule
* 3.6 Optimization Problems
* 3.7 Newton's Method
* 3.8 Antiderivatives
4. [**Chapter 4: Integrals**](#chapter4-en)
* 4.1 The Area and Distance Problems
* 4.2 The Definite Integral
* 4.3 The Fundamental Theorem of Calculus
* 4.4 Indefinite Integrals and the Net Change Theorem
* 4.5 The Substitution Rule
5. [**Appendix: Key Formulas & Study Tips**](#appendix-en)
* A.1 Table of Common Derivatives
* A.2 Tips for Studying Calculus
---
<a name="chapter1-en"></a>
## Chapter 1: Functions, Limits, and Continuity
The cornerstone of calculus is the idea of a "limit." It allows us to study the behavior of a function
"near" a point, which in turn enables us to define continuity and the derivative.
### 1.1 Review of Functions
* **Definition**: A function is a rule that assigns to each element in a set (the Domain) exactly
one element in another set (the Range). Denoted as $f: D \to R$.
* **Common Function Types**:
* **Polynomials**: $P(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$
* **Rational Functions**: $R(x) = \frac{P(x)}{Q(x)}$, where $P, Q$ are polynomials.
* **Power Functions**: $f(x) = x^a$
* **Exponential Functions**: $f(x) = a^x, (a > 0, a \neq 1)$
* **Logarithmic Functions**: $f(x) = \log_a(x)$
* **Trigonometric Functions**: $\sin(x), \cos(x), \tan(x), \dots$
* **Inverse Trigonometric Functions**: $\arcsin(x), \arccos(x), \arctan(x), \dots$
### 1.2 The Intuitive Concept of a Limit
* **Concept**: When we say the limit of $f(x)$ as $x$ approaches $c$ is $L$ (written as $\
lim_{x \to c} f(x) = L$), we mean that the value of $f(x)$ can be made "arbitrarily close" to $L$ by
taking $x$ "sufficiently close" to $c$, but not equal to $c$.
* **Key Point**: The limit is only concerned with the behavior of the function *near* $c$, not
the value of the function *at* $c$.
### 1.3 Computing Limits
* **Limit Laws**: Suppose $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$. Then:
1. **Sum/Difference**: $\lim_{x \to c} [f(x) \pm g(x)] = L \pm M$
2. **Constant Multiple**: $\lim_{x \to c} [k \cdot f(x)] = k \cdot L$
3. **Product**: $\lim_{x \to c} [f(x)g(x)] = L \cdot M$
4. **Quotient**: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$, (provided $M \neq 0$)
5. **Power**: $\lim_{x \to c} [f(x)]^n = L^n$
* **Direct Substitution**: If a function is continuous at $c$ (like polynomials and rational
functions within their domains), the limit can be found by direct substitution.
* **Simplification**: For indeterminate forms like $\frac{0}{0}$, try algebraic techniques like
factoring or rationalizing to eliminate the zero factor.
* **Example**: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \
lim_{x \to 2} (x+2) = 4$
* **The Squeeze Theorem**: If $g(x) \le f(x) \le h(x)$ for all $x$ near $c$ (except possibly at $c$),
and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.
* **Important Limit**: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$
### 1.4 One-Sided Limits and Infinite Limits
* **Left-Hand Limit**: $x$ approaches $c$ from the left side, denoted $\lim_{x \to c^-} f(x)$.
* **Right-Hand Limit**: $x$ approaches $c$ from the right side, denoted $\lim_{x \to c^+} f(x)$.
* **Existence of a Limit**: $\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = \lim_{x \to
c^+} f(x) = L$.
* **Infinite Limits**: $\lim_{x \to c} f(x) = \infty$ means that the values of $f(x)$ grow infinitely
large as $x$ approaches $c$. This corresponds to a **vertical asymptote** at $x=c$.
* **Limits at Infinity**: $\lim_{x \to \infty} f(x) = L$ means that $f(x)$ approaches $L$ as $x$
grows infinitely large. This corresponds to a **horizontal asymptote** at $y=L$.
### 1.5 The Precise Definition of a Limit ($\epsilon-\delta$ Definition)
$\lim_{x \to c} f(x) = L$ means: for every number $\epsilon > 0$, there exists a number $\delta >
0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$.
### 1.6 Continuity
* **Definition of Continuity at a Point**: A function $f(x)$ is continuous at a point $x=c$ if it
satisfies three conditions:
1. $f(c)$ is defined ($c$ is in the domain of $f$).
2. $\lim_{x \to c} f(x)$ exists.
3. $\lim_{x \to c} f(x) = f(c)$.
* **Continuity on an Interval**: A function is continuous on an interval if it is continuous at every
point in the interval.
* **Common Continuous Functions**: Polynomials, exponential functions, logarithmic functions,
and trigonometric functions are continuous on their respective domains.
### 1.7 The Intermediate Value Theorem (IVT)
* **Theorem**: Suppose $f(x)$ is continuous on the closed interval $[a, b]$ and $N$ is any
number between $f(a)$ and $f(b)$. Then there exists a number $c$ in the open interval $(a, b)$
such that $f(c) = N$.
* **Application**: Often used to prove the existence of roots of an equation.
---
<a name="chapter2-en"></a>
## Chapter 2: The Derivative
The derivative is one of the central concepts of calculus, describing the instantaneous rate of
change of a function.
### 2.1 The Derivative and the Tangent Line Problem
* **Average Rate of Change**: Over the interval $[x, x+h]$, the average rate of change of $f$ is
$\frac{\Delta y}{\Delta x} = \frac{f(x+h) - f(x)}{h}$.
* **Instantaneous Rate of Change / Derivative**: The derivative of a function $f$ at a number
$x$, denoted by $f'(x)$, is the limit of the average rate of change as $h \to 0$.
$$
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
$$
* **Geometric Interpretation**: $f'(a)$ is the slope of the tangent line to the curve $y=f(x)$ at
the point $(a, f(a))$.
* **Physical Interpretation**: If $s(t)$ is the position function, then $s'(t)$ is the instantaneous
velocity.
### 2.2 The Derivative as a Function
* For each $x$ value where the limit exists, we can compute $f'(x)$. This defines a new function
called the **derivative function** of $f$.
* **Differentiability and Continuity**: If a function is differentiable at a point, it must be
continuous at that point. The converse is not true (e.g., $f(x) = |x|$ is continuous at $x=0$ but
not differentiable).
* **Notations for the Derivative**: $f'(x)$, $y'$, $\frac{dy}{dx}$, $\frac{d}{dx}f(x)$.
### 2.3 Basic Differentiation Rules
* **Constant Rule**: $\frac{d}{dx}(c) = 0$
* **Power Rule**: $\frac{d}{dx}(x^n) = nx^{n-1}$
* **Constant Multiple Rule**: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$
* **Sum/Difference Rule**: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
* **Exponential Functions**: $\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(a^x) = a^x \ln(a)$
* **Logarithmic Functions**: $\frac{d}{dx}(\ln|x|) = \frac{1}{x}$, $\frac{d}{dx}(\log_a|x|) = \
frac{1}{x \ln a}$
* **Trigonometric Functions**:
* $\frac{d}{dx}(\sin x) = \cos x$
* $\frac{d}{dx}(\cos x) = -\sin x$
* $\frac{d}{dx}(\tan x) = \sec^2 x$
### 2.4 The Product and Quotient Rules
* **Product Rule**:
$$
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
$$
* **Quotient Rule**:
$$
\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
$$
### 2.5 The Chain Rule
* **Core Idea**: Used to find the derivative of composite functions. If $y = f(u)$ and $u = g(x)$,
then the derivative of $y$ with respect to $x$ is:
$$
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \quad \text{or} \quad [f(g(x))]' = f'(g(x)) \cdot
g'(x)
$$
* **Mnemonic**: "Differentiate the outer function, leaving the inner function alone, then
multiply by the derivative of the inner function."
### 2.6 Implicit Differentiation
* **When to Use**: Used when $y$ cannot be explicitly solved for in terms of $x$ (e.g., $x^2 +
y^2 = 25$).
* **Steps**:
1. Differentiate both sides of the equation with respect to $x$.
2. When differentiating a term involving $y$, apply the Chain Rule, multiplying by $\frac{dy}
{dx}$.
3. Solve the resulting equation for $\frac{dy}{dx}$.
### 2.7 Higher-Order Derivatives
* **Second Derivative**: $f''(x) = \frac{d}{dx}[f'(x)]$, denoted $\frac{d^2y}{dx^2}$.
* **Third Derivative**: $f'''(x)$
* **n-th Derivative**: $f^{(n)}(x)$
* **Physical Interpretation**: If $s(t)$ is position, then $s'(t)$ is velocity, and $s''(t)$ is
acceleration.
---
<a name="chapter3-en"></a>
## Chapter 3: Applications of Differentiation
The derivative is a powerful tool for analyzing the behavior of functions and solving real-world
problems.
### 3.1 Related Rates
* **Core Problem**: Studying the relationship between two or more variables that are changing
over time.
* **Problem-Solving Steps**:
1. Identify the variables and draw a diagram.
2. Find an equation that relates these variables.
3. Differentiate both sides of the equation with respect to time $t$ (using implicit
differentiation).
4. Substitute the known values and solve for the unknown rate of change.
### 3.2 Extreme Values of Functions
* **Absolute Maximum/Minimum**: The largest/smallest value of a function over its entire
domain.
* **Local (or Relative) Maximum/Minimum**: The largest/smallest value of a function in a small
neighborhood around a point.
* **Critical Point**: A number $c$ in the domain of $f$ is a critical point if either $f'(c) = 0$ or
$f'(c)$ does not exist.
* **Fermat's Theorem**: If $f$ has a local extremum at $c$ and $f'(c)$ exists, then $f'(c) = 0$.
* **The Extreme Value Theorem**: A continuous function on a closed interval $[a, b]$ must
attain an absolute maximum and an absolute minimum value. These values occur either at critical
points or at the endpoints of the interval ($a$ or $b$).
### 3.3 The Mean Value Theorem (MVT)
* **Theorem**: If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there is at
least one number $c$ in $(a, b)$ such that:
$$
f'(c) = \frac{f(b) - f(a)}{b - a}
$$
* **Geometric Interpretation**: There is at least one point on the curve where the tangent line
is parallel to the secant line connecting the endpoints.
* **Rolle's Theorem**: A special case of MVT where $f(a) = f(b)$. In this case, there must be a
point $c \in (a, b)$ such that $f'(c) = 0$.
### 3.4 Derivatives and the Shape of a Graph
* **First Derivative Test**:
* If $f'(x) > 0$ on an interval, then $f(x)$ is **increasing** on that interval.
* If $f'(x) < 0$ on an interval, then $f(x)$ is **decreasing** on that interval.
* The sign change of $f'(x)$ at a critical point determines if it's a local max, local min, or
neither.
* **Second Derivative and Concavity**:
* If $f''(x) > 0$ on an interval, the graph of $f(x)$ is **concave upward**.
* If $f''(x) < 0$ on an interval, the graph of $f(x)$ is **concave downward**.
* **Inflection Point**: A point where the concavity of the function changes. This typically occurs
where $f''(x)=0$ or $f''(x)$ does not exist.
* **Second Derivative Test**:
* If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$.
* If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$.
### 3.5 L'Hôpital's Rule
* **When to Use**: For computing limits of indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\
infty}$.
* **Rule**: If $\lim_{x \to c} \frac{f(x)}{g(x)}$ results in an indeterminate form, then:
$$
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
$$
provided the limit on the right exists (or is infinite).
### 3.6 Optimization Problems
* **Core Idea**: Using derivatives to find the maximum or minimum value of a function to solve
practical problems (e.g., maximizing profit, minimizing material cost).
* **Problem-Solving Steps**:
1. Understand the problem, define variables.
2. Write a primary equation for the quantity to be optimized and a secondary (constraint)
equation.
3. Express the primary equation as a function of a single variable.
4. Find the derivative, locate critical points, and determine the optimal value based on the
context of the problem.
### 3.8 Antiderivatives
* **Definition**: A function $F(x)$ is called an **antiderivative** of $f(x)$ if $F'(x) = f(x)$.
* **Indefinite Integral Notation**:
$$
\int f(x) \,dx = F(x) + C
$$
where $C$ is an arbitrary constant called the constant of integration.
* **Basic Antiderivative Formulas**: These are the reverse of basic derivative formulas. For
example, $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$).
---
<a name="chapter4-en"></a>
## Chapter 4: Integrals
The integral is the other central concept of calculus. It is the inverse operation of differentiation
and is primarily used to calculate accumulated quantities like area and volume.
### 4.1 The Area and Distance Problems
* **The Area Problem**: How do we find the area under a curve? We can approximate it by
dividing the region into an infinite number of infinitesimally thin rectangles and summing their
areas.
* **Riemann Sum**: This form of approximation is called a Riemann Sum.
$$
A \approx \sum_{i=1}^{n} f(x_i^*) \Delta x
$$
where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a sample point in the $i$-th subinterval.
### 4.2 The Definite Integral
* **Definition**: If the limit of the Riemann sum exists as $n \to \infty$, we call this limit the
**definite integral** of $f$ from $a$ to $b$.
$$
\int_a^b f(x) \,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x
$$
* **Geometric Interpretation**: If $f(x) \ge 0$, then $\int_a^b f(x) \,dx$ represents the area of
the region bounded by the curve $y=f(x)$, the x-axis, and the lines $x=a$ and $x=b$.
### 4.3 The Fundamental Theorem of Calculus (FTC)
This theorem connects the seemingly unrelated concepts of differentiation and integration,
standing as the greatest achievement of calculus.
* **FTC Part 1**: If $g(x) = \int_a^x f(t) \,dt$, then $g'(x) = f(x)$.
* **Meaning**: The derivative of an integral with a variable upper limit is simply the integrand
itself.
* **FTC Part 2**: If $F$ is any antiderivative of $f$ (i.e., $F'(x) = f(x)$), then:
$$
\int_a^b f(x) \,dx = F(b) - F(a)
$$
* **Meaning**: This part provides a powerful method for calculating definite integrals without
needing to compute the limit of a Riemann sum.
### 4.4 Indefinite Integrals and the Net Change Theorem
* **The Net Change Theorem**: The definite integral of a rate of change of a quantity gives the
net change in that quantity.
$$
\int_a^b F'(x) \,dx = F(b) - F(a)
$$
For example, the integral of the velocity function over $[t_1, t_2]$ is the net change in position
(displacement).
### 4.5 The Substitution Rule
* **Core Idea**: The reverse of the Chain Rule for integration, used to simplify integrals.
* **Method**:
1. Choose a suitable substitution $u = g(x)$.
2. Calculate its differential $du = g'(x) \,dx$.
3. Rewrite the entire integral in terms of $u$.
4. Evaluate the integral with respect to $u$.
5. Substitute $g(x)$ back in for $u$.
* **For Definite Integrals**: When making the substitution, the limits of integration must also be
changed to their corresponding $u$-values: $u_1 = g(a), u_2 = g(b)$. This eliminates the need to
substitute back to $x$.
$$
\int_a^b f(g(x))g'(x) \,dx = \int_{g(a)}^{g(b)} f(u) \,du
$$
---
<a name="appendix-en"></a>
## Appendix: Key Formulas & Study Tips
### A.1 Table of Common Derivatives
| Function $f(x)$ | Derivative $f'(x)$ |
| :--- | :--- |
| $c$ | $0$ |
| $x^n$ | $nx^{n-1}$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln|x|$ | $\frac{1}{x}$ |
| $\log_a|x|$ | $\frac{1}{x \ln a}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
| $\arcsin x$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\arctan x$ | $\frac{1}{1+x^2}$ |
### A.2 Tips for Studying Calculus
1. **Strengthen Your Algebra Skills**: Many mistakes in calculus are actually algebra mistakes.
Ensure you are proficient with functions, factoring, exponents, and logarithms.
2. **Understand Concepts, Don't Just Memorize Formulas**: Grasp the geometric and physical
meaning behind limits, derivatives, and integrals. This is far more important than rote
memorization.
3. **Practice Extensively**: Mathematics is a subject you learn by *doing*, not by *watching*.
Work through a wide variety of problems until the solution steps become second nature.
4. **Draw Pictures**: Visualize functions and problems whenever possible. Sketching graphs,
tangent lines, and areas can significantly deepen your understanding.
5. **Seek Help**: Don't be afraid to ask questions. Discussing problems with classmates,
professors, or teaching assistants is an incredibly effective way to learn.
6. **Review Regularly**: The concepts in calculus build upon one another. Periodically review
previous topics to ensure your foundation remains solid.
---
