MATH 150A - Calculus I: Lecture Notes

# MATH 150A - Calculus I: Lecture Notes

## Chapter 0: Pre-Calculus Review

### 0.1 Functions and Their Properties

A **function** $f$ is a rule that assigns to each element $x$ in a set $D$ exactly one element,

called $f(x)$, in a set $E$.

* **Domain**: The set $D$ of all possible input values for which the function is defined.

* **Range**: The set of all possible output values $f(x)$ as $x$ varies throughout the domain.

* **Vertical Line Test**: A curve in the xy-plane represents the graph of a function if and only if

no vertical line intersects the curve more than once.

* **Polynomial Functions**: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

* **Power Functions**: $f(x) = x^a$, where $a$ is a constant.

* **Rational Functions**: $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.

The domain excludes values of $x$ where $Q(x)=0$.

* **Exponential Functions**: $f(x) = b^x$, where $b > 0$ and $b \ne 1$.

* **Logarithmic Functions**: $f(x) = \log_b x$, the inverse of the exponential function.

* **Trigonometric Functions**: $\sin(x), \cos(x), \tan(x), \csc(x), \sec(x), \cot(x)$.

### 0.3 Trigonometric Identities Review

* **Pythagorean Identity**: $\sin^2(\theta) + \cos^2(\theta) = 1$

* **Angle Addition**:

## Chapter 1: Limits and Continuity

### 1.1 The Idea of a Limit

**Intuitive Definition**: We write

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

and say "the limit of $f(x)$, as $x$ approaches $a$, equals $L$" if we can make the values of $f(x)

$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $a$ (on either side of $a$) but

not equal to $a$.

**Example**: Investigate $\lim_{x \to 2} (x^2 - 1)$.

* Values as $x$ approaches 2 from the left:

* $f(1.9) = 2.61$

* $f(1.99) = 2.9601$

* Values as $x$ approaches 2 from the right:

* $f(2.1) = 3.41$

2. **Difference Law**: $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$

3. **Constant Multiple Law**: $\lim_{x \to a} [c f(x)] = c \lim_{x \to a} f(x)$

4. **Product Law**: $\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

5. **Quotient Law**: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}

$ if $\lim_{x \to a} g(x) \ne 0$

**Direct Substitution Property**: If $f$ is a polynomial or a rational function and $a$ is in the

domain of $f$, then $\lim_{x \to a} f(x) = f(a)$.

**Techniques for Indeterminate Forms (e.g., $\frac{0}{0}$)**:

* **Factoring and Canceling**:

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x-3)(x+3)}{x-3} = \lim_{x \to 3} (x+3) =

* **Left-Hand Limit**: $\lim_{x \to a^-} f(x)$ means $x$ approaches $a$ from values less than

$a$.

* **Right-Hand Limit**: $\lim_{x \to a^+} f(x)$ means $x$ approaches $a$ from values greater

than $a$.

**Theorem**: $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to

a^+} f(x) = L$.

### 1.4 Limits Involving Infinity

* **Horizontal Asymptotes**: The line $y=L$ is a horizontal asymptote of the curve $y=f(x)$ if

either

$$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$$

* **Vertical Asymptotes**: The line $x=a$ is a vertical asymptote if at least one of the following

is true:

$$\lim_{x \to a^+} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to a^-} f(x) = \pm\infty$$

2$$

As $x \to -1^-$, $f(x) \to \infty$. So, $x=-1$ is a vertical asymptote.

### 1.5 Continuity

**Definition**: A function $f$ is **continuous** at a number $a$ if:

1. $f(a)$ is defined.

2. $\lim_{x \to a} f(x)$ exists.

3. $\lim_{x \to a} f(x) = f(a)$.

**Intermediate Value Theorem (IVT)**: Suppose that $f$ is continuous on the closed interval $[a,

b]$ and let $N$ be any number between $f(a)$ and $f(b)$, where $f(a) \ne f(b)$. Then there

exists a number $c$ in $(a, b)$ such that $f(c) = N$.

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

if this limit exists.

The derivative $f'(a)$ represents the **instantaneous rate of change** of $y=f(x)$ with respect

to $x$ at $x=a$. Geometrically, it is the **slope of the tangent line** to the curve $y=f(x)$ at the

point $(a, f(a))$.

### 2.2 The Derivative as a Function

The derivative can also be a function itself:

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

Other Notations: If $y = f(x)$, the derivative can be written as $y'$, $\frac{dy}{dx}$, $\frac{df}{dx}

$, or $\frac{d}{dx}f(x)$.

**Differentiability implies Continuity**: If $f$ is differentiable at $a$, then $f$ is continuous at

$a$. The converse is not true (e.g., $f(x)=|x|$ at $x=0$).

2. **Power Rule**: $\frac{d}{dx}(x^n) = nx^{n-1}$ for any real number $n$.

* $\frac{d}{dx}(e^x) = e^x$

* $\frac{d}{dx}(a^x) = a^x \ln(a)$

### 2.4 The Product and Quotient Rules

* **Product Rule**:

$$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$$

or $(fg)' = fg' + gf'$.

* **Quotient Rule**:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$$

or $(\frac{f}{g})' = \frac{gf' - fg'}{g^2}$.

* $\frac{d}{dx}(\cot x) = -\csc^2 x$

### 2.6 The Chain Rule

If $y = f(u)$ and $u = g(x)$ are both differentiable functions, then the composite function $F(x) =

f(g(x))$ is differentiable and

$$F'(x) = f'(g(x)) \cdot g'(x)$$In Leibniz notation:$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}

{dx}$$

**Example**: Differentiate $y = \sin(x^2)$.

* Let $u = x^2$ and $y = \sin(u)$.

* $\frac{du}{dx} = 2x$ and $\frac{dy}{du} = \cos(u)$.

* $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(u) \cdot (2x) = \cos(x^2) \cdot 2x = 2x\

cos(x^2)$.

### 2.7 Implicit Differentiation

1. Differentiate both sides of the equation with respect to $x$.

**Example**: Find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$.

1. $\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)$

2. $2x + 2y \cdot \frac{dy}{dx} = 0$

3. $2y \frac{dy}{dx} = -2x \implies \frac{dy}{dx} = -\frac{x}{y}$

### 2.8 Related Rates

Used to find the rate of change of one quantity in terms of the rate of change of another.

**Problem-Solving Strategy**:

1. Identify all given quantities and quantities to be found. Assign variables.

### 3.1 Extreme Values of Functions

* **Absolute Maximum/Minimum**: The largest/smallest function value on its entire domain.

* **Local (or Relative) Maximum/Minimum**: The largest/smallest function value in some open

interval.

**The Extreme Value Theorem**: If $f$ is continuous on a closed interval $[a, b]$, then $f$

attains an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ at some

numbers $c$ and $d$ in $[a, b]$.

**Fermat's Theorem**: If $f$ has a local extremum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.

A **critical number** of a function $f$ is a number $c$ in the domain of $f$ such that either

$f'(c) = 0$ or $f'(c)$ does not exist. Extrema can only occur at critical numbers or endpoints.

2. Differentiable on the open interval $(a, b)$.

connecting the endpoints.

### 3.3 Derivatives and the Shape of a Graph

**Increasing/Decreasing Test**:

* If $f'(x) > 0$ on an interval, then $f$ is increasing on that interval.

* If $f'(x) < 0$ on an interval, then $f$ is decreasing on that interval.

**The First Derivative Test**: Suppose $c$ is a critical number of a continuous function $f$.

* If $f'$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$.

* If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$.

### 3.4 L'Hôpital's Rule

Suppose we have an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. That is, $\lim_{x \

to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$ (or both are $\pm\infty$). Then

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

if the limit on the right side exists (or is $\pm\infty$).

**Example**:

$$\lim_{x \to 0} \frac{\sin x}{x} \quad \left(\text{form } \frac{0}{0}\right)$$

$$= \lim_{x \to 0} \frac{\frac{d}{dx}(\sin x)}{\frac{d}{dx}(x)} = \lim_{x \to 0} \frac{\cos x}{1} = \

frac{\cos 0}{1} = 1$$

### 3.5 Optimization Problems

These problems involve finding the maximum or minimum value of a quantity.

3. Write a secondary (constraint) equation relating the variables.

### 3.6 Antiderivatives

A function $F$ is called an **antiderivative** of $f$ on an interval $I$ if $F'(x) = f(x)$ for all $x$ in

$I$.

The set of all antiderivatives of $f$ is given by $F(x) + C$, where $C$ is an arbitrary constant of

integration.

The indefinite integral $\int f(x) \,dx$ denotes the antiderivative of $f$.

**Common Antiderivatives**:

* $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1)$

### 4.1 The Definite Integral

The definite integral is the limit of a Riemann sum and can be interpreted as the **net area**

under a curve.

$$\int_a^b f(x) \,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$$

where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a sample point in the $i$-th subinterval.

### 4.2 The Fundamental Theorem of Calculus (FTC)

This theorem connects the concepts of differentiation and integration.

**Part 1 (FTC1)**: If $f$ is continuous on $[a, b]$, then the function $g$ defined by

$$g(x) = \int_a^x f(t) \,dt \quad a \le x \le b$$

is continuous on $[a, b]$ and differentiable on $(a, b)$, and $g'(x) = f(x)$.

This means the derivative of an integral function is the original function.

This part provides a practical method for evaluating definite integrals.

* Using FTC2:

$$\int_1^3 x^2 \,dx = \left[ \frac{x^3}{3} \right]_1^3 = F(3) - F(1) = \frac{3^3}{3} - \frac{1^3}{3}

= 9 - \frac{1}{3} = \frac{26}{3}$$

### 4.3 The Substitution Rule

The reversal of the Chain Rule for integration. If $u = g(x)$ is a differentiable function, then

$$\int f(g(x))g'(x) \,dx = \int f(u) \,du$$

For definite integrals:

$$\int_a^b f(g(x))g'(x) \,dx = \int_{g(a)}^{g(b)} f(u) \,du$$

### 4.4 Area Between Curves

If $f$ and $g$ are continuous functions with $f(x) \ge g(x)$ throughout $[a, b]$, then the area

$A$ of the region bounded by the curves $y=f(x)$, $y=g(x)$ and the lines $x=a$, $x=b$ is

$$A = \int_a^b [f(x) - g(x)] \,dx$$

This can be remembered as the integral of "(top function) - (bottom function)".

**Example**: Find the area of the region enclosed by $y=x$ and $y=x^2$.

1. **Find intersection points**: $x^2 = x \implies x^2 - x = 0 \implies x(x-1)=0$. They intersect at

$x=0$ and $x=1$.

2. **Determine top/bottom function**: In the interval $(0, 1)$, a test point like $x=0.5$ shows

$0.5 > (0.5)^2 = 0.25$. So, $y=x$ is the top function.

3. **Set up and evaluate the integral**:

$$A = \int_0^1 (x - x^2) \,dx = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1$$ $$= \left(\

frac{1^2}{2} - \frac{1^3}{3}\right) - (0) = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$$