# MATH250 - Calculus I Final Exam Review Notes
---
## Chapter 1: Limits and Continuity
### 1.1 The Concept of a Limit
The limit is the foundational concept of calculus. The limit of a function $f(x)$ as $x$ approaches
$c$ is $L$, written as $\lim_{x \to c} f(x) = L$. This means that as $x$ gets arbitrarily close to $c$
(but not equal to $c$), the value of $f(x)$ gets arbitrarily close to $L$.
- **Left-Hand Limit:** $\lim_{x \to c^-} f(x)$, means $x$ approaches $c$ from values less than
$c$.
- **Right-Hand Limit:** $\lim_{x \to c^+} f(x)$, means $x$ approaches $c$ from values greater
than $c$.
- A limit exists **if and only if** the left-hand and right-hand limits are equal: $\lim_{x \to c} f(x)
= L \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$.
### 1.2 Calculating Limits
**Limit Laws:**
- Sum/Difference Rule: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$
- Product Rule: $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
- Quotient Rule: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$
(provided $\lim_{x \to c} g(x) \neq 0$)
**Methods for Handling Indeterminate Forms:**
- **For type $\frac{0}{0}$:**
- Factoring and canceling
- Rationalizing the numerator or denominator
- L'Hôpital's Rule (see chapter on applications of derivatives)
- **For type $\frac{\infty}{\infty}$:**
- Divide numerator and denominator by the highest power of $x$
- L'Hôpital's Rule
> #### Squeeze Theorem
>
> If $g(x) \le f(x) \le h(x)$ for all $x$ in an open interval containing $c$ (except possibly at $c$
itself), and if $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.
>
> An important application: $\lim_{x \to 0} \frac{\sin x}{x} = 1$.
### 1.3 Continuity
A function $f(x)$ is continuous at a point $x=c$ if it meets all three of the following conditions:
1. $f(c)$ is defined.
2. $\lim_{x \to c} f(x)$ exists.
3. $\lim_{x \to c} f(x) = f(c)$.
> #### Intermediate Value Theorem (IVT)
>
> If a function $f(x)$ is continuous on the closed interval $[a, b]$, and $N$ is any number
between $f(a)$ and $f(b)$, then there is at least one number $c$ in $(a, b)$ such that $f(c) = N$.
>
> This theorem is often used to prove that an equation has a root within a certain interval.
---
## Chapter 2: Derivatives
### 2.1 Definition of the Derivative
The derivative of a function $f(x)$ at a point $x=a$, denoted $f'(a)$, is defined as:
$$
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
$$
The derivative $f'(a)$ represents the slope of the tangent line to the graph of $f(x)$ at the point $
(a, f(a))$, and also the instantaneous rate of change of the function at that point.
If a function is differentiable at a point, it must be continuous at that point. **Differentiability
implies continuity, but continuity does not imply differentiability.**
### 2.2 Basic Differentiation Rules
**Common Function Derivatives:**
- Constant Rule: $\frac{d}{dx}(c) = 0$
- Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
- 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 Function Derivatives:**
- $\frac{d}{dx}(\sin x) = \cos x$
- $\frac{d}{dx}(\cos x) = -\sin x$
- $\frac{d}{dx}(\tan x) = \sec^2 x$
- $\frac{d}{dx}(\cot x) = -\csc^2 x$
- $\frac{d}{dx}(\sec x) = \sec x \tan x$
- $\frac{d}{dx}(\csc x) = -\csc x \cot x$
### 2.3 Rules of Differentiation
- **Product Rule:**
$$
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
$$
- **Quotient Rule:**
$$
\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
$$
- **Chain Rule:** For differentiating composite functions.
If $y = f(u)$ and $u = g(x)$, then:
$$
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \quad \text{or} \quad [f(g(x))]' = f'(g(x)) \cdot
g'(x)
$$
**Core Idea:** Differentiate the "outside" function, leaving the inside function alone, then
multiply by the derivative of the "inside" function.
### 2.4 Implicit Differentiation and Related Rates
**Implicit Differentiation:**
Used when $y$ is an implicit function of $x$ (e.g., $x^2 + y^2 = 25$). To find $\frac{dy}{dx}$:
1. Differentiate both sides of the equation with respect to $x$.
2. When differentiating a term involving $y$, apply the Chain Rule. For example, $\frac{d}{dx}
[f(y)] = f'(y) \cdot \frac{dy}{dx}$.
3. Isolate all terms with $\frac{dy}{dx}$ on one side of the equation.
4. Solve for $\frac{dy}{dx}$.
**Related Rates:**
Problems where you find the rate of change of one quantity in terms of the rate of change of
another. The steps are:
1. Draw a diagram and identify all given quantities and rates.
2. Write an equation that relates the variables.
3. Differentiate both sides of the equation with respect to time $t$ (implicit differentiation).
4. Substitute all known values and solve for the unknown rate.
---
## Chapter 3: Applications of Differentiation
### 3.1 Extrema and Optimization
- **Critical Point:** A point $c$ in the domain of $f$ where either $f'(c)=0$ or $f'(c)$ does not
exist.
- **Local (Relative) Maxima/Minima:** These can only occur at critical points.
- **Absolute (Global) Maxima/Minima:** On a closed interval $[a,b]$, the absolute extrema are
found by comparing the function values at all critical points within the interval and at the
endpoints $a$ and $b$.
### 3.2 Derivatives and the Shape of a Graph
- **First Derivative Test:** Used to find local extrema.
- If $f'(x)$ changes from positive to negative at $c$, then $f(c)$ is a local maximum.
- If $f'(x)$ changes from negative to positive at $c$, then $f(c)$ is a local minimum.
- **Increasing/Decreasing 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.
- **Concavity:**
- If $f''(x) > 0$ on an interval, then the graph of $f(x)$ is concave up on that interval.
- If $f''(x) < 0$ on an interval, then the graph of $f(x)$ is concave down on that interval.
- **Inflection Point:** A point where the concavity of the function changes. Found where
$f''(x)=0$ or $f''(x)$ is undefined.
- **Second Derivative Test:** Used to classify critical points.
- If $f'(c)=0$ and $f''(c) < 0$, then $f(c)$ is a local maximum.
- If $f'(c)=0$ and $f''(c) > 0$, then $f(c)$ is a local minimum.
- If $f'(c)=0$ and $f''(c)=0$, the test is inconclusive; use the First Derivative Test.
> #### Mean Value Theorem (MVT)
>
> If a function $f(x)$ is continuous on the closed interval $[a, b]$ and differentiable on the open
interval $(a, b)$, then there is at least one point $c$ in $(a, b)$ such that:
> $$
> f'(c) = \frac{f(b) - f(a)}{b - a}
> $$
> Geometrically, this means there is a point where the slope of the tangent line is equal to the
slope of the secant line connecting the endpoints.
### 3.3 L'Hôpital's Rule
Used to evaluate limits of indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
> If $\lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0$ or $\pm\infty$, then:
> $$
> \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
> $$
> **Note:** Only apply this rule if you have an indeterminate form. You must also verify that the
limit on the right-hand side exists or is $\pm\infty$.
---
## Chapter 4: Integration
### 4.1 Antiderivatives and Indefinite Integrals
A function $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$.
The indefinite integral of $f(x)$ is the set of all its antiderivatives, denoted by:
$$
\int f(x) \,dx = F(x) + C
$$
Here, $C$ is the constant of integration.
### 4.2 The Definite Integral and Riemann Sums
The definite integral $\int_a^b f(x) \,dx$ represents the **net area** under the curve of $f(x)$
from $x=a$ to $x=b$. (Area above the x-axis is positive, area below is negative).
It is defined as the limit of a Riemann sum:
$$
\int_a^b f(x) \,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x
$$
where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a sample point in the i-th subinterval.
### 4.3 The Fundamental Theorem of Calculus (FTC)
> #### FTC Part 1:
>
> If $f$ is continuous on $[a,b]$, then the function $g$ defined by $g(x) = \int_a^x f(t) \,dt$ is
continuous on $[a,b]$ and differentiable on $(a,b)$, and $g'(x) = f(x)$.
>
> This part connects differentiation and integration as inverse processes.
> $$
> \frac{d}{dx} \int_a^x f(t) \,dt = f(x)
> $$
>
> #### FTC Part 2 (The Evaluation Theorem):
>
> If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$ (i.e., $F'(x)=f(x)$), then:
> $$
> \int_a^b f(x) \,dx = F(b) - F(a)
> $$
> This part provides a practical method for calculating definite integrals.
### 4.4 The Substitution Rule
This is the reverse of the Chain Rule for integration. It is used for integrating composite functions.
- **For Indefinite Integrals:**
$$
\int f(g(x)) g'(x) \,dx = \int f(u) \,du \quad (\text{by letting } u = g(x))
$$
- **For Definite Integrals:**
When using substitution, you must also change the limits of integration.
$$
\int_a^b f(g(x)) g'(x) \,dx = \int_{g(a)}^{g(b)} f(u) \,du
$$
**Common Integrals:**
- $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
- $\int \frac{1}{x} \,dx = \ln|x| + C$
- $\int e^x \,dx = e^x + C$
- $\int \sin x \,dx = -\cos x + C$
- $\int \cos x \,dx = \sin x + C$
- $\int \sec^2 x \,dx = \tan x + C$
- $\int \frac{1}{1+x^2} \,dx = \arctan x + C$
---
## Chapter 5: Applications of Integration
### 5.1 Area Between Curves
If $f(x)$ and $g(x)$ are continuous functions on the interval $[a,b]$ and $f(x) \ge g(x)$ for all $x$
in $[a,b]$, then the area of the region bounded by the curves is:
$$
A = \int_a^b [f(x) - g(x)] \,dx \quad (\text{Top Function} - \text{Bottom Function})
$$
If the curves are given as functions of $y$, such as $x=f(y)$ and $x=g(y)$, the area is:
$$
A = \int_c^d [f(y) - g(y)] \,dy \quad (\text{Right Function} - \text{Left Function})
$$
