Calculus I Lecture Notes: Functions, Limits, Derivatives, and Integration

Chapter 1: Functions

1.1 Objectives

• Introduce the concept of a function with examples and study different classes of functions.
• Discuss methods of finding the domain, range, zeros, and singularities of real-valued functions.
• Analyze injective (one-to-one), surjective (onto), and bijective functions.
• Explore inverses of functions and the composition of functions.
• Understand the differences and similarities between several types of functions.

1.2 Learning Outcomes

• Define a function; identify different types of functions and distinguish relations that are not functions.
• Determine the domain and range that makes a rule a function.
• Investigate the injectiveness, surjectiveness, and inverse of different functions.
• Establish the composition of two or more functions.

1.4 Introduction and Definitions

• A function is a means to describe the real world in mathematical terms (e.g., the area of circle $C$ as a function of its radius $r$).
• Definition 1.1: Function: Let $X$ and $Y$ be two nonempty sets. A function $f: X \rightarrow Y$ is a rule which associates to each element $x$ of set $X$ a unique element $y = f(x)$ in $Y$. Alternatively: a relation $R$ from $X$ to $Y$ is a function if for every $x \in X$, there corresponds exactly one $y \in Y$.
• Definition 1.2: Domain, Codomain, and Range:
  • Domain: The set $X$ of all allowed $x$-values. Denoted by $\text{Dom}(f)$ or $D(f)$.
  • Independent Variable: $x$, which takes values in the domain.
  • Dependent Variable: $y$, whose value depends on $x$.
  • Image: For a particular $x_1$, the value $y_1 = f(x_1)$ is the image of $x_1$.
  • Preimage: $x_1$ is the preimage of $y_1$.
  • Codomain: The set $Y$ in the definition $f: X \rightarrow Y$.
  • Range: The set of all admissible $y$-values as $x$ goes through its domain. Denoted by $\text{Ran}(f) = \{y \in Y : f(x) = y, x \in D(f)\}$.

1.5 Classes or Types of Functions

• 1. Linear Functions: $y = f(x) = mx + c$ (where $m$ and $c$ are constants). The graph is a straight line with gradient $m$ and $y$-intercept $c$.
• 2. Constant Functions: $y = f(x) = c$. A special case of linear functions where $m = 0$.
• 3. Quadratic Functions: $y = f(x) = ax^2 + bx + c$ (where $a, b, c$ are constants and $a \neq 0$).
• 4. Polynomial Functions: $y = P_n(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$ ($a_n \neq 0$). Constants $a_i$ are coefficients and $n$ is a non-negative integer.
• 5. Power Functions: $f(x) = x^r$ (where $r$ is any real number).
• 6. Rational Functions: Ratio of two polynomials $y = f(x) = \frac{P(x)}{Q(x)}$ (where $Q(x) \neq 0$).
• 7. Algebraic Functions: Roots of an equation of degree $n$ in $y$ whose coefficients are rational functions of $x$. Functions not algebraic are called Transcendental (includes trigonometric, logarithmic, exponential, and hyperbolic functions).
• 8. Trigonometric Functions: Involve ratios such as $\sin(x), \cos(x), \tan(x), \csc(x), \sec(x),$ or $\cot(x)$.
• 9. Inverse Trigonometric Functions: Involve inverses such as $\arcsin(x), \arccos(x),$ or $\arctan(x)$.
• 10. Exponential Functions: $f(x) = e^x$ or $f(x) = a^x$ ($a > 0$). Series expansion: $e^x = \sum_{n=0}^{\infty} (\frac{x^n}{n!})$.
• 11. Hyperbolic Functions:
  • $\sinh(x) = \frac{e^x - e^{-x}}{2}$
  • $\cosh(x) = \frac{e^x + e^{-x}}{2}$
  • $\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
  • $\text{csch}(x) = \frac{2}{e^x - e^{-x}}$, $\text{sech}(x) = \frac{2}{e^x + e^{-x}}$, $\text{coth}(x) = \frac{e^x + e^{-x}}{e^x - e^{-x}}$.
• 13. Modulus (Absolute Value) Function: $f(x) = |x| = \sqrt{x^2}$. Defined as $x$ if $x \geq 0$ and $-x$ if $x < 0$.
• 14. Characteristic Function: $\chi_S(x) = 1$ if $x \in S$ and $0$ if $x \notin S$.
• 15. Floor Function: $\lfloor x \rfloor$ is the greatest integer $\leq x$. Example: $\lfloor 2.34 \rfloor = 2$.
• 16. Ceiling Function: $\lceil x \rceil$ is the smallest integer $\geq x$. Example: $\lceil 2.34 \rceil = 3$.
• 18. Euler’s Form: $e^{ix} = \cos(x) + i\sin(x)$.
• 19. Periodic Functions: $f(x + T) = f(x)$ for a period $T > 0$. Smallest $T$ is the primitive period.
• 20. Odd Functions: $f(-x) = -f(x)$. Example: $x^3$ and $\sin(x)$.
• 21. Even Functions: $f(-x) = f(x)$. Example: $x^2$ and $\cos(x)$.

1.7 Monotonic Functions

• Increasing: $f(x_1) \leq f(x_2)$ whenever $x_1 < x_2$.
• Strictly Increasing: $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.
• Decreasing: $f(x_1) \geq f(x_2)$ whenever $x_1 < x_2$.
• Strictly Decreasing: $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.

1.8 Zeros and Singularities

• Zero: The value $x$ for which $f(x) = 0$.
• Singularity: For a rational function $f(x) = \frac{P(x)}{Q(x)}$, the value of $x$ for which $Q(x) = 0$.

1.9 Injective, Surjective, and Bijective Functions

• Injective (One-to-One): $f(x_1) = f(x_2) \implies x_1 = x_2$.
• Surjective (Onto): Every element in the codomain has a preimage; Range = Codomain.
• Bijective: Both injective and surjective.

1.10 Composite Functions

• Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$. The composition is $h(x) = (g \circ f)(x) = g(f(x))$.
• Example: If $g(x) = 2x + 3$ and $f(x) = x^2 + x$, then $g(f(x)) = 2(x^2 + x) + 3 = 2x^2 + 2x + 3$.

1.11 Inverse Functions

• A function $g = f^{-1}$ exists if $f$ is bijective. It satisfies $g(f(x)) = x$ and $f(g(y)) = y$.
• Guideline to find $f^{-1}$: Verify bijectivity, solve $y = f(x)$ for $x$ to get $x = f^{-1}(y)$, and verify the composite conditions.

Chapter 2: Limits and Continuity

2.4 Limit of a Function

• Left Hand Limit: $f(x) \rightarrow l$ as $x \rightarrow x_0^-$ if for $\epsilon > 0$, $\exists \delta > 0$ such that $|f(x) - l| < \epsilon$ when $x_0 - \delta < x < x_0$.
• Right Hand Limit: $f(x) \rightarrow l$ as $x \rightarrow x_0^+$ if for $\epsilon > 0$, $\exists \delta > 0$ such that $|f(x) - l| < \epsilon$ when $x_0 < x < x_0 + \delta$.
• Limit of a Function: $f(x)$ has limit $l$ at $x = x_0$ iff the left and right hand limits both exist and are equal.
• Definition 2.3: $\lim_{x \rightarrow x_0} f(x) = l$ if for any $\epsilon > 0$, $\exists \delta > 0$ such that $0 < |x - x_0| < \delta \implies |f(x) - l| < \epsilon$.

2.5 Properties of Limits

• 1. Limits are unique.
• 2. $\lim_{x \rightarrow a} k = k$ (constant).
• 3. Sum/Difference: $\lim [f(x) \pm g(x)] = \lim f(x) \pm \lim g(x)$.
• 4. Product: $\lim [f(x)g(x)] = (\lim f(x))(\lim g(x))$.
• 5. Scalar: $\lim [kf(x)] = k \lim f(x)$.
• 6. Quotient: $\lim [\frac{f(x)}{g(x)}] = \frac{\lim f(x)}{\lim g(x)}$ (given $\lim g(x) \neq 0$).

2.9 Continuity

• Definition 2.9: $f$ is continuous at $x = a$ if:
  • 1. $f(a)$ exists.
  • 2. $\lim_{x \rightarrow a} f(x)$ exists.
  • 3. $\lim_{x \rightarrow a} f(x) = f(a)$.
• Rational functions are continuous except at the zeros of the denominator.

Chapter 3: The Derivative

3.4 Fundamentals and Definitions

• The slope $m$ of the tangent line to the graph of $f$ at $P(a, f(a))$ is the derivative $f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}$.
• Finding the derivative this way is called differentiation from first principles.
• If $y = f(x)$, the derivative is $\frac{dy}{dx} = \lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$.

3.6 Rules of Differentiation

• Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$.
• Constant Multiple: $\frac{d}{dx}[cf(x)] = cf'(x)$.
• Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.
• Product Rule: $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$.
• Quotient Rule: $\frac{d}{dx}[\frac{u}{v}] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.
• Chain Rule: If $y = h(g(x))$, let $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$.
• Leibnitz Rule: For the $n$-th derivative of a product $uv$:     $D^n(uv) = D^n(u)v + \binom{n}{1}D^{n-1}(u)D(v) + \binom{n}{2}D^{n-2}(u)D^2(v) + \dots + uD^n(v)$.

3.7 Derivatives of Transcendental Functions

• Logarithmic: $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$. For general base: $\frac{d}{dx}(\log_a(u)) = \frac{1}{u \ln(a)} \frac{du}{dx}$.
• Exponential: $\frac{d}{dx}(e^x) = e^x$. For general base: $\frac{d}{dx}(a^u) = a^u \ln(a) \frac{du}{dx}$.
• Hyperbolic:
  • $\frac{d}{dx}(\sinh(u)) = \cosh(u) \frac{du}{dx}$
  • $\frac{d}{dx}(\cosh(u)) = \sinh(u) \frac{du}{dx}$
  • $\frac{d}{dx}(\tanh(u)) = \text{sech}^2(u) \frac{du}{dx}$
• Inverse Trigonometric:
  • $\frac{d}{dx}(\arcsin(u)) = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$
  • $\frac{d}{dx}(\arccos(u)) = \frac{-1}{\sqrt{1-u^2}} \frac{du}{dx}$
  • $\frac{d}{dx}(\arctan(u)) = \frac{1}{1+u^2} \frac{du}{dx}$

3.12 Applications of Differentiation

• L’Hospital’s Rule: If a limit $\frac{f(x)}{g(x)}$ results in $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$.
• Critical Points: Points where $f'(c) = 0$.
• Maximum and Minimum:
  • First Derivative Test: If $f'$ changes from positive to negative, it is a local maximum. If from negative to positive, a local minimum.
  • Second Derivative Test: If $f'(p) = 0$ and $f''(p) < 0$, there is a local maximum. If $f''(p) > 0$, a local minimum.
• Rolle’s Theorem: If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then $\exists p \in (a, b)$ such that $f'(p) = 0$.
• Lagrange’s Mean Value Theorem: $\frac{f(b) - f(a)}{b - a} = f'(p)$ for some $p \in (a, b)$.

Chapter 4: Integration

4.4 The Indefinite Integral

• Integration is finding the antiderivative. If $\frac{d}{dx}F(x) = f(x)$, then $\int f(x) dx = F(x) + C$.
• Standard Integrals:
  • $\int x^n dx = \frac{x^{n+1}}{n+1} + c$ (for $n \neq -1$)
  • $\int \frac{1}{x} dx = \ln|x| + c$
  • $\int e^x dx = e^x + c$
  • $\int \cos(x) dx = \sin(x) + c$
  • $\int \sin(x) dx = -\cos(x) + c$

4.5 Methods of Integration

• 1. Substitution: Let $x = x(t)$, then $\int f(x) dx = \int f(x(t)) \frac{dx}{dt} dt$.
• 2. Integration by Parts: Derived from the product rule: $\int u \, dv = uv - \int v \, du$.
• 3. Partial Fractions: Resolving a rational function $\frac{f(x)}{g(x)}$ into simpler fractions based on the factors of $g(x)$.
  • Linear factors: $\frac{A}{x-r}$.
  • Quadratic factors: $\frac{Bx+C}{x^2+px+q}$.
• 4. Reduction Formulas: Reducing higher powers to lower powers. Example for $I_n = \int \tan^n(x) dx$:     $I_n = \frac{\tan^{n-1}(x)}{n-1} - I_{n-2}$.

Chapter 5: The Definite Integral and Applications

5.4 Introduction and Area

• Definite Integral: The limit of a Riemann Sum: $\lim_{n \rightarrow \infty} \sum_{r=1}^n f(\xi_r) \delta_r = \int_a^b f(x) dx$.
• Fundamental Theorem: $\int_a^b f(x) dx = F(b) - F(a)$.
• The value represents the area under the curve $y = f(x)$ from $x = a$ to $x = b$.

5.5 Properties of Definite Integrals

• $\int_a^b f(x) dx = -\int_b^a f(x) dx$
• $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$
• $\int_0^a f(x) dx = \int_0^a f(a - x) dx$
• Derivative of an integral: $\frac{d}{dx} \int_a^x f(t) dt = f(x)$.

5.6 - 5.8 Geometric Applications

• Area between two curves: $A = \int_a^b [f(x) - g(x)] dx$.
• Arc Length: $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$.
• Solid of Revolution (Volume):
  • About $x$-axis: $V = \pi \int_a^b y^2 dx$.
  • About $y$-axis: $V = \pi \int_a^b x^2 dy$.
• Mean Value of a Function: $\bar{y} = \frac{1}{b - a} \int_a^b y \, dx$.

5.10 Moment of Inertia

• Definition: $I = \sum m_i r_i^2$. For continuous bodies, use integrals.
• Parallel Axis Theorem: $I = I_G + Ma^2$ (where $I_G$ is through mass center).
• Perpendicular Axis Theorem: $I_z = I_x + I_y$.
• Specific Results:
  • Thin Rod (length $2l$): $I_{center} = \frac{1}{3}Ml^2$.
  • Circular Disc (radius $a$): $I_{center} = \frac{1}{2}Ma^2$.
  • Sphere (radius $r$): $I = \frac{2}{5}Mr^2$.

5.11 Differential Equations

• Focuses on Ordinary Differential Equations (ODE) solvable by separation of variables.
• General Form: $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{1}{g(y)} dy = \int f(x) dx$.

5.12 - 5.14 Science, Engineering, and Economic Applications

• Displacement: $s(t) = \int |v(t)| dt$.
• Work Done: $W = \int_a^b F(s) ds$. Hooke’s Law for springs: $F = kx \implies W = \int kx \, dx$.
• Growth and Decay: $\frac{dp}{dt} = kp \implies p(t) = p_0 e^{kt}$.
  • Half-life is the time for a substance to decay to half its original amount.
• Economics:
  • Total Cost: $C(x) = \int C'(x) dx$ (where $C'(x)$ is marginal cost).
  • Total Revenue: $r(x) = \int M(x) dx$ (where $M(x)$ is marginal revenue).