MATH 1007C Calculus I: Comprehensive Study Notes and Formula Reference

Functions and Their Properties

Review of Functions

Functions are foundational objects in calculus, describing relationships between variables. Understanding their properties is essential for further study.

• Definition: A function f assigns to each element x in a set a unique element f(x).

• Types of Functions: Linear, quadratic, polynomial, rational, exponential, logarithmic, trigonometric, etc.

• Domain and Range: The set of all possible input values (domain) and output values (range).

• Composition of Functions: $(f \circ g)(x) = f(g(x))$

• Transformations: Shifts, reflections, and stretches/compressions of graphs.

• Symmetry: Even, odd, and periodic functions.

Example:

Given $f(x) = x^2$ and $g(x) = x + 1$, the composition $(f \circ g)(x) = f(g(x)) = (x + 1)^2$.

Limits and Continuity

Understanding Limits

Limits describe the behavior of a function as the input approaches a certain value. They are essential for defining derivatives and integrals.

• Limit Notation: $\lim_{x \to a} f(x)$

• One-Sided Limits: $\lim_{x \to a^+} f(x)$ and $\lim_{x \to a^-} f(x)$

• Special Limits: Trigonometric limits, limits at infinity, and indeterminate forms.

• Continuity: A function is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.

• Intermediate Value Theorem: If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = N$.

Example:

Evaluate $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

Derivatives and Differentiation

Definition and Interpretation

The derivative measures the instantaneous rate of change of a function. It is a central concept in calculus, with applications in physics, engineering, and beyond.

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

• Geometric Meaning: The slope of the tangent line to the curve at a point.

• Rules of Differentiation:

  • Sum Rule: $(f + g)' = f' + g'$

  • Product Rule: $(fg)' = f'g + fg'$

  • Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

  • Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$

• Special Derivatives: $\frac{d}{dx} x^n = nx^{n-1}$, $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$, $\frac{d}{dx} e^x = e^x$, $\frac{d}{dx} \ln x = \frac{1}{x}$

Example:

Find the derivative of $f(x) = x^3 + 2x$:

$f'(x) = 3x^2 + 2$

Applications of Derivatives

Critical Points and Optimization

Derivatives are used to find local maxima and minima, points of inflection, and to analyze the behavior of functions.

• Critical Points: Where $f'(x) = 0$ or $f'(x)$ does not exist.

• Second Derivative Test: If $f''(x) > 0$, local minimum; if $f''(x) < 0$, local maximum.

• Optimization: Finding the maximum or minimum values of a function in a given interval.

• Related Rates: Applications involving rates of change with respect to time.

Example:

Find the local extrema of $f(x) = x^2 - 4x + 3$:

Set $f'(x) = 2x - 4 = 0 \implies x = 2$. $f''(x) = 2 > 0$, so $x = 2$ is a local minimum.

Integrals and the Fundamental Theorem of Calculus

Definite and Indefinite Integrals

Integration is the reverse process of differentiation and is used to find areas under curves and solve accumulation problems.

• Indefinite Integral: $\int f(x) dx$ represents the family of all antiderivatives of $f(x)$.

• Definite Integral: $\int_a^b f(x) dx$ gives the net area under $f(x)$ from $x = a$ to $x = b$.

• Properties:

  • $\int_a^a f(x) dx = 0$

  • $\int_a^b f(x) dx = -\int_b^a f(x) dx$

  • $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$

• Fundamental Theorem of Calculus:

  • Part I: If $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$

  • Part II: If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$

Example:

Evaluate $\int_0^2 x dx = \left. \frac{1}{2}x^2 \right|_0^2 = 2$

Trigonometric and Logarithmic Differentiation

Special Techniques

Some functions require special techniques for differentiation, such as logarithmic differentiation and implicit differentiation.

• Logarithmic Differentiation: Useful for functions of the form $y = f(x)^{g(x)}$.

• Implicit Differentiation: Used when $y$ is defined implicitly by an equation involving $x$ and $y$.

Example:

For $y = x^x$, take $\ln$ of both sides: $\ln y = x \ln x$, then differentiate both sides to find $y'$.

Summary Table: Common Derivatives

Function	Derivative
$x^n$	$nx^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$a^x$	$a^x \ln a$

Additional info:

• This summary is based on the provided syllabus and formula sheet, covering the main topics of a first-semester Calculus course, including functions, limits, derivatives, applications of derivatives, and integrals.

• Students should refer to their course materials for more detailed examples and proofs.