University of Pittsburgh - Calculus 2 (MATH 0230)

Sequences

A sequence is an ordered list of numbers defined by a function whose domain is the set of natural numbers. In Calculus 2, we focus on sequences and their convergence behavior.

Notation

A sequence is typically denoted as {a_n} where a_n represents the nth term of the sequence.

Convergence of Sequences

A sequence {a_n} converges to a limit L if:

If no such finite limit exists, the sequence diverges.

Monotonic and Bounded Sequences

• Monotonic sequences: Either always increasing (a_n+1 > a_n) or always decreasing (a_n+1 < a_n)
• Bounded sequences: All terms lie within some finite interval
• Monotone Convergence Theorem: Every bounded, monotonic sequence converges

Limit Laws for Sequences

Standard limit laws apply to sequences:

• lim(n) (a_n + b_n) = lim(n) a_n + lim(n) b_n
• lim(n) (a_n b_n) = lim(n) a_n lim(n) b_n
• lim(n) (a_n / b_n) = lim(n) a_n / lim(n) b_n (if lim(n) b_n 0)

Series

An infinite series is the sum of the terms of an infinite sequence. Series play a crucial role in advanced mathematics and have numerous applications in science and engineering.

Basic Definitions

An infinite series is denoted as:

The partial sum S_n is the sum of the first n terms:

A series converges if its sequence of partial sums {S_n} converges to a finite limit.

Geometric Series

A geometric series has the form:

It converges if |r| < 1 and diverges otherwise. When it converges:

p-Series

A p-series has the form:

A p-series converges if p > 1 and diverges if p 1. The series with p = 1 is called the harmonic series.

Tests for Convergence

Calculus 2 introduces several tests to determine if series converge or diverge:

1. Divergence Test (nth Term Test)

If lim(n) a_n 0, then (n=1 to ) a_n diverges.

(Note: If lim(n) a_n = 0, the series may either converge or diverge.)

2. Integral Test

If f is a continuous, positive, decreasing function on [1,) with f(n) = a_n, then:

• (n=1 to ) a_n converges if ⁽⁾ f(x) dx converges
• (n=1 to ) a_n diverges if ⁽⁾ f(x) dx diverges

3. Comparison Tests

• Comparison Test: If 0 a_n b_n for all n, then:
  • If b_n converges, then a_n converges
  • If a_n diverges, then b_n diverges
• Limit Comparison Test: If a_n > 0, b_n 0, and lim(n)(a_n/b_n) = L where 0 < L < , then a_n and b_n either both converge or both diverge.

4. Ratio Test

For series a_n, let L = lim(n)|a_n+1/a_n|. Then:

• If L < 1, the series converges absolutely
• If L > 1, the series diverges
• If L = 1, the test is inconclusive

5. Root Test

For series a_n, let L = lim(n)|a_n|. Then:

• If L < 1, the series converges absolutely
• If L > 1, the series diverges
• If L = 1, the test is inconclusive

6. Alternating Series Test

An alternating series (-1)ⁿ⁺¹a_n or (-1)ⁿa_n converges if both conditions are met:

• a_n+1 a_n for all n (terms are decreasing)
• lim(n) a_n = 0

Power Series

A power series is an infinite series of the form:

Power series are functions of x that converge for certain values of x. The interval of convergence consists of all x values for which the series converges.

Taylor and Maclaurin Series

Taylor series represent functions as infinite polynomials centered at a point a:

Maclaurin series are Taylor series centered at 0 (a = 0).

Common Maclaurin Series

• e^x = (n=0 to ) xⁿ/n!
• sin(x) = (n=0 to ) (-1)ⁿx²ⁿ⁺¹/(2n+1)!
• cos(x) = (n=0 to ) (-1)ⁿx²ⁿ/(2n)!
• ln(1+x) = (n=1 to ) (-1)ⁿ⁺¹xⁿ/n for |x| < 1
• (1+x)^k = (n=0 to ) [k(k-1)(k-2)...(k-n+1)/n!]xⁿ for |x| < 1

Vectors

Vectors are mathematical objects that have both magnitude and direction. In Calculus 2, we explore vector algebra and its applications to geometry and motion in space.

Vector Notation

A vector in 3D space can be written in component form as:

where i, j, and k are the standard unit vectors in the x, y, and z directions respectively.

Vector Operations

• Scalar multiplication: cv = cv, cv, cv
• Vector addition: u + v = u+v, u+v, u+v
• Magnitude (length): |v| = (v + v + v)
• Dot product: u v = uv + uv + uv
• Cross product: u v = uv - uv, uv - uv, uv - uv

Properties of the Dot Product

• The dot product is a scalar quantity
• u v = |u||v|cos(), where is the angle between u and v
• If u v = 0 and u, v are non-zero vectors, then they are orthogonal (perpendicular)
• u u = |u|

Properties of the Cross Product

• The cross product is a vector quantity perpendicular to both u and v
• u v = -(v u)
• |u v| = |u||v|sin(), which represents the area of the parallelogram formed by u and v
• If u v = 0 and u, v are non-zero vectors, then they are parallel

Lines in Space

A line through point P(x, y, z) with direction vector v = a, b, c can be expressed as:

Vector equation:

Parametric equations:

Symmetric equations:

Planes in Space

A plane through point P(x, y, z) with normal vector n = a, b, c has equation:

Vector-Valued Functions

A vector-valued function in 3D can be written as:

The derivative of a vector-valued function is:

This derivative represents the tangent vector to the curve at point r(t).

Curvature and Motion

• Velocity: v(t) = r'(t)
• Speed: |v(t)| = |r'(t)|
• Acceleration: a(t) = r''(t)
• Arc length: s = _a^b |r'(t)| dt

Differential Equations

Differential equations involve derivatives and model many real-world phenomena. Calculus 2 introduces basic techniques for solving first-order differential equations and some second-order linear equations.

Basic Terminology

• Order: The highest derivative in the equation
• Linear vs. Nonlinear: Depends on whether the function and its derivatives appear linearly
• Homogeneous vs. Nonhomogeneous: Whether all terms contain the function or its derivatives
• Initial value problem: A differential equation with specific condition(s)

First-Order Differential Equations

Separable Equations

A differential equation is separable if it can be written as:

These are solved by integrating both sides:

Linear First-Order Equations

These have the form:

To solve these:

1. Find the integrating factor: (x) = e^P(x)dx
2. Multiply both sides by (x)
3. Integrate and solve for y

Exact Equations

A differential equation M(x,y)dx + N(x,y)dy = 0 is exact if M/y = N/x.

Exact equations can be solved by finding a function F(x,y) such that F/x = M and F/y = N. The solution is F(x,y) = C.

Applications of First-Order Differential Equations

• Exponential growth and decay: dP/dt = kP
• Cooling laws (Newton's Law of Cooling): dT/dt = k(T - T_amb)
• Falling bodies with air resistance: m(dv/dt) = mg - kv
• Chemical reactions: dC/dt = kC

Second-Order Linear Differential Equations

Calculus 2 typically focuses on homogeneous second-order equations with constant coefficients:

The solution technique involves:

1. Finding the auxiliary equation: r + ar + b = 0
2. Solving for r (roots may be real and distinct, real and repeated, or complex conjugates)
3. Writing the general solution based on the roots

Three Cases Solutions

• Distinct real roots (r, r): y = Ce^rx + Ce^rx
• Repeated real root (r): y = Ce^rx + Cxe^rx
• Complex conjugate roots ( i): y = e^x(Ccos(x) + Csin(x))

Resources and Study Tips

Recommended Textbooks

• Calculus: Early Transcendentals by James Stewart
• Thomas' Calculus by George B. Thomas
• University Calculus by Hass, Weir, and Thomas

Online Resources

• Khan Academy's Calculus 2 videos and practice exercises
• PatrickJMT's Calculus tutorials on YouTube
• Paul's Online Math Notes (Lamar University)
• MIT OpenCourseWare for single-variable calculus
• Wolfram Alpha for checking calculations and exploring concepts

Study Strategies

• Practice integration daily: Integration techniques require regular practice to master
• Focus on understanding convergence tests: Create a flowchart or summary sheet of when to use each test
• Memorize key series: Remember the common Taylor/Maclaurin series and their intervals of convergence
• Visualize vector operations: Use diagrams to understand the geometric interpretation
• Solve applications problems: Differential equations are best learned through applications
• Work in study groups: Explaining concepts to peers reinforces understanding
• Attend office hours: Your instructor can clarify confusing points
• Review Calculus 1 concepts: Differentiation rules are fundamental to integration techniques

Exam Preparation Tips

• Review all past homework and quiz problems
• Practice problems under timed conditions
• Create a formula sheet to review before the exam
• Understand the theory behind the techniques, not just how to apply them mechanically
• Get enough sleep before the exam to ensure clear thinking

University of Pittsburgh Specific Resources

• Math Assistance Center (MAC) for free tutoring
• Course-specific recitation sections for additional practice
• Previous exam samples available through instructors
• Study groups formed during class sections