Differential Equations and Exponential Growth

Exponential Growth and Decay

Exponential growth and decay models describe processes where the rate of change of a quantity is proportional to the quantity itself. These are commonly used in population dynamics, radioactive decay, and other natural phenomena.

• General Form: , where is the initial amount, is the growth (or decay) rate, and is time.

• Half-life: The time required for a quantity to reduce to half its initial value. For decay, where is the half-life.

• Example: If a bacteria culture starts with 200 cells and grows to 360 in 30 minutes, you can solve for and predict future population sizes.

Solving Differential Equations

Differential equations relate a function to its derivatives. Common types include separable and linear equations.

• Separable Equations: Can be written as and solved by integrating both sides.

• Example: has solution .

Applications of Integration

Area Between Curves

The area between two curves and from to is:

• Example: Area between and .

Volumes of Solids of Revolution

Volumes can be found using the disk/washer or shell methods.

• Disk/Washer Method:

• Shell Method:

• Example: Volume generated by rotating and about .

Arc Length and Surface Area

• Arc Length:

• Surface Area of Revolution:

Work and Fluid Problems

• Work: where is the force at position .

• Example: Pumping water from a tank or lifting objects with variable force.

Integration Techniques

Integration by Parts

• Formula:

Trigonometric Integrals and Substitutions

• Use identities to simplify integrals involving , , , etc.

• Trigonometric substitution is useful for integrals involving , , or .

Partial Fractions

• Decompose rational functions into simpler fractions for easier integration.

Improper Integrals

• Integrals with infinite limits or discontinuous integrands. Convergence must be checked.

• Example: diverges.

Sequences and Series

Sequences

• A sequence is a list of numbers in a specific order, often defined by a formula .

• Limits of sequences determine their long-term behavior.

Series and Convergence Tests

• A series is the sum of the terms of a sequence: .

• Convergence Tests:

  • Root Test:

  • Ratio Test:

  • Alternating Series Test: For , converges if decreases to 0.

  • Comparison Test: Compare with a known convergent or divergent series.

  • Integral Test: Relates convergence of a series to an improper integral.

Power Series and Taylor/Maclaurin Series

• Power Series:

• Radius of Convergence: The interval where the series converges.

• Taylor Series:

• Maclaurin Series: Taylor series centered at .

• Example:

Parametric and Polar Coordinates

Parametric Equations

• Curves defined by , for in an interval.

• Eliminate the parameter to find a Cartesian equation.

• Derivatives:

• Arc Length:

Polar Coordinates

• Points are given by , where is the radius and is the angle.

• Conversion: ,

• Area:

• Arc Length:

• Surface Area: (for revolution about the polar axis)

Selected Table: Convergence of Series (Inferred from Problems)

Additional Info

• Some problems reference specific textbook sections (e.g., Section 7.2, 8.1, 10.1), indicating standard Calculus II topics.

• Problems cover a wide range of Calculus II material, including applications, integration techniques, sequences and series, and polar/parametric equations.