Overview
This lecture reviews the essentials of calculus needed for physics, covering derivatives, integration, and their applications to motion and work problems.
Derivatives and the Power Rule
- The derivative of ( x^n ) with respect to ( x ) is ( n x^{n-1} ).
- Example: ( \frac{d}{dx} x^5 = 5 x^4 ).
- When a constant multiplies a function, multiply the constant by the derivative (e.g., ( \frac{d}{dx} 5x^8 = 40x^7 )).
Integration and the Power Rule
- Integration is the reverse process of differentiation.
- The indefinite integral of ( x^n ) is ( \frac{x^{n+1}}{n+1} + C ).
- Example: ( \int x^3 dx = \frac{1}{4}x^4 + C ).
- For constants: ( \int 7x^5 dx = \frac{7}{6} x^6 + C ).
Calculus Connections in Physics
- Multiplying variables (e.g., force × displacement for work) involves integration for variable quantities.
- Dividing variables (e.g., displacement/time for velocity) involves differentiation.
- If force or acceleration is not constant, use integrals to compute work or change in velocity.
Work and Variable Forces
- Work by a variable force is given by ( W = \int_{a}^{b} F(x) dx ).
- The area under the force vs. displacement curve represents the work done.
Velocity, Acceleration, and Differentiation
- Instantaneous velocity is the derivative of position with respect to time: ( v(t) = \frac{dx}{dt} ).
- Average velocity is displacement divided by time: ( \bar{v} = \frac{\Delta x}{\Delta t} ).
- Acceleration is the derivative of velocity with respect to time: ( a(t) = \frac{dv}{dt} ).
- Average acceleration: ( \bar{a} = \frac{v_{f} - v_{i}}{t_{2} - t_{1}} ).
- Instantaneous acceleration requires calculating the tangent (derivative) at a point on a velocity vs. time graph.
Tangent and Secant Lines
- A tangent line touches a curve at one point and represents instantaneous rate of change (derivative).
- A secant line touches at two points and represents average rate of change.
Key Terms & Definitions
- Derivative — Measures the instantaneous rate of change of a function.
- Integral — Represents the accumulated quantity, often the area under a curve.
- Constant of Integration (C) — An arbitrary constant added after indefinite integration.
- Tangent Line — Line touching a curve at a single point (instantaneous rate).
- Secant Line — Line intersecting a curve at two points (average rate).
Action Items / Next Steps
- Practice finding derivatives and integrals using the power rule.
- Review the relationship between differentiation/integration and physical formulas in your textbook.
- Complete assigned physics problems involving variable force and motion.