Overview
This lecture introduces the basic concepts and rules of differentiation and integration, focusing on essential techniques and formulas useful for introductory calculus-based physics.
The Derivative (Differentiation)
- The derivative measures the rate at which a function changes with respect to its variable; it represents the slope of the function.
- The slope between two points on a curve is estimated as the difference in function values (ฮy) divided by the difference in input values (ฮx).
- The true derivative is defined as the limit as this difference (h) approaches zero:
( f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} )
- Notation: ( f'(x) ), ( \frac{df}{dx} ), ( \frac{d}{dx} )
- For polynomials, apply the power rule: bring down the exponent and subtract one from it.
- Example: ( \frac{d}{dx} x^n = n x^{n-1} )
- The derivative of a sum or difference is the sum or difference of derivatives; constants multiply through but have zero derivative.
- Key derivatives:
- ( \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} )
- Chain rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.
- Product rule: The derivative of a product ( f(x) = g(x) p(x) ) is ( f'(x) = g'(x) p(x) + g(x) p'(x) ).
The Integral (Integration)
- The integral is the reverse of differentiation; it is sometimes called the antiderivative.
- Geometrically, the integral represents the area under a curve.
- Indefinite integrals (no limits) find general antiderivatives plus a constant ( C ).
- Power rule: ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ) (for ( n \neq -1 ))
- Always add ( +C ) since the derivative of a constant is zero.
- Definite integrals compute the net area under a curve between two bounds ( a ) and ( b ):
( \int_a^b f(x) dx = F(b) - F(a) ), where ( F(x) ) is an antiderivative of ( f(x) ).
- Key integrals:
- ( \int \cos x , dx = \sin x + C )
- ( \int \sin x , dx = -\cos x + C )
- ( \int e^x , dx = e^x + C )
- ( \int \frac{1}{x} dx = \ln x + C )
- The power rule does not apply for ( \int x^{-1} dx ).
Key Terms & Definitions
- Derivative โ The instantaneous rate of change of a function with respect to a variable.
- Integral โ The accumulation of quantities, often representing area under a curve.
- Antiderivative โ A function whose derivative is the given function.
- Power Rule โ For differentiation: ( x^n \rightarrow n x^{n-1} ); for integration: ( x^n \rightarrow \frac{x^{n+1}}{n+1} ).
- Chain Rule โ Used to differentiate composite functions.
- Product Rule โ Used to differentiate the product of two functions.
- Definite Integral โ Integral evaluated between upper and lower bounds.
- Indefinite Integral โ General antiderivative including a constant.
Action Items / Next Steps
- Practice differentiating and integrating polynomials and key special functions.
- Review the power, chain, and product rules and their applications.
- Attempt example problems involving definite and indefinite integrals.
- Prepare questions for the next class as needed.