Final: General Concepts

Functions

• domain, range
• function definition (table, graph, formula, words)
• types of functions (polynomial, rational, trig, power, root, exponential, etc.)
• symmetry (even, odd)
• asymptotes (vertical, horizontal, slant)
• increasing, decreasing
• concavity
• continuities and discontinuities (connectedness)
• differentiability (smoothness)
• one-to-one, invertible
• compositions of functions
• shifts and stretches of functions

Limits

• limit laws
• tangent versus secant lines
• limits at infinity, and infinite limits
• continuity and discontinuity
• Intermediate value theorem
• pinching or squeeze theorem

Derivatives and differentiation

• Limit definition:

  

• relationship between the size and sign of the derivative of f and the shape of the graph of f
• sum, difference, product, quotient rules
• chain rule (derivatives of compositions)
• implicit differentiation
• related rates problems
• optimization problems
• antiderivatives
• higher derivatives
• linearization
• Mean value theorem (Rolle's theorem on a tilt)
• Newton's method

Optimization

• extrema (maxima, minima; local and global)
• first and second derivative tests
• check endpoints

Integrals and Integration

• Motivation: areas
• using approximation rules to estimate areas (left, right, midpoint, trapezoidal)
• taking a limit and defining the area exactly
• Fundamental Theorem of Calculus
• substitution rule (the chain rule in reverse)
• integrals as more general than simply areas: so

  

  means that in computing quantity S (area, length, volume, time, dollars, etc.) we simply add up lots of tiny amounts (infinitesimals) of S. Examples:

  • Volume calculations:

    

  • The time to arrive at State route 126 from Exit 3 on I-75:

    

    where v(x) is the average automobile velocity at every point along the road from point a to point b. (dt was calculated using the DiRT formula - don't forget the dirt formula!).

• using symmetry to help calculate integrals
• areas between two curves
• definite versus indefinite integrals