March 11 2018

Calculus - calculus studies motion and change - it allows for the understanding of nature - by means of mathematical machinery of great and sometimes mysterious power - calculus revolves around two generic problems - differential calculus is the problem of finding tangents - integral calculus is the problem of finding areas - the Fundamental Theorem of Calculus says in effect that the solution of the tangent problem can be used to solve the area problem - this is certainly the most important theorem in the whole of mathematics - it allows the two halves of calculus to be weld together into a problem solving art of astonishing power and versatility Differentiation - calculating the geometric slope of a tangent - we must be prepared to consider f'(x) purely as a function - leaving the geometric motivation, the derivative is - f'(x) = lim dx>0 f(x + dx) - f(x) / dx - this is the fundamental operation of calculus - upon which everything else depends - we follow the 3-steps specified above 1) expand f(x + dx) - f(x) and simplify dx 2) divide by dx 3) evaluate the limit as dx > 0 - we can also write difference quotient f(x + dx) - f(x) / dx as dy/dx - where dy is the specific change that results when x changes by dx - then f'(x) = dy/dx or f'(x) = d/dx f(x) Rules of Differentiation - the three step rule is slow and clumsy - we now develop a small number of formal rules - that will allow us to differentiate large classes of functions by purely mechanical procedures - d/dx c = 0 - d/dx xn = n x n-1 - d/dx c xn = c n x n-1 - d/dx (u + v) = du/dx + dv/dx - d/dx (uv) = u dv/dx + v du/dx (product) - d/dx (u/v) = v du/dx - u dv/dx / v2 (quotient) - dy/dx = dy/du du/dx (chain) - d/dx un = n u n-1 du/dx - implicit differentiation - where x and y are entangled with each other - differentiate without solving for y - applications of differentiation - maxima, minima, concavity, points of inflection, related rates - Newton's method for approximating derivatives Integration - many problems in geometry and physics depend on anti-derivatives or integration - we give individual meaning to the pieces dy and dx as differentials, changes in quantities of y and x - Indefinite Integrals - the anti-derivative of f(x) is called the integral of f(x) - if we can find one anti-derivative F(x), then all others have the form F(x) + c - d/dx F(x) = f(x) - S f(x) dx = F(x) + c - S xn dx = x n+1 / n + 1, n != 1 - S c f(x) dx = c S f(x) dx - S[ f(x) + g(x) ] dx = S f (x) dx + S g(x) dx - the S ... dx can be seen together as the opposite of d/dx - or can think in explicitly in terms of differentials, d F(x) = f(x) dx - where the integral sign S acts on a differential to produce the function itself - Differential Equations - an equation involving an unknown function and one or more of its derivatives - these equations dominate the study of nature - Definite Integrals - if an area can be calculated by exhaustion - then it can be computed easily using anti-derivatives - Sab f(x) dx = lim dx > 0 E f(x) dx The Fundamental Theorem of Calculus - if S f(x) dx = F(x) - Sab f(x) dx = F(b) - F(a) - variable limits of integration - it may be very difficult to calculate the indefinite integral of a function - but it always exists, if we use a variable limit of integration - F(x) = Sax f(t) dt - applications of integration - areas, volumes, arc lengths - force, work, energy