**Summary**

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