Calc Study Notes

1LS3 McMaster
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   Arcsin (sin -1 ) vs. Csc: arcsin(x) is the inverse of sin(x). csc(x) is the reciprocal of sin(x) or 1/[sin(x)]. If sin(x) = 0, then x = arcsin(0) = nπ, n = any integer.   Variable: Measurement that can change during experiment Parameter: measurement that is constant during experiment but can change between experiments Relation: set of all pairs of values of independent & dependent variables Function:  a unique assignment of a # y [of f(x)]  to a #x.    Vertical line test: verifies that a relation is a function    Horizontal line test: verifies that a function has an inverse    Types of functions: o    Algebraic: obtained from polynomials using elementary algebraic operations (+, -, ÷ , *) and not functions o   Transcendental: non-algebraic functions o   Linear: y=mx+b, where m & b are real #’s      Ratio= output/input not constant    Domain:   o   Natural domain: largest set of values where  f   is defined   o   Given domain: subset of natural domain that is explicitly stated    Graph: set of all points (x, f(x)) where x is in the domain of  f    o   Semi-log graph: plots logarithm of output against input to reduce range of a function    Composition mop of functions m &  p : (m o p)(x) = m(p(x)) Discrete-Time Dynamical System: DTDS:    Consists of an initial value and a rule that transforms the present system state one step into the future    m t+1 = f(m t  ), where the solution is the sequence of values m t   given at regularly spaced intervals (i.e., data set)    Limits of DTDS sequences: associated function m(t) that is defined for all t  ∈ R . If m(t) has a limit at infinity, then sequence shares this limit (v.v not always true)    Exponential DTDS:   o   if b t+1 =rb t   with initial condition b 0   b t  =b 0 r t    o   radioactive decay: m t =m 0 e kt    o   per capita production r : #offspring produced per member of a population    New population   = p.c.p x old population    P t+1 =r *P t        Behavior of population:    r>1 Population increases    r=1 Population constant    r<1 Population decreases (limited population model)     Additive DTDS:   o   if h t+1  =h t  +a with initial condition h 0     h t  =h 0 +at Updating function  f  :    the rule which states how output (m t+1 )is observed from input (m t  )     f updates the measurement by one time step     f m t+1 =f(m t  ) Inverse function  f  -1 :    Undoes the action of updating function (go backwards one time-step into the past)    If  f is a function with domain D & range R, then  f  -1   has domain R and range D &  f -1 (b)=a if  f (a) = b Cancellation formulas: for a function and its inverse:     f -1 (f(x))=x for all x in domain of f     f(f -1 (x))=x for all x in domain of  f -      Cobwebbing: Using the graph of the updating function (m t+1  vs. m t  ) to determine the behavior of the solutions of a DTDS    up/down from x-axis toward function  move left/right sideways toward diagonal    solution= points on the diagonal Equilibria m * :    The value(s) that doesn’t change under a DTDS      Points of Intersection b/w updating function f(m t  ) and the diagonal m t+1 =m t       Value of m* such that f(m * )=m *      Stable equilibrium: solutions move closer to the equilibrium    Unstable equilibrium: solutions move away from the equilibrium  Continuous-Time Dynamical System: CTDS:      Values given for all values of the variable (i.e., continuous curve)    Differential equations describing measurements that are collected continuously    Defined in terms of limits via continuity, derivatives, & integrals Rate of change: How dependent variable changes with a change in independent variable  Avg. rate of change  f  : slope of secant line (line that intersects 2 points on a curve) f(t  2 ) – f(t  1 )/(t  2 -t  1 ) or f(t  o   + Δt) -f(t  o )/ ((t  o   + Δt) - t  o )  Instantaneous rate of change  f ’  : slope of tangent line at point P of curve is the limit of the secant lines (slopes of secants PQ as Q approaches P, but Q ≠  P) Derivative of f at t  0 :    f ’(t  0 )= lim Δt   0 (Δf/ Δt)      f ’(t  0 )= lim Δt   0 (  f  (t  0 + Δt) -  f  (t  0 )/ Δt   or  f ’  (a)= lim h  0 (  f  (a+ h)-  f  (a)/ h Derivative of f w.r.t to x in the domain of f ’(x)      df/dx    = f ’(x)= lim h  0 (  f  (x+ h)-  f  (x)/ h Relationship b/w f ’ & f on an interval (c,d)      If  f increasing:  f ’   (+),  f rate of change (+), slope of tangent (+)    If  f decreasing:  f ’   (-),  f rate of change (-), slope of tangent (-) Limit of function lim x   a  f  (x)= L   To define how close we wish the values f(x) (y-values) to be to L , we take an interval around L , where we can make the values of   f   fall within these intervals if we pick x close enough to a , but not equal to a. How  f is defined near a (  f may or may not be defined at x=a) Limit Laws (where c is a constant and limits exist) The Squeeze Theorem : f   (  x  ) is squeezed b/w 2 other functions having same limit L . for all  x   close to a , except perhaps for  x = a . If then, lim x  a  f  (x)= L Direct substitution rule: For algebraic and some transcendental functions (exponential, logarithmic, trigonometric, inverse trigonometric), lim x  a f(x)=f(a) if a is in the domain of f(x) where base a>0   Evaluating Limits via Direct Substitution: Infinity ( ±∞ )  limit DNE #/0 = ±∞     limit DNE   Real #   limit = real # Indeterminate   simplify before doing direct sub. Indeterminate forms:   0/0 ∞/∞   ∞ 0  1 ∞  0 o   ∞ - ∞   0* ∞   Equal Limits rule: If  f  (x) =  g (x) when x ≠ a, and the limits exist, then lim x  a  f  (x) = lim x  a  g (x) Limits (end behavior, long-term behavior) at Infinity    Vertical Asymptote: A vertical line x=a is a V.A. of the graph of y=f(x) if lim x  a  f(x) = ±∞   as x    a from either side      Horizontal Asymptote y: A horizontal line y= L  is a H.A of the graph y=f(x) if lim x  ∞  f(x) = L or lim x  ∞ -  f(x) = L   o   If deg. of numerator > deg. of denominator      No H.A      As x approaches ± ∞ , limit DNE   o   If deg. of numerator < deg. of denominator      H.A. = x-axis (line y=0)      As x approaches ± ∞ , limit = 0   o   If deg. of numerator = deg. of denominator      H.A. = Divide coefficient of highest powers of x in numerator by denominator Comparing Functions that Approach ∞ at ∞  When lim x  ∞  f  (x)= ∞  & lim x  ∞    g (x)= ∞       f  (x) faster to ∞ if lim x  ∞  f  (x)/  g (x) = ∞       f  (x) slower to ∞  if lim x  ∞  f  (x)/  g (x) = 0      Same rate to ∞  if lim x  ∞  f  (x)/  g (x) = L (where L is finite, not 0)  Comparing Functions that Approach 0 at ∞   When lim x  ∞  f  (x)= 0 & lim x  ∞    g (x)= 0       f  (x) faster to 0 if lim x  ∞  f  (x)/  g (x) = 0     f  (x) slower to 0 if lim x  ∞  f  (x)/  g (x) = ∞      Same rate to 0 if lim x  ∞  f  (x)/  g (x) = L (where L is finite, not 0)  Continuity: function  f is continuous if it’s continuous at every point in its domain;    Point a  is continuous if: 1.   Lim x  a  f  (x) exists


Jul 23, 2017


Jul 23, 2017
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