Calculus AB Notes

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Calculus AB Notes
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Limits

• F has limit L as x approaches c if for any
  positive z there is a positive k so that
• for all x, where c and L are real numbers
• Notation
• It is known that

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Properties of Limits

• (these are still true if x ? c is replaced by x ?
  /- infinity)
• If L, M, c, k are real s and lim(x?c) f(x) L
  and lim(x?c) g(x) M, Then
• Sum rule lim(x?c)(f(x) g(x)) L M
• Difference Rule lim(x?c)(f(x) g(x)) L - M
• Product Rule lim(x?c)(f(x) g(x)) L M
• ? constant multiple rule lim(x?c)(k f(x))
  k L
• Quotient Rule lim(x?c)(f(x) / g(x)) L / M, M ?
  0
• Power Rule (if r and s are integers, s is not
  zero, and Lr/s is a real
• lim(x?c) (f(x))r/s Lr/s
• Polynomial functions
• if f(x) anxn an-1xn-1 a0 is a
  polynomial function and c is a real , then
• lim(x?c) f(x) f(c) ancn an-1cn-1 a0
  (i.e. plug in c for x)

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Continuity

• A function is continuous if, for every x in the
  domain, there is a real value of f(x)
• A function can be continuous on an interval
• Function y f(x) is continuous at an interior
  point c of its domain if lim(x?c)f(x) f(c)
• Function y f(x) is continuous at
• A left endpoint a of its domain if lim(x?a)f(x)
  f(a)
• A right endpoint b of its domain if lim(x?b-)f(x)
  f(b)
• If f is not continuous at xc, then c is a point
  of discontinuity

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Properties of Continuity

• If functions f and g are continuous at c, then
  fg, f-g, fg, kf for any k, f/g if g(c) ?
  0, are all continuous at c
• Composites if f is continuous at c and g is
  continuous at f(c), then composite g o f is
  continuous at c

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Derivative of a Function

• Derivative of a function f with respect to
  variable
• limh?0
• Derivative at a point xa (instantaneous rate of
  change at xa)
• limx?a
• Right-hand derivative at a limh?0
• Left-hand derivative at b limh?0-
• Many calculators use a different formula

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Differentiability

• Differentiable function derivative exists at
  every point of the functions domain
• A function is differentiable on a closed interval
  if it has
• A derivative at every interior point of the
  interval
• A right-hand derivative at the left endpoint of
  the interval
• A left-hand derivative at the right endpoint of
  the interval
• Differentiability implies
• Local linearity f resembles its own tangent
  line when close to the point of differentiation
• continuity

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How Derivatives May Fail to Exist at a Point

• Corner one-sided derivatives differ
• Ex.) at x 0
• Cusp slopes of secant lines approach infinity
  from one side and negative infinity from the
  other
• Ex.) at x 0
• Vertical tangent slopes of secant lines
  approach either infinity or negative infinity
  from both sides
• Ex.) at x 0
• Discontinuity (removable, jump, etc.)

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Extreme Value Theorem?

• If f is continuous on a closed interval a,b,
• then there is a maximum and a minimum value of f
  for the interval
• (open endpoints and unbound ends are not included)

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Mean Value Theorem

• If f is continuous on the closed interval a,b
  and differentiable on interior (a,b),
• Then there exists at least one point c in (a,b)
  where
• i.e. instantaneous rate average rate of change
  somewhere in a,b

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Rules for Differentiation

• Derivative of a constant (c) function
• Power rule n constant , and if nlt0 the
  derivative does not exist at x 0
• Constant multiple rule
• Sum and difference rule if u and v are
  differentiable with respect to x, then their sum
  and difference are differentiable when v and u
  are, and

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Rules for Differentiation (continued)

• Product rule
• Quotient rule provided v ? 0

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Extreme Values (Extrema)

• (ffunction, Ddomain of f)
• Absolute extreme values (these are also local
  extrema)
• Absolute (global) maximum value on D at c,if and
  only if f(c) f(x) for all x in D
• Absolute (global) minimum value on D at c if and
  only if f(c) f(x) for all x in D
• Relative extrema (local extreme values)
  (c interior point in f in D)
• Local maximum value at c iff f(c) f(x) for all
  x in open interval containing c
• Local minimum value at c iff f(c) f(x) for all
  x in an open interval containing c
• 83211

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Extrema
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Increasing/Decreasing/Constant

• Relations to the derivative f, if f is
  continuous on a, b, and differentiable on (a,
  b)
• Increasing
• if fgt0 at every point in (a, b), then f
  increases on a, b
• Decreasing
• if flt0 at every point in (a, b), then f
  decreases on a, b
• Constant
• if f0 for each point in an interval, then there
  is a k for which f(x)k for all x in the interval
• If f(x)g(x) for each point in an interval,
  then there is a k where f(x) g(x) k for all
  x in the interval

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First Derivative Test for Local Extrema

• For continuous function f(x)
• At a crtical point c
• If f changes from to -, then f(x) has a local
  maximum at c
• If f changes from to , then f(x) has a local
  minimum at c
• If f does not change sign, then f(x) has no
  local extreme value at c
• At a left endpoint a
• If flt0 for xgta, f has a local maximum value at
  a
• If fgt0 for xgta, f has a local minimum value at
  a
• At a right endpoint b
• If flt0 for xltb, f has a local minimum value at
  b
• If fgt0 for xgtb, f has a local maximum value at b

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Second Derivative Tests, Concavity

• On an open interval I, the graph of a
  differentiable function yf(x) is
• concave up
• if y is increasing on I
• if y gt 0 on I
• concave down
• if y is decreasing on I
• if y lt 0 on I
• Point of inflection where the graph of a
  function has a tangent and concavity changes
• ? a point where the second derivative is zero

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Second Derivative Test for Local Extrema

• If f(c) 0 and f(c) lt 0, then f has a local
  maximum at x c
• If f(c) 0 and f(c) gt 0, then f has a local
  minimum at x c

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End Behavior Models

• g(x) is a right end behavior model for f if and
  only if lim(x?8)
• g(x) is a left end behavior model for f if and
  only if lim(x?8)
• g(x) is an end behavior model for f(x) if it is
  both a right and a left end behavior model
• The term containing the highest power of x in a
  polynomial function is the end behavior model for
  that function
• In a function that is the quotient of two
  polynomials, the end behavior model is given by
  the term containing the highest power in the
  numerator, divided by the term containing the
  highest power in the denominator

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Chain rule

• If f is differentiable at the value g(x) and g is
  differentiable at x,
• Then the composite function (f o g)(x) f(g(x))
  is differentiable at x and
• (f o g)(x) f(g(x)) (g(x))

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Differentiating Parametrized Curves

• Parametrized curve (x(t), y(t))
• differentiable at t if x and y are both
  differentiable at t
• if all three derivatives exist and
  , then

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Implicit Differentiation

• Differentiate a function (with x only) with
  respect to x, using
• Then solve for by collecting terms on
  one side, factoring out and isolating it on
  one side

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Separable Differential Equations

• Separable differential equation a differential
  equation that can be expressed as the product of
  a function of x and a function of y
• If and
  , then
• Each side can then be evaluated seperately, to
  find an equation with x and y

Integrate with respect to x
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Derivatives of Trigonometric Functions
Functions that begin with c have a negative
derivative
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Derivatives of Inverse Functions

• Derivatives of inverse functions
• if f is differentiable on every point in an
  interval I, and on I (i.e.
  interval cannot have extrema other than
  endpoints)
• then f has an inverse f -1 and is
  differentiable on every point on I
• The derivative of a functions invers is the
  reciprocal of the original functions derivative
• Ex if f(1) 2 and f(1) 4
• then f-1(2) 1 and f(2) 1/4

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Inverse Trigonometric Functions

• Inverse function inverse cofunction identities
• Calculator conversion identities

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Derivatives of Inverse Trigonometric Functions
Domain all real s
Domain all real s
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Derivatives of Exponential and Logarithmic
Functions
Lim(h?0)
and
If u is a differentiable function of x
If a gt 0 and a ? 1
If u is a differentiable function of x and ugt0
If a gt 0 and a ? 1
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Power Rule for Arbitrary Real Powers

• If u is a differentiable function of x and n is
  real,
• Then un is a differentiable function of x and

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Solving Optimization Problems

• Optimize to maximize or minimize some aspect
• Pick a variable for the quantity to be maximized
  or minimized
• Write a function for the variable, whose extreme
  value is the information sought
• Graph / determine reasonable domain
• Use derivative to find critical points and
  endpoints, apply to problem

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Optimization with Economic Applications

• r(x) revenue from selling x items, dr/dx
  marginal revenue
• c(x) cost of producing x items, dc/dx
  marginal cost
• p(x) r(x) c(x) profit from selling x items,
  dp/dx marginal profit
• Maximum profit if any exist, it occurs at a
  production level where marginal revenue
  marginal cost
• Minimum average cost if any exist, it occurs at
  a production level where average cost marginal
  cost
• When only whole values of the variable are
  reasonable, the possible values directly above
  and below the non-whole answer should be checked

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Related Rates

• Relationships between rates of change problem
  solving
• If using t for time, assume all variables are
  differentiable functions of t
• Restate given information in terms of chosen
  variables
• Find an equation relating the variables
• Differentiate with respect to t
• Substitute known values into the differential
  equation obtained and evaluate

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Linearization and Newtons Method

• Linearization of f at a (if f is differentiable
  at a)
• L(x) f(a) f(a) (x-a), an approximation
• (standard linear approximation of f at a)
• L(x) is the equation of the tangent to the curve
  at a
• Center of the approximation is the point x a
• Newtons Method
• Guess a first approximate solution of the
  equation f(x) 0 (using graph, etc.)
• Use first approximation to get successive
  approximations with

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Slope Fields(also called Direction Fields)

• For , a slope field is a
  plot of short line segments with slopes f(x,y) on
  a lattice of points (x,y) in a plane
• Each segment shows the original functions slope
  at that point
• Slope fields of differential equations can reveal
  the family of functions that has that derivative
  (antiderivative)
• When using slope fields
• Horizontal lines show where the derivative 0
  the relationship between x and y at such points
  can be useful
• Where the derivative (segment pattern) is
  positive or negative can be important

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Antiderivative

• Antiderivative (sometimes represented by F)
  g(x) antiderivative of f(x) if g(x) f(x) for
  all x in the domain
• There are an infinite number of antiderivatives
  for a given function
• These possible antiderivatives, which differ by a
  constant, are the only antiderivatives of a the
  function
• Substitution of a point of the antiderivative may
  be necessary to find the antiderivative asked for
  in a problem

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Riemann Sums - Explanation

• Let f(x) be a continuous function defined on
  a,b
• Partition a,b into n subintervals by choosing
  n-1 points such that a lt x1 lt x2 lt lt xn-1 lt b
  (x is a point)
• a x0 and b xn a partition of a,b P
  x0, x1, x2, xn
• Lengths of intervals are ?x1, ?x2, ?xn
  therefore the kth subinterval has a length ?xk
• A number is chosen from each interval ck is the
  number from the kth interval
• Each subinterval has area f(ck) ?xk so
  f(ck) determines if it is , -, or 0
• Sum of products a Riemann sum for f on the
  interval a,b
• Value depends on partition and the numbers chosen
  for ck

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Riemann Sums

• All Riemann sums for a given interval on a given
  function converge to a common value if lengths of
  subintervals tend to zero
• Norm of a partition P longest subinterval
• If this tends to zero, then lengths of all
  subintervals tend to zero

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Types of Riemann Sums
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Indeterminate Forms
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Indeterminate Forms and LHôpitals Rule

• LHôpitals Rule
• (for dealing with 0/0) if f(a) g(a) 0, f and
  g are differentiable on an open interval I
  containing a and g(x) ? 0 on I if x ? a (for
  dealing with 0/0)
• Or, (for dealing with 8 / 8 , 8 0, 8 8) if
  f(x) and g(x) both approach 8 as x?a
• then
• This can be used for one-sided intervals as well
  by choosing I as an open interval with a as an
  endpoint

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Other Indeterminate Forms

• For the indeterminate forms 18, 00, 80, the
  logarithm of the function can be evaluated using
  LHopitals Rule and the result can be
  exponentiated to find the limit of the function
• Because
• (additionally, a can be finite or infinite)

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Volumes

• Volume of a solid with known integrable cross
  section area
• If A(x) the area of cross sections
  perpendicular to the x-axis, for a solid from a
  to b, then the volume from xa to xb is
• Can integrate with respect to y if equations are
  in the form x f(y), integrating from y a to y
  b
• Cavalieris volume theorem solids of equal
  height with identical cross sections at each
  height have equal volume
• Solids of revolution are formed by revolving a
  function around a given line cross sections can
  be circular
• Solids with holes can be calculated by finding
  volume without holes, then subtracting the volume
  of the hole

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Volume by Cylindrical Shells

• Can be used on bundt cake shapes
• Cut cylindrical shells from the inner hole out
  (vertical cuts perpendicular to base of cake)
• Cylindrical shells can be rolled out to become
  rectangular slabs
• Thickness of each slab is ?x, height is f(x),
  length is 2 r
• Region of integration is between the
  intersections of f(x) with the cake base, so from
  xa to xb
• Can be done with respect to y from y a to y b
  if the function is in the form x f(y)
• The variable with respect to which it is
  integrated is the variable of the axis along
  which the radius changes (base of bundt cake)

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