CHAPTER THREE: APPLICATIONS OF THE DERIVATIVE

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CHAPTER THREEAPPLICATIONS OF THE DERIVATIVE

• Hare Krsna Hare Krsna Krsna Krsna Hare Hare
• Hare Rama Hare Rama Rama Rama Hare Hare
• Jaya Sri Sri Radha Vijnanasevara (Lord Krsna, the
  King of Math and Science)
• KRSNA CALCULUS PRESENTS

Edited on October 7, 2003
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HARE KRISHNA!

• Welcome back, dear students! I pray that you are
  all well.
• In order to understand this chapter, you will
  have to know the following
• Definition of the derivative (at least the
  concept of it difference quotient.), derivatives
  of polynomials, trig functions, implicit
  differentiation, chain rule, and the rectilinear
  motion relationships (position, velcoity, and
  acceleration.)
• If you do not understand this, please review the
  last chapter (Chapter 2) and keep practicing that
  chapter until you understand those topics above.

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WHERE WE HAVE GONE?

• As one mathematician said, Calculus is a
  journey. A good tourist will think about the
  past here and use it to him or her in the
  future.
• Before we move onto the next chapter, lets take
  a step back and see what we have covered so far,
  just to help us remember important things.

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SUMMARY OF PREVIOUSLY STUDIED MATERIAL

• In chapter 1, we had studied limits. An informal
  and most commonly used definition of the limit is
  simply the question, When I make x closer and
  closer and closer to some number c, what f(x) am
  I approaching to?
• Asymptotes were explored using limits. Limits
  help determine end-behavior.
• Continuity was briefly mentioned. Continuity is
  present at xc is f(c) exists, if the limit as x
  approaches c from both sides exist and are the
  same value, and if the limit as x approaches c
  and f(c) are equal.

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SUMMARY

• With the help of limits, we were able to
  calculate instantaneous rates, and the slope of
  the line tangent to the curve at a certain point.
  By letting the change in x be infinitesimally
  small in the slope formula (difference quotient),
  we are able to get derivatives.
• A differentiable function is continuous function,
  but a continuous function is not always
  differentiable.

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DERIVATIVES.

• Practice of derivative rules and formulas must be
  emphasized. Derivatives and Integrals (next
  chapter) will be the backbone to calculus.
• Definitely study the differentiation rules from
  the last chapter!!!!!!

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APPLICATIONS OF THE DERIVATIVE

• This chapter will underscore the uses of
  derivatives in the following cases.
• How to graph functions by knowledge of critical
  points and derivatives.
• How to differentiate inverse, exponential and
  logarithmic functions.
• Differentials
• Related rates and optimization problems

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CURVE SKETCHING

• You can easily graph any function by knowing
  three things.
• 1) ZEROS AND UNDEFINED SPOTS
• 2) MAXIMUM AND MINIMUM POINTS
• 3) CONCAVITY AND INFLECTION POINTS.

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CRITICAL POINTS

• Lets find the points where the function is zero
  or undefined.
• We want to graph f(x)x32x2x.
• Obviously, there are no undefined spots, due to
  the fact that this is a polynomial function. All
  polynomial functions are continuous, thus
  differentiable.
• To find the zeros, set the entire function to 0.
• 0 x32x2x ? Original function
• 0x(x22x1) ? Factored
• 0x 0(x1)(x1) ? Factored the trinomial and
  set both factors equal to zero.
• x0, -1 ? x is solved for.
• Thus x0 and x-1

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MAXIMUM OR MINIMUM POINTS

• Here is the graph of the f(x).

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MAXIMUM AND MINIMUM

• Notice that if you draw tangent lines at the
  maximum and the minimum, you will see that you
  will get horizontal lines. That means the slope
  is zero. Thus, the derivative is zero. In other
  words, at a maximum or minimum, the derivative is
  zero.

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MAXIMUM AND MINIMUM POINTS

• Since the f(x)0, we can solve for the maximum
  and minimum points.
• First, find the derivative of f(x)x32x2x.
• f(x) 3x24x1 ? Differentiation using power
  rule and sum/difference rule.
• 0 3x24x1 ? Max/Min always have f(x)0.
• 0(3x1)(x1)?Factoring to solve for x.
• 03x1 0x1 ? Setting factors equal to 0.
• x-1/3 and x-1 ? Solve for two answers of x.
• We know that x -1/3 and x-1 are the possible
  points, however, without a graph, we cannot
  determine which is the max and which one is the
  min.

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SIGN ANALYSIS TEST FOR MAX/MIN.

• You can do a sign chart and determine the sign
  between possible zeros. Look how it is set up.
• What the sign of the two factors to the right of
  x-1/3?
• What is the sign of the two factors between -1
  and -1/3?
• What is the sign of the two factors to the left
  of -1?
• Since they are factors, you multiply them.
  Account for overall sign.

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DERIVATIVE OF SIGNS

• If the derivative goes from negative slope to
  zero to positive slope, then x is where the
  minimum is.
• If the derivative goes from positive slope to
  zero to minimum slope, then x is where the
  maximum is.

Negative slope
Positive Slope
zero
zero
Positive Slope
Negative slope
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WHAT WE KNOW

• We know that this is a cubic function, with x-1
  and x0 as the two zeros. No undefined points
• We know that the maximum is at x -1 and the
  minimum is at x -1/3.

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HORIZONTAL TANGENTS

• Be careful when solving for x when f(x) is 0.
  Perhaps you will have such function where the
  slope is zero at a certain point, but that point
  is neither maximum nor minimum. y(x)x3 for
  example. Once you differentiate that, you get
  y(x)3x2. y(x)0. Therefore, you equate 0 and
  3x2. You will see that x 0. However, by doing a
  sign analysis chart, you will see that before and
  after 0, there is no sign change. The slope
  continues to be positive.

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CONCAVITY

• Concavity is very difficult to define.
• A good way to define concavity is using a spoon.
  When you hold a spoon the right way (where the
  milk is in the spoon), the spoon is said to be
  concave up.
• When the spoon is down (when the milk is not in
  the spoon the spoon is not in the right way),
  then the spoon is said to be concave down.

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CONCAVITY

• An upward parabola is concave up. A downward
  parabola is concave down.
• The f(x) is above the tangent line when it is
  concave up. The f(x) graph is under the tangent
  line when it is concave down.
• The concavity is the rate of slope change. Hint
  hint!! RATE OF CHANGE!! The rate of change of the
  slope, means the derivative of the slope The
  slope is the derivative of the function. So the
  concavity is the derivative of the derivative.
  The first derivative of slope and the second
  derivative of the function.

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CONCAVITY GRAPHS
Both function have negative slopes. f(x)lt0. The
blue function is concave up, since it is above
the tangent line. The red function is concave
down, since it is below its respective tangent
line.
Both function have positive slopes.
f(x)gt0. However, the red function is concave up,
since it is above the tangent line. The blue
function is concave down, since it is below its
tangent lines.
RED f(x)gt0 f(x)gt0
BLUE f(x)gt0 f(x)lt0
BLUE f(x)gt0 f(x)gt0
RED f(x)gt0 f(x)lt0
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ZERO CONCAVITY

• Remember when the function, f(x), was at a
  maximum or minimum, the derivative, the slope,
  f(x) was 0.
• When the slope is at its maximum or minimum, the
  derivative of the slope is 0.
• In order words, when f(x) is at its maximum or
  minimum, then the derivative, f(x), the
  concavity will be 0.
• Concavity is the slope of the slope. ?!!

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Side note Vertical tangents.

• Lets examine for a moment, the vertical line
  graph.
• There is no such function that when graphed has a
  vertical line.
• If you look at the slope of a line (Dy/Dx). You
  will see that the difference in y is some number.
  However, since there is no change in x, that will
  be equal to zero. n/0 is infinitely big that is
  undefined.
• However, there are functions with vertical
  tangents. The cube root of x at x0 has a
  vertical tangent. Therefore, the derivative of
  the cube root function will be defined everywhere
  except x0.
• Just be aware of that!

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CONCAVITY AND INFLECTION POINTS

• The point where the change in derivative sign
  (positive vs. negative slope) is called extrema
  (i.e. max/min points) f(max) and f(min), their
  derivatives at those points will both be 0.
• The point where the change in concavity occurs is
  called the inflection point.
• Since this is where the slope, f(x), is the
  maximum, the steepest or the flattest, the
  derivative of slope, f(x), will be 0.
• After solving for x and doing a sign analysis,
  you will find the inflection points.

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FINDING INFLECTION POINTS

• The original function was x32x2x.
• Since we want to find the inflection point, we
  must obtain the second derivative.
• f(x) x32x2x. ? GIVEN
• f(x) 3x24x1. ? FIRST DERIVATIVE
• f(x)6x4 ? SECOND DERIVATIVE
• Since the inflection points are found by solving
  for x when f(x) 0.
• 06x4 ?f(x) 0
• x -2/3 ? SOLVED FOR X.
• Now we must do sign analysis.
• Everything right of -2/3 for the factor 6x4 is
  positive. Everything left of it is negative.
  Thus, x-2/3 is an inflection point for x32x2x.
• Everything right of x-2/3 is concave up.
  Everything the to left of x-2/3 is concave down.

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When sketching this curve

• Make sure your zeros are at the x values.
• Make sure your max, mis, and horizontal tangents
  are drawn at the proper spots.
• Make sure your inflection point is accurately
  drawn.
• A good way of thinking about inflection points is
  when the tangent lines switch sides with respect
  to the graph.

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YOUR GRAPH
f(x)gt0 f(x)gt0
f(x)0 f(x)0 f(x)lt0
ZERO
MAXIMUM
ZERO
f(x)lt0 f(x)gt0
f(x)0
f(x)0 f(x)gt0
f(x)lt0 f(x)lt0
MINIMUM
f(x)gt0 f(x)lt0
INFLECTION POINT
f(x)lt0 f(x)0
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ROLLES THEOREM

• Some people hate calculus a lot. For example,
  they might like to find instantaneous velocity by
  saying distance/time. But we know that is not
  true.
• Rolle was a mathematician who tried to disprove
  calculus. He disliked it very much. He wanted to
  a lot of geometry and algebra. Well, you know, if
  you hate something a lot, it is just as much as
  loving it a lot. Similarly, Rolle hates calculus
  so much, that he had a theorem that is very
  helpful IN calculus.
• This theorem will not be tested very frequently
  in exams, but its best to know it anyway.

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ROLLES THEOREM (MEAN VALUE THEOREM)

• There will always be an average rate line that
  will be parallel to the tangent line of a certain
  point. This means they have the same slope.
• You can see that the average rate line between
  x-2 and x2 is parallel to the tangent line of
  x0. Thus, you can conclude, that they have the
  same slope.

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DIFFERENTIATION OF INVERSE FUNCTIONS

• An inverse of a function is found by reflecting
  the graph over the line yx.
• The function is found simply by switching the y
  with the x and vice verse.. For example If y
  3x2 with coordinates (0,2) as one possible
  point, then its inverse is x3y2 with
  coordinates (2,0). I just switched the x and y
  around. Thats all.
• Sometimes, a function may have an inverse that is
  not a function. yx2 is function and xy2 is not.
  Remember, a function is a function if for any x,
  there is one and only one y value. The vertical
  line test determines that. (Course III Info for
  N.Y.S students).

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DIFFERENTIATION OF INVERSE FUNCTIONS

• From pre-calculus, you know that if f(x) has its
  inverse g(x), then the composition f(g(x)) is x.
• If f(x) has g(x) as its inverse function, then
  the formula for finding the derivative of a
  function is.

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DERIVATIVE OF THE INVERSE OF yx2.

• f(x)
• g(x)
• Composition of f and g
• dy/dx of f(x)
• Composition of f and g.
• Inv. Deriv. Formula
• Substitution.

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CHECK

• You can check it also using the power rule.

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DERIVATIVE OF EXPONENTIAL FUNCTIONS

• The natural exponential function, f(x)ex, has
  the base e. e is a transcendental number (just
  like 16,108, p, f...). It is named after its
  founder, Leonard Euler. e is derived and found
  many ways. e is a very special number in
  calculus. You will see why in a moment. e is
  approximately 2.718 If you define it using
  limits, e

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DERIVATIVE OF THE EXPONENTIAL FUNCTION

• Derivative definition
• f(x)ex
• Law of exponents, exh(ex)(eh)
• Factor out ex.
• Previously proven limit
• Product Rule of limits
• Derivative

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THE DERIVATIVE OF THE EXPONENTIAL FUNCTION

• This is one of the two functions whose derivative
  is its own function! d(ex)ex! Amazing how
  transcendental functions work! Jaya! Jaya!

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THE NATURAL LOGARITHM FUNCTION yln x

• The natural logarithm is NOT taking a piece of
  log naturally from a tree, play rhythms on it! ?
• The natural logarithm is the inverse of the
  natural exponential function. It has e as its
  base. There are many ways yln x is derived by.
• yln x is defined for all xgt0.
• We can find the derivative using the inverse
  differentiation rule since ln x is the inverse of
  ex.

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DIFFERENTIATING yln x.

• f(x)
• g(x) ? inverse of f(x)
• Composition of f(x) and g(x)
• dy/dx of f(x)
• Composition of f(x) and g(x)
• If f(g(x)) f(g(x)), and if f(g(x)) x, then
  f(g(x))x.
• Inverse Differentiation Rule
• Replacing f(g(x)) with x.

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THEREFORE

• I find it pretty cool that an algebraic function
  like y 1/x is the derivative of transcendental
  function y ln x. Transcendental functions are
  functions that cannot be derived simply by
  algebra.

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INVERSE TRIGONOMETRIC FUNCTIONS

• The inverse of the function y sin x is y
  arcsin x.
• NOTE arcsin x is NOT a function. For example,
  arcsin (½) p/3, 5p/6, -7p/6, -11p/6, so on
  since one x value produced an infinite number of
  y values, you know that this is not a function
• However, Arcsin x IS a function (notice capital
  A, compared to previous lower-case a). Arcsin x
  is restricted from -p/2, p/2, just as Arccos x
  is restricted from 0,p and Arctan x is
  restricted from (-p/2, p/2). Note that the ()
  parenthesis represent the open interval excluding
  endpoints. Remember than tan (p/2) 8.
• If you look at the inverse cosine function, you
  will notice that it is just the inverse sine
  function reflected over the x axis. Therefore,
  the slope, the derivative, of the inverse cosine
  function will be negative the slope, derivative,
  of the inverse sine function. Same thing will
  occur with tangent and cotangent, secant and
  cosecant functions.
• d(arcsin x)-d(arccos x)
• d(arctan x)-d(arccot x)
• d(arcsec x)-d(arccsc x)

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DIFFERENTIATION OF THE ARCSINE FUNCTION

• f(x)
• g(x) ? inverse function
• Composition of f and g.
• dy/dx of f
• Composition of f and g.
• Derivative formula for finding inverses of
  functions
• Applying f(g(x)).
• What is cos(arcsin x)?

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The denominator

• To simplify the denominator to more practical
  terms, we know that yarcsin x is the same as
  sin(arcsin x) sin y x. Then to solve for cos
  y.

Pythagorean Trig Identity sin y x Solved for
cos y.
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AFTER APPLYING cos y IN THE DENOMINATOR

• We see that

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USING THE SAME PROCESS

• If you use the derivative of an inverse rule for
  the arctangent and the arcsecant functions, you
  will see that

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MISCELLANEOUS THINGS TO KNOW

• Some textbooks will make great emphasis on this.
  Other textbooks wont talk about it too much.
  Since the AP and many college calculus I courses
  do not test this topic greatly, I wont discuss
  it in great depth.
• You can treat dy/dx as a fraction of the
  infinitesimally small change in y over the
  infinitesimally small change in x.
• You can consider y x2, ?? dy 2x dx since
  dy/dx 2x. dy is said, a differential amount of
  y. Differential means an infinitesimal change in
  a specific direction.

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EXAMPLE DIFFERENTIAL PROBLEM

• If you take a cube with each side having length
  8, what is the differential volume if you have a
  cube that has each side having length 7.99?

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PROBLEM WORKED OUT

• The volume of a cube is given
• dV/dx is the differential amount of volume over
  the differential amount of change in length. The
  change in length is simply 8-7.99.01dx.
  Multiply both sides by it so you isolate dV. Then
  plug in 7.99 for x, and plug in 0.01 for dx and
  calculate the differential amount of volume.

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APPLICATION

• At the Sri Sri Gaura Nitai temple, they want to
  make the rectangular temple room such that the
  perimeter is 1000 feet. Find the maximum length
  and width so that everyone can have the fortune
  to the Sri Sri Gaura Nitai. In addition, find the
  area.

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OPTIMIZATION

• Optima- what???
• Optimization problems is a fancy way of saying
  maximum minimum problems.
• The best way to start out is to draw a picture to
  help yourself visualize the problem.

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THEN..

• Since we have two variables, ?l and w, we have to
  solve everything in terms of one variable.
• We know that A l w and Prect 2 l 2w.
• So we can solve for l in either equation. Most
  preferably use the P equation.

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And finally

• Substite the expression for l into the area
  formula.

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According to the problem

• P 1000 ft so
• But to find maximum w, the derivative will have
  to equal 0.
• Then solve for w.

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Then

• Plug w in for the expression for length.
• Coincidentially, the width and length are the
  same. Therefore, the temple room will be square
  so that everyone could see the Lord Sri Sri Gaura
  Nitai ? Jaya!

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FIND THE AREA

• Simple.. The length is 250 ft. The width is 250
  ft. Multiply them together..
• A 625,000 sq ft.
• Nice big temple aint it?? HARI HARI BOL!!!!

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RELATED RATES

• Remember how I constantly emphasized, RATE
  DERIVATIVE
• Rate specifically deals with time.
• Here is an example
• The diameter and height of a paper cup in the
  shape of a cone are both 4 in. and water is
  leaking out at the rate of 0.5 in3/ sec. Find the
  rate at which the water level is dropping when
  the diameter of the surface is 2 in.

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VARIABLES SOUND NICE.. (laughs)

• V is volume in cubic inches
• h is height.
• t is time
• Volume formula (known)
• 2r h, since diameterh.

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Differentiate with respect to time

• Remember now we are differentiating with time!!!
  So, we must differentiate implicitly!! Remember
  chapter two??
• Since we know that the rate of leak is ½ cu in/
  sec, dV/dt is ½ since we are decreasing.
• Solve for the rate of change for the height.
• Since there is no variable for time, the rate of
  change for the height is constant for any value
  of t.
• Replace the h with 2 in and the rate of the water
  level is dropping with the height and time is
  1/2p in/sec.

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SUMMARY

• We covered a good amount of material in this
  chapter. Lets go over what we learned.
• You can find the maximum or minimum of f(x) by
  setting f(x) 0. Solve for x and do a sign
  analysis.
• You can find concavity of f(x) by setting
  f(x)0. Solve for x and do a sign analysis.

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DERIVATIVES

• We learned how to differentiate more functions
  using the inverse rule.
• COMMIT THESE TO MEMORY AS WELL AS THE FUNCTIONS
  IN CHAPTER TWO!!! IF YOU DO NOT REMEMBER ANY
  FUNCTIONS DERIVATIVE, DO NOT GO ON TO CHAPTER
  FOUR!! COMMIT EVERYTHING TO MEMORY!!!!

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OPTIMIZATION/RELATED RATES

• For optimization problems, relate everything to
  one variable. For example, area and perimeter
  with sides a and b. Solve for a, for example, and
  plug the a expression in for the area formula so
  you have an area function A(b). Then
  differentiate that and set it equal to 0. DONT
  FORGET SIGN ANALYSIS!!!
• For related rate problems, gather all of your
  variables together and use any geometry formulas
  you need. Differentiate with respect to TIME. You
  most likely will need to use implicit
  differentiation. It
• It helps in both scenarios to draw a picture of
  the problem to get a better understanding.

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TEST

• Please visit the following website to take a
  practice exam and please look over the answers.
• If you have any difficulty, please e-mail me at
  kksongs_1_at_hotmail.com.
• Please read help statement prior to e-mailing me.
• Good luck! Hare Krsna!

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THE END