B.3.5 - Graphical Differentiation

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B.3.5 - Graphical Differentiation

• Calculus - Santowski

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Lesson Objectives

• 1. Given the equation of a function, graph it and
  then make conjectures about the relationship
  between the derivative function and the original
  function
• 2. From a function, sketch its derivative and
  from a derivative, graph an original function
• 3. Given a mathematical statement about a
  function and its derivative, give a contextual
  interpretation of the mathematical statement
• 4. Determine graphically where a function is and
  isnt differentiable

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Fast Five

• 1. Find f(x) if f(x) -x2 2x
• 2. Sketch a graph whose first derivative is
  always negative
• 3. The fuel consumption (measured in litres per
  hour) of a car traveling at a speed of v km/hr is
  c f(v). What is the meaning of f (v)? What are
  its units? Write a sentence that explains the
  meaning of the equation f (30) -0.05.
• 4. Graph the derivative of the function shown in
  the next graph

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(A) Important Terms

• Recall the following terms as they were presented
  in previous lessons
• turning point points where the direction of the
  function changes
• maximum the highest point on a function
• minimum the lowest point on a function
• local vs absolute a max can be a highest point
  in the entire domain (absolute) or only over a
  specified region within the domain (local).
  Likewise for a minimum.
• "end behaviour" describing the function values
  (or appearance of the graph) as x values getting
  infinitely large positively or infinitely large
  negatively or approaching an asymptote

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(A) Important Terms

• increase the part of the domain (the interval)
  where the function values are getting larger as
  the independent variable gets higher if f(x1) lt
  f(x2) when x1 lt x2 the graph of the function is
  going up to the right (or down to the left)
• decrease the part of the domain (the interval)
  where the function values are getting smaller as
  the independent variable gets higher if f(x1) gt
  f(x2) when x1 lt x2 the graph of the function is
  going up to the left (or down to the right)
• concave up means in simple terms that the
  direction of opening is upward or the curve is
  cupped upward
• An alternative way to describe it is to visualize
  where you would draw the tangent lines gt you
  would have to draw the tangent lines underneath
  the curve

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(A) Important Terms

• Concavity is best defined with graphs
• (ii) concave down means in simple terms that
  the direction of opening is downward or the
  curve is cupped downward
• An alternative way to describe it is to visualize
  where you would draw the tangent lines ? you
  would have to draw the tangent lines above the
  curve

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(B) New Understanding of Concavity

• In keeping with the idea of concavity and the
  drawn tangent lines, if a curve is concave up and
  we were to draw a number of tangent lines and
  determine their slopes, we would see that the
  values of the tangent slopes increases (become
  more positive) as our x-value at which we drew
  the tangent slopes increase
• This idea of the increase of the tangent slope
  is illustrated on the next slides

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(B) New Term Concave Up
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(B) New Term Concave Down
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(C) Functions and Their Derivatives

• In order to see the connection between a graph
  of a function and the graph of its derivative, we
  will use graphing technology to generate graphs
  of functions and simultaneously generate a graph
  of its derivative
• Then we will connect concepts like max/min,
  increase/decrease, concavities on the original
  function to what we see on the graph of its
  derivative

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(D) Example 1
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(D) Example 1

• Points to note
• (1) the fcn has a minimum at x2 and the
  derivative has an x-intercept at x2
• (2) the fcn decreases on (-8,2) and the
  derivative has negative values on (-8,2)
• (3) the fcn increases on (2,8) and the
  derivative has positive values on (2,8)
• (4) the fcn changes from decrease to increase at
  the min while the derivative values change from
  negative to positive
• (5) the function is concave up and the derivative
  fcn is an increasing fcn

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(E) Example 2
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(E) Example 2

• f(x) has a max. at x -3.1 and f (x) has an
  x-intercept at x -3.1
• f(x) has a min. at x -0.2 and f (x) has a root
  at 0.2
• f(x) increases on (-?, -3.1) (-0.2, ?) and on
  the same intervals, f (x) has positive values
• f(x) decreases on (-3.1, -0.2) and on the same
  interval, f (x) has negative values
• At the max (x -3.1), the fcn changes from being
  an increasing fcn to a decreasing fcn ? the
  derivative changes from positive values to
  negative values
• At a the min (x -0.2), the fcn changes from
  decreasing to increasing ? the derivative changes
  from negative to positive
• f(x) is concave down on (-?, -1.67) while f (x)
  decreases on (-?, -1.67)
• f(x) is concave up on (-1.67, ? ) while f (x)
  increases on (-1.67, ?)
• The concavity of f(x) changes from CD to CU at x
  -1.67, while the derivative has a min. at x
  -1.67

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(F) Internet Links

• Watch the following animations which serve to
  illustrate and reinforce some of these ideas we
  saw in the previous slides about the relationship
  between the graph of a function and its
  derivative
• (1) Relationship between function and derivative
  function illustrated by IES
• (2) Moving Slope Triangle Movie

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(G) Matching Function Graphs and Their Derivative
Graphs

• To further visualize the relationship between the
  graph of a function and the graph of its
  derivative function, we can run through some
  exercises wherein we are given the graph of a
  function ? can we draw a graph of the derivative
  and vice versa

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(G) Matching Function Graphs and Their Derivative
Graphs
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(G) Matching Function Graphs and Their Derivative
Graphs - Answer
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(G) Matching Function Graphs and Their Derivative
Graphs Working Backwards
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(G) Matching Function Graphs and Their Derivative
Graphs Working Backwards
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(H) Continuity and Differentiability

• Recall the fundamental idea that a derivative at
  a point is really the idea of a limiting sequence
  of secant slopes (or tangent line) drawn to a
  curve at a given point
• Now , if a function is discontinuous at a given
  point, try drawing secant lines from the left and
  secant lines from the right and then try drawing
  a specific tangent slope at the point of
  discontinuity in the following diagrams
• Conclusion ? you can only differentiate a
  function where is it is continuous

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(H) Continuity and Differentiability
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(H) Continuity and Differentiability

• Follow this link to One-sided derivatives from
  IES Software
• And then follow this link to Investigating
  Differentiability of Piecewise Functions from D.
  Hill (Temple U.) and L. Roberts (Georgia College
  and State University

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(H) Continuity and Differentiability

• One other point to add that comes from our study
  of the last two examples gt even if a function is
  continuous, this does not always guarantee
  differentiability!!!!
• If a continuous function as a cusp or a corner in
  it, then the function is not differentiable at
  that point gt see graphs on the next slide and
  decide how you would draw tangent lines (and
  secant lines for that matter) to the functions at
  the point of interest (consider drawing
  tangents/secants from the left side and from the
  right side)
• As well, included on the graphs are the graphs of
  the derivatives (so you can make sense of the
  tangent/secant lines you visualized)

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(H) Continuity and Differentiability

• Continuous functions are non-differentiable under
  the following conditions
• The fcn has a corner (ex 1)
• The fcn has a cusp (ex 2)
• The fcn has a vertical tangent (ex 3)
• This non-differentiability can be seen in that
  the graph of the derivative has a discontinuity
  in it!

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(I) Internet Links

• Work through these interactive applets from
  maths online Gallery - Differentiation 1 wherein
  we are given graphs of functions and also graphs
  of derivatives and we are asked to match a
  function graph with its derivative graph

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(J) Interpretation of Derivatives

• What follows in the next slides are various
  questions which involve interpretations of
  derivatives ? what do they really mean in the
  context of word problems
• Here you are expected to verbally convey your
  understanding of derivatives along the lines of
  rates of change, rate functions, etc...
• Most of the following applications are
  independent of algebra and graphs

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(J) Interpretation of Derivatives

• Ex. 1. The cost C in dollars of building a house
  of A square feet in area is given by the function
  C f(A). What is the practical interpretation of
  the function dC/dA or f (A)? One option on
  interpreting is to consider units (dollars per
  square foot).
• Ex. 2 You are told that water is flowing through
  a pipe at a constant rate of 10 litres per
  second. Interpret this rate as the derivative of
  some function.

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(J) Interpretation of Derivatives

• Ex. 3. If q f(p) gives the number of pounds of
  sugar produced when the price per pound is p
  dollars, then what are the units and meaning of f
  (3) 50?
• Ex. 4. The number of bacteria after t hours in a
  controlled experiment is n f(t). What is the
  meaning of f (4)? Suppose that there is an
  unlimited amount of space and nutrients for the
  bacteria. Which is larger f (4) or f (8)? If
  the supply of space and nutrients were limited,
  would that affect your conclusion?

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(J) Interpretation of Derivatives

• Ex. 5. The fuel consumption (measured in litres
  per hour) of a car traveling at a speed of
  v km/hr is c f(v). What is the meaning of f
  (v)? What are its units? Write a sentence that
  explains the meaning of the equation f (30)
  -0.05.
• Ex. 6. The quantity (in meters) of a certain
  fabric that is sold by a manufacturer at a price
  of p dollars per meter is Q f(p). What is the
  meaning of f (16). Its units are? Is f (16)
  positive or negative? Explain.

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(J) Interpretation of Derivatives

• Ex. 7. A company budgets for research and
  development for a new product. Let m represent
  the amount of money invested in RD and T be the
  time until the product is ready to market.
• (A) Give reasonable units for T and m. What is T
  ' in these units.
• (B) What is the economic interpretation of T '?
• (C) Would you expect T ' to be positive or
  negative? Explain?

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(J) Interpretation of Derivatives

• Ex. 8 Interpret each sentence as a statement
  about a function and its derivative. In each
  case, clearly indicate what the function is, what
  each variable means and appropriate units. Make a
  sketch of a graph that best reflects the context.
• (A) The price of a product decreases as more of
  it is produced.
• (B) The increase in demand for a new product
  decreases over time.
• (C) The work force is growing more slowly than it
  was five years ago.
• (D) Health care costs continue to rise but at a
  higher rate than 4 years ago.
• (E) During the past 2 years, Canada has continued
  to cut its consumption of imported oil

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(K) Homework

• Textbook, p201-204
• Q1,2,4,6 (explanations required)
• Q7-14 (sketches required)
• Q1-19 (word problems)
• photocopy Hughes-Hallett p 115