Curve Sketching: Chapter 3

Chapter 3 Overview: Curve Sketching

• The chapter explores the relationship between a function's derivative and its graph shape
• It allows one to use derivatives to identify key graph features and find optimal real-world values.

Chapter 3 Learning Goals

• Graphically and numerically determine intervals where a smooth function's instantaneous rate of change is positive, negative, or zero
• Describe the behavior of the instantaneous rate of change around local maxima and minima
• Solve problems involving derivatives of polynomial, sinusoidal, exponential, rational, radical, and composite functions using the product and chain rules
• Sketch the derivative function's graph from a continuous function's graph, identifying inflection points
• Recognize the second derivative as the rate of change of the rate of change; sketch first and second derivative graphs
• Algebraically find the second derivative f"(x) of polynomial or rational functions f(x) and connect its features to the original and first derivative graphs using technology
• Describe key polynomial function features from first and/or second derivatives; sketch consistent graphs and explain infinite possibilities
• Sketch polynomial functions from equations using strategies, and verify using technology
• Solve optimization problems from real-world situations involving polynomial, rational, and exponential functions
• Solve real-world problems with mathematical models and derivatives; interpret and communicate results.

Increasing and Decreasing Functions

• Knowing where quantities increase or decrease helps analyze productivity, cost, population, and service needs.
• The Geometer's Sketchpad and graphing calculators may help one identify the intervals

Increasing/decreasing Intervals w Graphing

• Graph f(x) and f'(x).
• Values where f(x) is increasing?
• Where is f(x) decreasing?
• What is true about the graph of f'(x) when f(x) is increasing?
• What is true about the graph of f'(x) when f(x) is decreasing?
• What do these answers mean re: slope of the tangent to f(x)?
• Compare f(x) and f'(x) at vertical lines where tangent slope is zero.
• What is the behaviour of f'(x) when f(x) is increasing/decreasing?

Increasing/decreasing Intervals w Graphing Calculator

• The first derivative of a smooth function f(x) determines intervals where it increases or decreases.
• f(x) increases where f'(x) > 0 and decreases where f'(x) < 0.

How to find the intervals of increase/decrease:

• Find f'(x).
• The function f(x) is increasing when 6x² + 6x - 36 > 0
• Determine the values of x when we have a solution/zero

Use the Graph of f'(x) to find intervals of increase/decrease:

• Determine f'(x), then graph it.
• Use the graph of f'(x) to determine the intervals on which the derivative is positive or negative.
• The graph of f'(x) is above the *x-*axis when *x < -*3 and x > 2, so f'(x) > 0 when *x < -*3 and when x > 2.
• The function f(x) is increasing on the intervals *x < -*3 and x > 2 and decreasing on the interval -3<x<2.

Using the First Derivative to Sketch a Function

• If f'(x) is constant at -2, then f(x) has a constant slope of slope --2.
• Given the graph of the derivative: -Positive derivative means increasing function. -Negative derivative means decreasing function. -Derivative equal to zero indicates potential local extrema.

Temperature Increase as Example

• Graphically or algebraically determine the intervals for the function, where the temperature increase is increasing
• T'(d) > 0
• Determine values where T'(d) is zero
• Determine values of expression to see where temperature is increasing/decreasing

Definitions for Graphs

• A local maximum occurs where the derivative changes from positive to zero to negative. This means all y-coordinates nearby are lower in vicinity
• A local minimum occurs where the derivative goes from negative to zero to positive, meaning surrounding y-coordinates are smaller.
• Absolute Maximum: at a if f(a) =f(x) for all x in the domain.
• Absolute Minimum: at a if f(a)≤f(x) for all x in the domain.

Maxima and Minima Key Definitions

• A critical number is a value a in the function's domain where f'(a) = 0 or f'(a) doesn't exist
• The point (a, f(a)) is a critical point
• To find absolute max/min values on an interval: find critical numbers, substitute into f(x), and substitute interval endpoints.

Local Extrema

• To find Local Extrema, perform first derivative test
• If f'(x) changes from positive to zero to negative as x increases from x < a to x > a, then (a, f(a)) is a local maximum value.
• If f'(x) changes from negative to zero to positive as x increases from x < a to x > a, then (a, f(a)) is a local minimum value.

Absolute/Local Extrema or Turning Points?

• If f'(x) = 0, then there must be a local maximum or minimum
• The maximum value in an interval always occur when f'(x) = 0

Concavity and Inflection Points

• Concave Up: graph curves upward
• Concave Down: graph curves downward
• A point of inflection occurs where the graph changes from concave up to concave down (or vice versa).

How the Tangent Slope and Curve relates

• Concavity relates to tangent line position: concave up means tangents below the curve, concave down means tangents above.
• Using The Geometer's Sketchpad or graphing it helps

Second Derivative

• The second derivative aids critical point classification and concavity analysis.
• From that perform 2nd derivative test
• If f'(a) = 0 and f'"(a) > 0, local minimum exists
• If f'(a) = 0 and f'"(a) < 0, local maximum exists.
• If f'"(a) = 0 and point of inflection exists

Using First and Second Derivative Tests

• Use first and second derivatives determine the intervals over which a function is concave up or concave down.
• How does the second derivative relate to the concavity of a function?

Example Problem

• For the function f(x) = x^4-6x^2-5, find the points of inflection and the intervals of concavity.
• f'(x) = 4x³-12x
• f'"(x) =12x²-12
• To find Point of Inflection, determine where the second derivative is zero
• The concavity of the graph changes at (*-1, -*10) and at (1, ,-*10), so these are the points of inflection. -The function is concave up to the left of *x = -*1 and to the right of x = 1. -The function is concave down between these *x-*values.

Analyzing f(X) based on f'(x)

• Given f'(x), f(x) concave down where f'(x) is decreasing and concave up where f'(x) is increasing.
• This is only one of the possible graphs that satisfy the given conditions.

Simple Rational Functions

• Rational functions show up as relationships between velocity/time or temperature/distance. *
• Asymptotes aren't f(x) but used as boundary
• x = a is a vertical asymptote if f(x) → ± ∞ as x → a from the left and/or right
• Can graph using chart, graphing calculator and looking at limits

Limit and Rational F(x) Behavior

• As x approaches -2 from the left, f(x) approaches a large positive number.
• As x approaches –2 from the right, f(x) approaches a large negative number

End Behaviour

• Vertical asymptotes must be considered when finding intervals of concavity or intervals of increase or decrease.
• Use patterns to determine how a function behaves in the vicinity of a vertical asymptote.
• To find the x intercepts - set the equation to 0
• To find the y intercepts - sub in 0 to x

Determining Concavity

• If concavity can occur where 2nd derivative is non existent and is not sufficient

The Steps Involved in Optimization Questions

• Identify what the question is asking.
• Define the variables; draw a diagram if it helps.
• Identify the quantity to be optimized and write an equation.
• Define the independent variable. Express all other variables in terms of the independent variable.
• Define a function in terms of the independent variable.
• Identify and state any restrictions on the independent variable.
• Differentiate the function.
• Determine and classify the critical points.
• Answer the question posed in the problem.
• The context of a problem often dictates the interval or domain to be considered.
• Answers should be verified to ensure they make sense, given the context of the question.