5.5 Notes Solutions
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Bixby Hs**We aren't endorsed by this school
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CALC 101
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Mathematics
Date
Jan 12, 2025
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7
Uploaded by MinisterNeutron22921
Lesson 5: Curve SketchingCurve Sketching is a procedure for analyzing a function and its behavior without the aid of a graphing utility. We can use the relationships of ݂,݂ᇱ,and ݂ᇱᇱover specific intervals to quickly sketch the graph of the function ݕ=݂(ݔ). We already know how to find domain, range, intercepts and asymptotes. In the last two lessons we learned how the first and second derivatives can be used to determine four basic behaviors and curvature of a function. If we know the sign of ݂ᇱand ݂ᇱᇱover some given interval, we know exactly what the graph of ݂(ݔ)looks like on that interval.EX #1: Complete the table below by sketching the general shape of the curve when each condition is met.EX #2: Sketch a possible graph of the function ℇ =ࢍ(࢞)that satisfies the following conditions: i. ݃ᇱݔ> 0 on −∞, −1 ∪3, ∞and ݃ᇱݔ< 0 on −1, 3ii. ݃ᇱᇱݔ> 0 on (−∞, −3) ∪ (−1, 3)and ݃ᇱᇱݔ< 0on (−3, −1) ∪ (3, ∞)iii. andTopic 5.8: Sketching Graphs of Functions and Their Derivativesࢌᇱ࢞>ࢌᇱ࢞<ࢌᇱᇱ࢞>ࢌᇱᇱ࢞<lim௫→ିஶ݃ݔ= −4lim௫→ ஶ݃ݔ= 4© 2021 Jean Adams Flamingo Math.com
Sketching Function Graphs from the Derivative GraphThe Function Behavior Chart from our previous lesson can be a visual aid to using proper vocabulary as well as helping us make graphical representations from verbal descriptions. Discovering Key Concepts of Analyzing f’ GraphsEX #3: Can you sketch a possible graph of fgiven ݂ᇱ?1. Zeros of ݂ᇱ(ݔ)may be ____________________________ of ݂ݔ.2. Extrema of ݂ᇱݔare possible __________________________________________ for ݂(ݔ).3. When ݂ᇱݔis above the x-axis then ݂ݔis ______________________________________.4. When ݂ᇱݔis below the x-axis then ݂ݔis ______________________________________.5. If ݂ᇱ(ݔ)is increasing then ݂ᇱᇱݔ> 0 , so ݂(ݔ)is _____________________________________________.6. If ݂ᇱ(ݔ)is decreasing then ݂ᇱᇱݔ< 0 , so ݂(ݔ)is _____________________________________________.Function Behaviorࢌ(࢞)ࢌᇱ(࢞)ࢌᇱᇱ(࢞)SymbolVocabularyPositiveNegativeIncreasingDecreasingConcave UpConcave Down݂ᇱ(ݔ)© 2021 Jean Adams Flamingo Math.com
Reading Function Graphs and BehaviorsTopic 5.9: Connecting a Function, Its First Derivative, and Its Second DerivativeEX #4: A. Suppose the graph shown above is the graph of a function ࢟=ࢌ(࢞)on the interval [−ૡ,ૡ]:i. On what open interval(s) is ݂(ݔ)both increasing and concave up?ii. On what open interval(s) is ݂(ݔ)both decreasing and concave down?iii. At what x-value(s) does ݂(ݔ)have inflection points?B.If the graph above is the graph of the derivative ࢟=ࢌᇱ(࢞)on −ૡ,ૡ: i. On what open interval(s) is ݂(ݔ)decreasing ? Justify.ii. At what x-value(s) does ݂ݔhave a local maximum ? Justify.iii. On what open interval(s) is ݂(ݔ)concave down? Justify.iv. At what x-value(s) does ݂(ݔ)have an inflection point? Justify.C.Suppose the graph shown above is the graph of a function ࢟=ࢌᇱᇱ(࢞)on the interval [−ૡ,ૡ]:i. On what open interval(s) is ݂(ݔ)concave down? Justifyii. At what x-value(s) does ݂ݔhave an inflection point? Justify.© 2021 Jean Adams Flamingo Math.com
Sketching Graphs of Functions and Their DerivativesEX #4 (continued)D. Suppose the graph below is the graph of a function ࢟=ࢌ࢞on the interval −ૡ,ૡ.On the same set of axes, sketch and label a possible graph of ࢌᇱ(࢞)in red.E. If the graph below is the graph of the derivative ࢟=ࢌᇱ(࢞)on −ૡ,ૡ. Sketch and label a possible graph of ࢟=ࢌ(࢞)given that ࢌ−ૡ= −.© 2021 Jean Adams Flamingo Math.com
Guidelines for Analyzing the Graph of a Function:1. Determine the domain, intercepts, asymptotes, symmetry of the graph.2. Find critical points and intervals where the function is increasing and decreasing.3. Determine local maximum and minimum points.4. Determine concavity and find points of inflection.5. Sketch the curve.EX #5: Analyze and sketch the graph of ݃ݔ=ݔ+ 2ݔ− 1ଶA. Find the xand y-intercepts.B. Find the first and second derivatives.C. Complete a sign chart for ݃ᇱݔto find intervalswhere ݃ݔis increasing or decreasing.D. Determine any relative extrema.E. Complete a sign chart for ݃ᇱᇱݔto find intervals where ݃ݔis concave up or concave down.F. Identify any points of inflection.G. Sketch the graph of ݃(ݔ).© 2021 Jean Adams Flamingo Math.com 0=Xi-3++2x(x-3+2)X(+-2)(+-1)
EX #6: The graph of ࢟=ࢌᇱ(࢞)is shown below, on the interval [−ૡ,ૡ]. If ࢌ−ૡ=. On the same set of axes, sketch a possible graph of the function ࢟=ࢌ(࢞).EX #7: The function ࢌ(࢞)is defined and differentiable on the closed interval −,.The graph of ࢟=ࢌᇱ(࢞), the derivative of ࢌ, consists of three line-segments and a semicircle, shown in the figure at right. For each question below find the values for ࢟=ࢌ࢞on the interval −<࢞<and justify your reason. A. Find the x-coordinate of each critical point for ݕ=݂(ݔ).B. Find the x-coordinate of each relative extrema of ݕ=݂(ݔ)and label as a maximum or minimum. C. Find the open interval(s) over which the function ݕ=݂(ݔ)is increasing or decreasing.D. Find the x-coordinate of each point of inflection for ݕ=݂(ݔ).E. Find the interval(s) over which the function ݕ=݂(ݔ)is concave up or concave down.© 2021 Jean Adams Flamingo Math.com
EX #8: Let ࢎ(࢞) be a function that is continuous on the interval [,]. The function ࢎis twice differentiable except at ࢞=, where the derivative of ࢎdoes not exist . The function and its derivatives have the properties indicated in the table below. Use the table to sketch a graph of ࢟=ࢎ࢞.EX #9: The function ݂is defined and differentiable on theclosed interval [−6, 5]. The graph of ݕ=݂ᇱ(ݔ)the derivative of ݂, is shown in the figure below and has horizontal tangent lines at ݔ= −3 and ݔ= 2.A.Find the x-coordinate(s) of each relative extrema on−6 <ݔ< 5. Identify as a max or min. Justify.B.Find the x-coordinate of each point of inflection forݕ=݂(ݔ). Justify.C.Let ݃ݔ=ݔଷ−݂(ݔ). Find ݃ᇱ(−3).࢞00 <ݔ< 222 <ݔ< 333 <ݔ< 555 <ݔ< 6ࢎ(࢞)3Positive0Negative−3Negative0Positiveࢎᇱ(࢞)Negative−3NegativeDNEPositive1Positiveࢎᇱᇱ(࢞)−2Negative 0PositiveDNENegative2Positive© 2021 Jean Adams Flamingo Math.com −1