Section 3.6 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 𝒇, 𝒇′, and 𝒇′′ Graphs Section 3.6 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Review What are the differences of critical number(s) and point of inflection(s)? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Fill in the Blank Table 𝒇 𝒇′ 𝒇′′ Tangent Line 𝒇′ INCREASING CRITICAL NUMBERS, POSSIBLE EXTREMA DECREASING CONCAVE UP POSSIBLE POINT(S) OF INFLECTION CONCAVE DOWN 𝒇′ 𝒇′′ Tangent Line 𝒇′ ABOVE 𝑿–AXIS (increasing) ZERO or UND BELOW 𝑿–AXIS (decreasing) INCREASING DECREASING 𝒇′ ABOVE 𝑿–AXIS (increasing) ZERO or UND BELOW 𝑿–AXIS (decreasing) INCREASING DECREASING 𝒇′′ Tangent Line BELOW graph (Rel Min) ABOVE graph (Rel Max) 𝒇′ ABOVE 𝑿–AXIS (increasing) ZERO or UND BELOW 𝑿–AXIS (decreasing) INCREASING DECREASING 𝒇′′ 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Concave Up or Concave Down 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Graphing a Derivative’s Graph Apply the equation, 𝐥𝐢𝐦 𝒉→𝟎 = 𝒇 𝒙+𝒉 −𝒇(𝒙) 𝒉 to determine the derivative Identify the zero slopes of the original function or the highest and lowest (also known as peaks and valley) points of the derivative graph Sketch based on the slope based on the 𝑿-AXIS from the original function POSITIVE Slope: Above the 𝒙-axis (Increasing) NEGATIVE Slope: Below the 𝒙-axis (Decreasing) 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Link http://www.cengage.com/math/discipline_content/stewartcalcet7/2008/14_cengage_tec/publish/deployments/transcendentals_7e/v2_8.html# 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Steps From the 𝒇′ graph, identify any and all relative minimums and relative maximums From the 𝒇′ graph, identify any concavity Sketch the graph 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Review Sketch the derivative of the function given. 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 1 Sketch a possible 𝒇 graph of the function when given the 𝒇′ graph. Concave Up Concave Up Concave Down 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 2 Sketch a possible 𝒇 graph of the function when given the 𝒇′ graph. 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn Sketch a possible 𝒇 graph of the function when given the 𝒇′ graph. 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 3 The figure below shows the graph of 𝒇′, the derivative of 𝒇. If 𝒇 𝟎 =𝟎, which of the following could be a graph of 𝒇? Concave Up Concave Up Concave Down Concave Down Relative Max Relative Min 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn The figure below shows the graph of 𝒇′, the derivative of 𝒇. Which of the following could be a graph of 𝒇? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Direction of a Derivative from a Graph A graph which is differentiable is continuous A graph which is continuous, is not always differentiable A graph is neither continuous or differentiable is discontinuous Hole Vertical Asymptotes Jump discontinuities Kinks/Cusps 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 4 Given this graph of 𝒚= 𝒇 ′ 𝒙 below. On what intervals of which 𝒇 is increasing and decreasing? At which of the 𝒙 values have a local max and values of local min? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn Given this graph of 𝒚= 𝒇 ′ 𝒙 below. On what intervals of which 𝒇 is increasing and decreasing? At which of the 𝒙 values have a local max and values of local min? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 5 Given this graph of 𝒚= 𝒇 ′ 𝒙 below. On what intervals of which 𝒇 is concaving up and concaving down? At which of the 𝒙 values correspond to the possible of inflection points? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 6 Given this graph of 𝒚= 𝒇 ′ 𝒙 below. On what intervals of which 𝒇 is concaving up and concaving down? At which of the 𝒙 values correspond to the possible of inflection points? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 7 Given this graph of 𝒚= 𝒇 ′ 𝒙 below. At what values does 𝒇 have a local maximum and/or a local minimum? On what intervals of which 𝒇 is concaving up and concaving down? At which of the 𝒙 values correspond to the possible of inflection points? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 7 Given this graph of 𝒚= 𝒇 ′ 𝒙 below. At what values does 𝒇 have a local maximum and/or a local minimum? On what intervals of which 𝒇 is concaving up and concaving down? At which of the 𝒙 values correspond to the possible of inflection points? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn Given this graph of 𝒚= 𝒇 ′ 𝒙 below. At what values does 𝒇 have a local maximum and/or a local minimum? On what intervals of which 𝒇 is concaving up and concaving down? At which of the 𝒙 values correspond to the possible of inflection points? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn Given this graph of 𝒚= 𝒇 ′ 𝒙 below. At what values does 𝒇 have a local maximum and/or a local minimum? On what intervals of which 𝒇 is concaving up and concaving down? At which of the 𝒙 values correspond to the possible of inflection points? 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 8 Let 𝒇 be a function that is even and continuous on the closed interval −𝟑, 𝟑 . The function 𝒇 and its derivatives have the properties indicated in the table below: Find the 𝒙-coordinate of each point at which 𝒇 attains an absolute max value or an abs. min value. For each 𝒙-coordinate you give, state whether 𝒇 attains an abs. max or abs. min. What are the 𝒙–coordinates of all points of inflection 𝒇(𝒙) on the graph of 𝒇. Justify. 𝒙 𝟎 𝟎, 𝟏 𝟏 (𝟏, 𝟐) 𝟐 𝟐, 𝟑 𝒇(𝒙) Positive Negative −𝟏 𝒇′(𝒙) Undefined 𝒇′′(𝒙) 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 8a EVEN: 𝒇(−𝒙) = 𝒇(𝒙) Let 𝒇 be a function that is even and continuous on the closed interval −𝟑, 𝟑 . The function 𝒇 and its derivatives have the properties indicated in the table below: Find the 𝒙-coordinate of each point at which 𝒇 attains an absolute max value or an abs. min value. For each 𝒙-coordinate you give, state whether 𝒇 attains an abs. max or abs. min. 𝒙 −𝟑, −𝟐 −𝟐 (−𝟐, −𝟏) −𝟏 −𝟏,𝟎 𝟎 𝟎, 𝟏 𝟏 (𝟏, 𝟐) 𝟐 𝟐, 𝟑 𝒇(𝒙) − + 𝒇′(𝒙) Und. 𝒇′′(𝒙) 𝒙 𝟎 𝟎, 𝟏 𝟏 (𝟏, 𝟐) 𝟐 𝟐, 𝟑 𝒇(𝒙) Positive Negative −𝟏 𝒇′(𝒙) Undefined 𝒇′′(𝒙) 𝒙 𝒙=−𝟑 𝒙=𝟎 𝒙=±𝟏 𝒙=±𝟐 𝒙=𝟑 NA 1 –1 𝒙 𝒙=−𝟑 𝒙=𝟎 𝒙=±𝟏 𝒙=±𝟐 𝒙=𝟑 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Example 8b Let 𝒇 be a function that is even and continuous on the closed interval −𝟑, 𝟑 . The function 𝒇 and its derivatives have the properties indicated in the table below: B. What are the 𝒙–coordinates of all points of inflection 𝒇(𝒙) on the graph of 𝒇. Justify. 𝒙 −𝟑, −𝟐 −𝟐 (−𝟐, −𝟏) −𝟏 −𝟏,𝟎 𝟎 𝟎, 𝟏 𝟏 (𝟏, 𝟐) 𝟐 𝟐, 𝟑 𝒇(𝒙) − + 𝒇′(𝒙) Und. 𝒇′′(𝒙) 𝒙 𝟎 𝟎, 𝟏 𝟏 (𝟏, 𝟐) 𝟐 𝟐, 𝟑 𝒇(𝒙) Positive Negative −𝟏 𝒇′(𝒙) Undefined 𝒇′′(𝒙) 𝒙 𝒙=−𝟏 𝒙=𝟏 Negative to Positive Positive to Negative 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn A function 𝒇(𝒙) is continuous on the closed interval −𝟑, 𝟑 such that 𝒇(−𝟑)=𝟒 and 𝒇 𝟑 =𝟏. The functions 𝒇 ′ 𝒙 and 𝒇 ′′ 𝒙 have properties in the table below: What are the 𝒙-coordinates for all absolute maximum and minimum points of 𝒇 𝒙 on the interval [−𝟑, 𝟑]? What are the 𝒙–coordinates of all points of inflection 𝒇 𝒙 on the graph of 𝒇 on the interval −𝟑, 𝟑 . Justify. 𝒙 (−𝟑, −𝟏) 𝒙=−𝟏 (−𝟏, 𝟏) 𝒙=𝟏 (𝟏, 𝟑) 𝒇′(𝒙) Positive Fails to Exist Negative 𝟎 𝒇′′ 𝒙 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn A function 𝒇(𝒙) is continuous on the closed interval −𝟑, 𝟑 such that 𝒇(−𝟑)=𝟒 and 𝒇 𝟑 =𝟏. The functions 𝒇 ′ 𝒙 and 𝒇 ′′ 𝒙 have properties in the table below: What are the 𝒙-coordinates for all absolute maximum and minimum points of 𝒇(𝒙) on the interval [−𝟑, 𝟑]? 𝒙 (−𝟑, −𝟏) 𝒙=−𝟏 (−𝟏, 𝟏) 𝒙=𝟏 (𝟏, 𝟑) 𝒇′(𝒙) Positive Fails to Exist Negative 𝟎 𝒇′′ 𝒙 𝒙 𝒙=−𝟑 𝒙=−𝟏 𝒙=𝟏 𝒙=𝟑 𝟒 REL MAX POS to NEG DNE 𝟏 𝒙 𝒙=−𝟑 𝒙=−𝟏 𝒙=𝟏 𝒙=𝟑 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Your Turn A function 𝒇(𝒙) is continuous on the closed interval −𝟑, 𝟑 such that 𝒇(−𝟑)=𝟒 and 𝒇 𝟑 =𝟏. The functions 𝒇 ′ 𝒙 and 𝒇 ′′ 𝒙 have properties in the table below: B) What are the 𝒙–coordinates of all points of inflection 𝒇(𝒙) on the graph of 𝒇 on the interval −𝟑, 𝟑 . Justify. 𝒙 (−𝟑, −𝟏) 𝒙=−𝟏 (−𝟏, 𝟏) 𝒙=𝟏 (𝟏, 𝟑) 𝒇′(𝒙) Positive Fails to Exist Negative 𝟎 𝒇′′ 𝒙 𝒙 𝒙=−𝟏 𝒙=𝟏 Positive to Positive Positive to Negative 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

AP Multiple Choice Practice Question 1 (non-calculator) The function 𝒇 has the property that 𝒇 𝒙 , 𝒇 ′ 𝒙 , 𝒇 ″ 𝒙 are negative for all real values of 𝒇. Which of the following could be the graph of 𝒇? (A) (C) (B) (D) 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

AP Multiple Choice Practice Question 1 (non-calculator) The function 𝒇 has the property that 𝒇 𝒙 ,  𝒇 ′ 𝒙 ,  𝒇 ″ 𝒙  are negative for all real values of 𝒇. Which of the following could be the graph of 𝒇? Vocabulary Connections and Process Answer and Justifications 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs

Assignment Worksheet 12/3/2018 7:02 PM §3.6: 𝑓, 𝑓′, and 𝑓′′ graphs