Integral Defined Functions Day 2 & Day 3 AP Calculus AB

Example 1a The graph given is 𝑓 𝑡 =2 on [0,𝑥]. Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡=2𝑥 for different values of 𝑥. 𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡 Using your answer from part (A), fill in the following table 𝒙 −𝟐 (−𝟒) −𝟏 (−𝟐) 𝟎 (𝟎) 𝟏 (𝟐) 𝟐 (𝟒) 𝟑 (𝟔) 𝐴(𝑥) −8 −4 4 8 12 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑 𝐴(𝑥)

Example 1a: Continued 𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡 𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 𝐴(𝑥) −8 −4 4 8 12 𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡 Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥) 𝐴 𝑥 =4𝑥

Complete the Table: Follow Ex 1a Parts (A)-(C) Examining an Upper Bound of 2𝑥 𝑓(𝑡) Equation, Interval 𝐴(𝑥) Integral Defined Function Equation 𝐴′(𝑥) 𝑓 𝑡 =2 [−2, 2𝑥] 𝐴 𝑥 = −2 2𝑥 2 𝑑𝑡 𝐴 𝑥 =4𝑥+4 𝐴 ′ 𝑥 =4 0, 2𝑥 𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡 𝐴 𝑥 =4𝑥 [2, 2𝑥] 𝐴 𝑥 = 2 2𝑥 2 𝑑𝑡 𝐴 𝑥 =4𝑥−4

Example 1b The graph given is 𝑓 𝑡 =2 on [0,𝑥]. Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡= 𝑥 2 for different values of 𝑥. 𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡 Using your answer from part (A), fill in the following table 𝒙 −𝟐 (𝟒) −𝟏 (𝟏) 𝟎 (𝟎) 𝟏 𝟐 𝟑 (𝟗) 𝐴(𝑥) 8 2 18 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑 𝐴(𝑥)

Example 1b: Continued 𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡 𝒙 −𝟐 (𝟒) −𝟏 (𝟏) 𝟎 (𝟎) 𝟏 𝟐 𝟑 (𝟗) 𝐴(𝑥) 8 2 18 𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡 Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥) 𝐴 𝑥 =2 𝑥 2

Complete the Table: Follow Ex 1b Parts (A)-(C) Examining an Upper Bound of 𝑥 2 𝑓(𝑡) Equation, Interval 𝐴(𝑥) Integral Defined Function Equation 𝐴′(𝑥) 𝑓 𝑡 =2 [−1, 𝑥 2 ] 𝐴 𝑥 = −1 𝑥 2 2 𝑑𝑡 𝐴 𝑥 =2 𝑥 2 +2 𝐴 ′ 𝑥 =4𝑥 0, 𝑥 2 𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡 𝐴 𝑥 =2 𝑥 2 [2, 𝑥 2 ] 𝐴 𝑥 = 2 𝑥 2 2 𝑑𝑡 𝐴 𝑥 =2 𝑥 2 −4

Example 1c The graph given is 𝑓 𝑡 =2 on [0,𝑥]. Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡=𝑥+1 for different values of 𝑥. 𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡 Using your answer from part (A), fill in the following table 𝒙 −𝟐 (−𝟏) −𝟏 (𝟎) 𝟎 (𝟏) 𝟏 (𝟐) 𝟐 (𝟑) 𝟑 (𝟒) 𝐴(𝑥) −2 2 4 6 8 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑 𝐴(𝑥)

Example 1c: Continued 𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡 𝒙 −𝟐 (−𝟏) −𝟏 (𝟎) 𝟎 (𝟏) 𝟏 (𝟐) 𝟐 (𝟑) 𝟑 (𝟒) 𝐴(𝑥) −2 2 4 6 8 𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡 Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥) 𝐴 𝑥 =2𝑥+2

Complete the Table: Follow Ex 1c Parts (A)-(C) Examining an Upper Bound of 𝑥+1 𝑓(𝑡) Equation, Interval 𝐴(𝑥) Integral Defined Function Equation 𝐴′(𝑥) 𝑓 𝑡 =2 [−1,𝑥+1] 𝐴 𝑥 = −1 𝑥+1 2 𝑑𝑡 𝐴 𝑥 =2𝑥 𝐴 ′ 𝑥 =2 0, 𝑥+1 𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡 𝐴 𝑥 =2𝑥+2 [2,𝑥+1] 𝐴 𝑥 = 2 𝑥+1 2 𝑑𝑡 𝐴 𝑥 =2𝑥−2

Example 1d The graph given is 𝑓 𝑡 =2 on [0,𝑥]. Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡= 1 2 𝑥 for different values of 𝑥. 𝐴 𝑥 = 0 1 2 𝑥 2 𝑑𝑡 Using your answer from part (A), fill in the following table 𝒙 −𝟐 (−𝟏) −𝟏 − 𝟏 𝟐 𝟎 (𝟎) 𝟏 𝟏 𝟐 𝟐 (𝟏) 𝟑 𝟑 𝟐 𝐴(𝑥) −2 −1 1 2 3 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑 𝐴(𝑥)

Example 1d: Continued 𝐴 𝑥 = 0 1 2 𝑥 2 𝑑𝑡 𝐴 𝑥 = 0 1 2 𝑥 2 𝑑𝑡 Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥) 𝐴 𝑥 =𝑥 𝒙 −𝟐 (−𝟏) −𝟏 − 𝟏 𝟐 𝟎 (𝟎) 𝟏 𝟏 𝟐 𝟐 (𝟏) 𝟑 𝟑 𝟐 𝐴(𝑥) −2 −1 1 2 3

Complete the Table: Follow Ex 1d Parts (A)-(C) Examining an Upper Bound of 1 2 𝑥 𝑓(𝑡) Equation, Interval 𝐴(𝑥) Integral Defined Function Equation 𝐴′(𝑥) 𝑓 𝑡 =2 [−1, 1 2 𝑥] 𝐴 𝑥 = −1 1 2 𝑥 2 𝑑𝑡 𝐴 𝑥 =𝑥+2 𝐴 ′ 𝑥 =1 0, 1 2 𝑥 𝐴 𝑥 = 0 1 2 𝑥 2 𝑑𝑡 𝐴 𝑥 =𝑥 [2, 1 2 𝑥] 𝐴 𝑥 = 2 1 2 𝑥 2 𝑑𝑡 𝐴 𝑥 =𝑥−4

Example 1e The graph given is 𝑓 𝑡 =2 on [0,𝑥]. Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡=3𝑥−1 for different values of 𝑥. 𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡 Using your answer from part (A), fill in the following table 𝒙 −𝟐 (−𝟕) −𝟏 (−𝟒) 𝟎 (−𝟏) 𝟏 (𝟐) 𝟐 (𝟓) 𝟑 (𝟖) 𝐴(𝑥) −14 −8 −2 4 10 16 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑 𝐴(𝑥)

Example 1e: Continued 𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡 𝒙 −𝟐 (−𝟕) −𝟏 (−𝟒) 𝟎 (−𝟏) 𝟏 (𝟐) 𝟐 (𝟓) 𝟑 (𝟖) 𝐴(𝑥) −14 −8 −2 4 10 16 𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡 Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥) 𝐴 𝑥 =6𝑥−2

Complete the Table: Follow Ex 1e Parts (A)-(C) Examining an Upper Bound of 3𝑥−1 𝑓(𝑡) Equation, Interval 𝐴(𝑥) Integral Defined Function Equation 𝐴′(𝑥) 𝑓 𝑡 =2 [−1, 3𝑥−1] 𝐴 𝑥 = −1 3𝑥−1 2 𝑑𝑡 𝐴 𝑥 =6𝑥 𝐴 ′ 𝑥 =6 0,3𝑥−1 𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡 𝐴 𝑥 =6𝑥−2 [2,3𝑥−1] 𝐴 𝑥 = 2 3𝑥−1 2 𝑑𝑡 𝐴 𝑥 =6𝑥−6

Table Analysis In your groups, answer the following questions based upon the data collected in the table. Find a pattern to determine the explicit equation of 𝐴(𝑥) from the integral defined function representation of 𝐴(𝑥). Hint: How does changing the lower bound affect 𝐴(𝑥)? Find a pattern to determine 𝐴′(𝑥) from the integral defined function representation of 𝐴 𝑥 . Hint: Consider the upper bound and the integrand.

Example 2: Application From AP Central Let 𝐹 𝑥 = 2 𝑥 𝑓 𝑡 𝑑𝑡 . The graph of 𝑓 on the interval [−2, 6] consists of two line segments and a quarter of a circle, as shown on the right. Find 𝐹(0) and 𝐹(4) Determine the interval where 𝐹(𝑥) is increasing. Justify your answer. Find the x-coordinates of the inflection points of 𝐹(𝑥). Justify your answer.

Example 2: Application From AP Central Find 𝐹(0) and 𝐹(4) 𝐹 0 = 2 0 𝑓 𝑡 𝑑𝑡 =− 0 2 𝑓 𝑡 𝑑𝑡 =−2 𝐹 4 = 2 4 𝑓 𝑡 𝑑𝑡 = 𝜋 Determine the interval where 𝐹(𝑥) is increasing. Justify your answer. 𝐹(𝑥) is increasing on 0, 4 & (4, 6) because 𝐹 ′ 𝑥 =𝑓(𝑥) is positive on those intervals. Find the x-coordinates of the inflection points of 𝐹(𝑥). Justify your answer. 𝐹(𝑥) has inflection points at 𝑥=2 and 𝑥=4 because 𝑓 ′ 𝑥 = 𝐹 ′′ 𝑥 changes from a positive to negative slope and vice versa at those points.

Example 3: Application From AP Central Let 𝐻 𝑥 = 0 𝑥+2 𝑓 𝑡 𝑑𝑡 , where 𝑓 is defined on the interval [−5, 5] and the graph 𝑓 consists of three line segments, as shown at the right. Determine the domain of 𝐻(𝑥) Determine the x-coordinates of and classify the relative extrema of 𝐻(𝑥). Justify your answer.

Example 3: Application From AP Central Determine the domain of 𝐻(𝑥) Domain: [−7, 3] , −5≤𝑥+2≤5 Determine the x-coordinates of and classify the relative extrema of 𝐻(𝑥). Justify your answer. 𝐻 ′ 𝑥 =𝑓 𝑥+2 =0, 𝑥=−2 is a critical # At 𝑥=−2, 𝐻(𝑥) has a rel. max because 𝐻 ′ 𝑥 =𝑓(𝑥+2) changes from negative to positive at this point.

Exit Ticket for Feedback Let 𝐺 𝑥 = 0 𝑥 𝑓 𝑡 𝑑𝑡 , where 𝑓 is defined on the interval [0, 17] and the graph 𝑓 consists of two semicircles and one line segment. Horizontal tangents are located at 𝑥=2 and 𝑥=8, and a vertical tangent is located at 𝑥=4. Find 𝐹(2), 𝐹′(2), and 𝐹′′(2). For what values of 𝑥 does 𝑓(𝑥) have a relative minimum value? Justify your answer.