1. Second Fundamental Theorem of Calculus Day 2
Section 4.4B
Calculus AP/Dual, Revised ©2017
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

2. §4.4B: Second Fundamental Theorem of Calculus
Equation
The Second FTC: 𝒂 𝒃 𝒇′ 𝒙 𝒅𝒙=𝒇 𝒃 −𝒇 𝒂 OR 𝒂 𝒃 𝒇′′ 𝒙 𝒅𝒙= 𝒇 ′ 𝒃 − 𝒇 ′ 𝒂
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

3. §4.4B: Second Fundamental Theorem of Calculus
Example 1
If 𝒇 𝟏 =𝟏𝟐, 𝒇′ is continuous, and 𝟏 𝟒 𝒇 ′ 𝒙 𝒅𝒙=𝟏𝟕 , what is the value of 𝒇 𝟒 ?
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

4. §4.4B: Second Fundamental Theorem of Calculus
Example 2
Given the values of the derivative 𝒇 ′ 𝒙 in the table and that 𝒇 𝟎 =𝟏𝟎𝟎, estimate 𝒇 𝒙 for 𝒙=𝟐. Use a right Riemann Sum. 𝒙 𝟎 𝟐 𝟒 𝟔
𝒇 ′ 𝒙 𝟏𝟎 𝟏𝟖 𝟐𝟑 𝟐𝟓
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

5. §4.4B: Second Fundamental Theorem of Calculus
Example 3
Find the value of 𝑭 𝟏 where 𝑭 ′ 𝒙 = 𝒆 − 𝒙 𝟐 and 𝑭 𝟎 =𝟐.
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

6. §4.4B: Second Fundamental Theorem of Calculus
Example 4
If 𝟐 𝟓 𝟐𝒇 𝒙 +𝟑 𝒅𝒙=𝟏𝟕 , find 𝟐 𝟓 𝒇 𝒙 𝒅𝒙 ? Hint: Break it up and establish 𝟐 𝟓 𝒇 𝒙 𝒅𝒙 as the missing variable
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

7. §4.4B: Second Fundamental Theorem of Calculus
Example 4
If 𝟐 𝟓 𝟐𝒇 𝒙 +𝟑 𝒅𝒙=𝟏𝟕 , find 𝟐 𝟓 𝒇 𝒙 𝒅𝒙 ? Hint: Break it up and establish 𝟐 𝟓 𝒇 𝒙 𝒅𝒙 as the missing variable
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

8. §4.4B: Second Fundamental Theorem of Calculus
Your Turn
If 𝟎 𝟓 𝒇 𝒙 𝒅𝒙=𝟒 , find 𝟎 𝟓 𝒇 𝒙 +𝟐 𝒅𝒙 .
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

9. Interpreting Behavior
In integral calculus, talk is now centered about functions and their antiderivatives.
Let 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . 𝒈 is an antiderivative of 𝒇. In differential calculus we would write this as 𝒈′=𝒇. Since 𝒇 is the derivative of 𝒈, we can reason about properties of 𝒈 in similar to what we did in differential calculus.
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

10. Using the Second Fundamental Theorem of Calculus
𝒈 𝒙 = 𝒇 𝒙 Area
𝒈 ′ 𝒙 =𝒇 𝒙 Graph
𝒈 ′′ 𝒙 =𝒇′ 𝒙 Slope
Spell “AGS”
The graph of the function of 𝒈 is SUBSTITUTION graph of 𝒇
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

11. §4.4B: Second Fundamental Theorem of Calculus
Example 5
Given this graph of 𝒇, identify where the function 𝒇 is positive and 𝒈 is increasing.
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

12. §4.4B: Second Fundamental Theorem of Calculus
Example 6
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Solve for 𝒈 𝟎 , 𝒈 −𝟏 , 𝒈 𝟐 , and 𝒈 𝟓 . Shown is the graph of 𝒇.
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

13. §4.4B: Second Fundamental Theorem of Calculus
Example 6a
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Solve for 𝒈 𝟎 .
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

14. §4.4B: Second Fundamental Theorem of Calculus
Example 6b
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Solve for 𝒈 −𝟏 .
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

15. §4.4B: Second Fundamental Theorem of Calculus
Example 6c
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Solve for 𝒈 𝟐 .
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

16. §4.4B: Second Fundamental Theorem of Calculus
Example 6d
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Solve for 𝒈 𝟓 .
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

17. §4.4B: Second Fundamental Theorem of Calculus
Example 7
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Find all the values of 𝒙 on the open interval −𝟏, 𝟓 at which 𝒈 has a relative maximum. Justify response.
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

18. §4.4B: Second Fundamental Theorem of Calculus
Example 8
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Find all the values of 𝒙 on the open interval −𝟏, 𝟓 at which 𝒈 has the point of inflection. Justify response.
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

19. §4.4B: Second Fundamental Theorem of Calculus
Example 9
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . What is the average rate of change of 𝒈 in the interval from 𝟎, 𝟐 .
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

20. §4.4B: Second Fundamental Theorem of Calculus
Your Turn
The graph of a function 𝒇 consists of a quarter circle and line segments. Let 𝒈 be the function given by: 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . Solve for 𝒈 𝟑 and find all values of 𝒙 on the open interval at which 𝒈 has a relative minimum. Justify your answers.
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

21. §4.4B: Second Fundamental Theorem of Calculus
Example 10
The graph of 𝒇 has horizontal tangents at 𝒙=𝟑 and 𝒙=𝟔. If 𝒈(𝒙)= 𝟎 𝟐𝒙 𝒇 𝒕 𝒅𝒕 , what is the value of  𝒈 ′ 𝟑 ?
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

22. §4.4B: Second Fundamental Theorem of Calculus
Your Turn
The graph of 𝒇 consists of three line segments. If 𝒈(𝒙)= 𝟏 𝟐𝒙 𝒇 𝒕 𝒅𝒕 , then find the instantaneous rate of change of 𝒈, with respects to 𝒙, at 𝒙=𝟐?
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus

23. AP Multiple Choice Practice Question 1 (non-calculator)
The graph of the function 𝒇 shown below consists of two line segments and a semicircle. Let 𝒈 be defined by 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . What is the value of 𝒈 𝟓 ? (A) 𝟎 (B) −𝟏.𝟓+𝟐𝝅 (C) 𝟐𝝅 (D) 𝟏.𝟓+𝟐𝝅
2/27/ :23 AM
§4.2B: Limits of Riemann's Sum

24. AP Multiple Choice Practice Question 1 (non-calculator)
The graph of the function 𝒇 shown below consists of two line segments and a semicircle. Let 𝒈 be defined by 𝒈 𝒙 = 𝟎 𝒙 𝒇 𝒕 𝒅𝒕 . What is the value of 𝒈 𝟓 ?
Vocabulary
Process and Connections
Answer
2/27/ :23 AM
§4.2B: Limits of Riemann's Sum

25. AP Multiple Choice Practice Question 2 (non-calculator)
The graph of the piecewise linear function 𝒇 is shown below. If 𝒈 𝒙 = −𝟐 𝒙 𝒇 𝒕 𝒅𝒕 , which of the following values is the greatest? (A) 𝒈 −𝟑 (B) 𝒈 𝟎 (C) 𝒈 𝟏 (D) 𝒈 𝟐
2/27/ :23 AM
§4.2B: Limits of Riemann's Sum

26. AP Multiple Choice Practice Question 2 (non-calculator)
The graph of the piecewise linear function 𝒇 is shown below. If 𝒈 𝒙 = −𝟐 𝒙 𝒇 𝒕 𝒅𝒕 , which of the following values is the greatest?
Vocabulary
Process and Connections
Answer
2/27/ :23 AM
§4.2B: Limits of Riemann's Sum

27. §4.4B: Second Fundamental Theorem of Calculus
Assignment
Worksheet
2/27/ :23 AM
§4.4B: Second Fundamental Theorem of Calculus