Download presentation
Presentation is loading. Please wait.
1
Second Fundamental Theorem of Calculus Day 2
Section 4.4B Calculus AP/Dual, Revised Β©2017 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
2
Β§4.4B: Second Fundamental Theorem of Calculus
Equation The Second FTC: π π πβ² π π
π=π π βπ π OR π π πβ²β² π π
π= π β² π β π β² π 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
3
Β§4.4B: Second Fundamental Theorem of Calculus
Example 1 If π π =ππ, πβ² is continuous, and π π π β² π π
π=ππ , what is the value of π π ? 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
4
Β§4.4B: Second Fundamental Theorem of Calculus
Example 2 Given the values of the derivative π β² π in the table and that π π =πππ, estimate π π for π=π. Use a right Riemann Sum. π π π π π π β² π ππ ππ ππ ππ 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
5
Β§4.4B: Second Fundamental Theorem of Calculus
Example 3 Find the value of π π where π β² π = π β π π and π π =π. 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
6
Β§4.4B: Second Fundamental Theorem of Calculus
Example 4 If π π ππ π +π π
π=ππ , find π π π π π
π ? Hint: Break it up and establish π π π π π
π as the missing variable 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
7
Β§4.4B: Second Fundamental Theorem of Calculus
Example 4 If π π ππ π +π π
π=ππ , find π π π π π
π ? Hint: Break it up and establish π π π π π
π as the missing variable 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
8
Β§4.4B: Second Fundamental Theorem of Calculus
Your Turn If π π π π π
π=π , find π π π π +π π
π . 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
9
Interpreting Behavior
In integral calculus, talk is now centered about functions and their antiderivatives. LetΒ π π = π π π π π
π . πΒ is an antiderivative ofΒ π. In differential calculus we would write this asΒ πβ²=π. SinceΒ πΒ is the derivative ofΒ π, we can reason about properties ofΒ πΒ in similar to what we did in differential calculus. 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
10
Using the Second Fundamental Theorem of Calculus
π π = π π Area π β² π =π π Graph π β²β² π =πβ² π Slope Spell βAGSβ The graph of the function of π is SUBSTITUTION graph of π 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
11
Β§4.4B: Second Fundamental Theorem of Calculus
Example 5 Given this graph of π, identify where the function π is positive and π is increasing. 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
12
Β§4.4B: Second Fundamental Theorem of Calculus
Example 6 The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Solve for π π , π βπ , π π , and π π . Shown is the graph of π. 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
13
Β§4.4B: Second Fundamental Theorem of Calculus
Example 6a The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Solve for π π . 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
14
Β§4.4B: Second Fundamental Theorem of Calculus
Example 6b The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Solve for π βπ . 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
15
Β§4.4B: Second Fundamental Theorem of Calculus
Example 6c The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Solve for π π . 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
16
Β§4.4B: Second Fundamental Theorem of Calculus
Example 6d The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Solve for π π . 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
17
Β§4.4B: Second Fundamental Theorem of Calculus
Example 7 The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Find all the values of π on the open interval βπ, π at which π has a relative maximum. Justify response. 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
18
Β§4.4B: Second Fundamental Theorem of Calculus
Example 8 The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Find all the values of π on the open interval βπ, π at which π has the point of inflection. Justify response. 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
19
Β§4.4B: Second Fundamental Theorem of Calculus
Example 9 The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . What is the average rate of change of π in the interval from π, π . 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
20
Β§4.4B: Second Fundamental Theorem of Calculus
Your Turn The graph of a function π consists of a quarter circle and line segments. Let π be the function given by: π π = π π π π π
π . Solve for π π and find all values of π on the open interval at which π has a relative minimum. Justify your answers. 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
21
Β§4.4B: Second Fundamental Theorem of Calculus
Example 10 The graph ofΒ π has horizontal tangents atΒ π=πΒ andΒ π=π. IfΒ π(π)= π ππ π π π
π , what is the value ofΒ π β² π ? 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
22
Β§4.4B: Second Fundamental Theorem of Calculus
Your Turn The graph ofΒ πΒ consists of three line segments. IfΒ π(π)= π ππ π π π
π , then find the instantaneous rate of change of π, with respects to π, at π=π? 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
23
AP Multiple Choice Practice Question 1 (non-calculator)
The graph of the function π shown below consists of two line segments and a semicircle. Let π be defined by π π = π π π π π
π . What is the value of π π ? (A) π (B) βπ.π+ππ
(C) ππ
(D) π.π+ππ
2/27/ :23 AM Β§4.2B: Limits of Riemann's Sum
24
AP Multiple Choice Practice Question 1 (non-calculator)
The graph of the function π shown below consists of two line segments and a semicircle. Let π be defined by π π = π π π π π
π . What is the value of π π ? Vocabulary Process and Connections Answer 2/27/ :23 AM Β§4.2B: Limits of Riemann's Sum
25
AP Multiple Choice Practice Question 2 (non-calculator)
The graph of the piecewise linear function π is shown below. If π π = βπ π π π π
π , which of the following values is the greatest? (A) π βπ (B) π π (C) π π (D) π π 2/27/ :23 AM Β§4.2B: Limits of Riemann's Sum
26
AP Multiple Choice Practice Question 2 (non-calculator)
The graph of the piecewise linear function π is shown below. If π π = βπ π π π π
π , which of the following values is the greatest? Vocabulary Process and Connections Answer 2/27/ :23 AM Β§4.2B: Limits of Riemann's Sum
27
Β§4.4B: Second Fundamental Theorem of Calculus
Assignment Worksheet 2/27/ :23 AM Β§4.4B: Second Fundamental Theorem of Calculus
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.