009B Sample Midterm 2, Problem 2

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This problem has three parts:

a) State the Fundamental Theorem of Calculus.
b) Compute   .
c) Evaluate .


Foundations:  
1. What does Part 1 of the Fundamental Theorem of Calculus say about 
Part 1 of the Fundamental Theorem of Calculus says that 
2. What does Part 2 of the Fundamental Theorem of Calculus say about where are constants?
Part 2 of the Fundamental Theorem of Calculus says that where is any antiderivative of

Solution:

(a)

Step 1:  
The Fundamental Theorem of Calculus has two parts.
The Fundamental Theorem of Calculus, Part 1
Let be continuous on and let
Then, is a differentiable function on and
Step 2:  
The Fundamental Theorem of Calculus, Part 2
Let be continuous on and let be any antiderivative of
Then,

(b)

Step 1:  
Let The problem is asking us to find
Let and
Then,
Step 2:  
If we take the derivative of both sides of the last equation, we get by the Chain Rule.
Step 3:  
Now, and by the Fundamental Theorem of Calculus, Part 1.
Since we have

(c)

Step 1:  
Using the Fundamental Theorem of Calculus, Part 2, we have
Step 2:  
So, we get
Final Answer:  
(a)
The Fundamental Theorem of Calculus, Part 1
Let be continuous on and let
Then, is a differentiable function on and
The Fundamental Theorem of Calculus, Part 2
Let be continuous on and let be any antiderivative of
Then,
(b)  
(c)

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