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:  
Review the Fundamental Theorem of Calculus.

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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