Difference between revisions of "009B Sample Midterm 2, Problem 2"
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(Created page with "<span class="exam"> This problem has three parts: ::<span class="exam">a) State the Fundamental Theorem of Calculus. ::<span class="exam">b) Compute <math style="ve...") 

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−    +  '''1.''' What does Part 1 of the Fundamental Theorem of Calculus say about <math style="verticalalign: 15px">\frac{d}{dx}\int_0^x\sin(t)~dt?</math> 
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+  ::Part 1 of the Fundamental Theorem of Calculus says that <math style="verticalalign: 15px">\frac{d}{dx}\int_0^x\sin(t)~dt=\sin(x).</math>  
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+  '''2.''' What does Part 2 of the Fundamental Theorem of Calculus say about <math style="verticalalign: 15px">\int_a^b\sec^2x~dx,</math> where <math style="verticalalign: 5px">a,b</math> are constants?  
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+  ::Part 2 of the Fundamental Theorem of Calculus says that <math style="verticalalign: 15px">\int_a^b\sec^2x~dx=F(b)F(a),</math> where <math style="verticalalign: 0px">F</math> is any antiderivative of <math style="verticalalign: 0px">\sec^2x.</math>  
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Revision as of 15:14, 8 April 2016
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 

2. What does Part 2 of the Fundamental Theorem of Calculus say about where are constants? 

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) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\pi/4}\sec^2 x~dx\,=\,1} . 