3.3  連續基本定理

 

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3.3  連續基本定理

 

 

3-3 連續基本定理

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Balzano定理(又稱勘根定理):

若滿足下列兩條件:

(1) 為連續

(2)

則至少有一點,使得

 

  

連續之中值定理(Intermediate Value Theorem)

(1) 為連續

(2) ,且間之任意實數

則至少有一點,使得

 

 

1.      Show that the equation  has at least one real solution. (10%)

 

解答:

                            

                  

                

Balzano定理知,在間至少有一實根。

 

 

 

 

 

2. Show that the function  has a zero between

解答:

 

 

 

 

3.If , show that there is a number  such that .     

 

解答:

 is continuous on  

, and .

Since , there is a number  in (31,32) such that  by Intermediate Value Theorem.

 

 

 

  

4. Use the Intermediate Value Theorem to show that there is a root of the given function in the specified interval.

 , (1,2).      

 

解答:

 is continuous on  

, and .

Since, there is a number  in (1,2) such that  by Intermediate Value Theorem.

Thus, there is a root of the function in the interval (1,2).

 

 

 

   

5. Use the Intermediate Value Theorem to show that there is a root of the given function in the specified interval.

 , (0,1).      

 

解答:

 is continuous on  

, and .

Since, there is a number  in (0,1) such that  by Intermediate Value Theorem.

Thus, there is a root of the function in the interval (0,1).

 

 

  

 

6. Use the Intermediate Value Theorem to show that there is a root of the given function in the specified interval.

 , (0,1).      

 

解答:

 is continuous on  

, and .

Since, there is a number  in (0,1) such that  by Intermediate Value Theorem.

Thus, there is a root of the function  in the interval (0,1).

 

 

 

  

7. Use the Intermediate Value Theorem to show that there is a root of the given function in the specified interval.

 , (1,1.4).      

 

解答:

 is continuous on [1, 1.4].

, and .

Since, there is a number  in (1,1.4) such that  by Intermediate Value Theorem.

Thus, there is a root of the function  in the interval (1,1.4).

 

 

  

 

8.Prove the equation has at least one real root of the given function.

        

 

解答:

 is continuous on [1, 2].

, and .

Since, there is a number  in (1,2) such that  by Intermediate Value Theorem.

Thus, there is a root of the function  in the interval (1,2).

 

 

 

 

9.Prove the equation has at least one real root of the given function.

        

 

解答:

 is continuous on [-1, 0].

, and .

Since, there is a number  in (-1,0) such that  by Intermediate Value Theorem.

Thus, there is a root of the function  in the interval (-1,0).

 

 

 

 

 

 

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