Quiz 1

Quiz-1, Math 140

1. (40%) True or False (40%)
2. If   , where b is a finite constant, then the function y= f(x) has a tangent at x=a with the slope b.
3. __  __ if  f(x) =  f(x), then f is continuous at x=c. (f can have a hole at x=c)
4. _  __If the , then  does not exist.
5. ______ For any statements A and B, if A implies B, then B always implies A.
6. ______ If f(x) =L, and f is continuous, then L=f(c)
7. ______ If f is continuous at x=c, then f(x) =  f(x)
8. If the  exists, then f must be continuous.
9. ____ If f(x) is always bounded between the two functions, h(x)=x²+2, and g(x)=2x +1, then f(x) =3.
10. _ ___ If f(x)g(x) exists then both  f(x) and  g(x) exist, regardless of the conditions of individual functions.
11. __ __ If f(x) =L then L=f(c)
12. __ __ If f is continuous at x=a, then the tangent at x=a exists.
13.           if f is a function and f(a) = f(b) then a = b.
14. A tangent line exists everywhere for the function
15. If f is discontinuous at x=a then  does not exit.
16. _ if f(x)=x² + 1, and g(x)=1/x, then does not exist because  =  does not exist.
17. __ if  and , then .
18.   If f is a continuous function and f(x₁) and f(x₂) have opposite signs, then the function f has a root in the interval (x₁, x₂).
19.   It is possible to find an exact area under the graph f(x)=1/x, bounded by x=0 and x=1.
20.  You can always use bisection algorithm to find a root of a continuous function.
21. ___ __ If a continuous function f(x) has a root at x=a, then there exists a number ε such that f(a- ε) and f(a+ ε) have opposite sign.

 

(You must show your work to get full credit for the problems below)

 

2. (10%) Consider the statement “If a creature is a human, then it has 2 legs”.

(1) What is the contrapositive of the above statement:

(2) What is the converse of the above statement:

(3) Which one, (1), or (2), above is always true:

 

 

 

 

3. (20%) Let f(x)=x⁴+2x³-4x²-10x-5

• Find a small interval on x-axis that contain a root of the function f in the interval (0, 3).
• Use bisecting method to find the root in (a) to an accuracy of 0.05. That is, if the true root is “r” then a solution “s” satisfying the condition |s-r| 05 is considered satisfactory.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4. (15) Find the limit:

 

 

 

 

 

 

 

 

 

 

 

 

5. (15%) Show the function does not have a limit at x=0, either using the definition of a limit, or by showing that the left limit is not the same as the right limit.