# Quiz 1

Quiz-1, Math 140

- (40%) True or False (40%)
__If , where b is a finite constant, then the function y= f(x) has a tangent at x=a with the slope b.______ ____if f(x) = f(x), then f is continuous at x=c. (f can have a hole at x=c)___ ____If the , then does not exist.- ______ For any statements A and B, if A implies B, then B always implies A.
- ___
_______ If f(x) =L, and f is continuous, then L=f(c) __________If f is continuous at x=c, then f(x) = f(x)__If the exists, then f must be continuous.__- ____ If f(x) is always bounded between the two functions, h(x)=x
^{2}+2, and g(x)=2x +1, then f(x) =3. - _ ___ If f(x)g(x) exists then both f(x) and g(x) exist, regardless of the conditions of individual functions.
- __ __ If f(x) =L then L=f(c)
____ ____If f is continuous at x=a, then the tangent at x=a exists.__A tangent line exists everywhere for the function____If f is discontinuous at x=a then does not exit._____ if f(x)=x__^{2}+ 1, and g(x)=1/x, then does not exist because = does not exist.______if and , then .__If f is a continuous function and f(x___{1}) and f(x_{2}) have opposite signs, then the function f has a root in the interval (x_{1}, x_{2}).__It is possible to find an exact area under the graph f(x)=1/x, bounded by x=0 and x=1.____You can always use bisection algorithm to find a root of a continuous function._______ ____ 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)

- (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:

- (20%) Let f(x)=x
^{4}+2x^{3}-4x^{2}-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.

- (15) Find the limit:

- (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.