Math

h.w2.docx

 Sketch by hand the graph of a function that satisfies: (a) (2) = 3 and (b) imit as (x to 2) f(x) = 4 . Is the function (x) continuous at = 2 ? Explain.

 Sketch by hand the graph of a function that satisfies: (a) (1) = 2 and (b) imit as (x to 1+) f(x) = 4 and(c) imit as (x to 1-) f(x) = 2 . Is the function (x) continuous from the left at = 1 ? Is the function (x) continuous at = 1 from the right? Is the function (x) continuous at = 1 ? Explain.

 Sketch by hand the graph of a function that satisfies: (a) (0) is not defined, (b) imit as (x to 0-) f(x) does not exist and (c) imit as (x to 0+) f(x) = 2 . Is the function (x) continuous from the left at = 0 ? Is the function (x) continuous at = 0 from the right? Is the function (x) continuous at = 0 ? Explain.

 The function (x) = [x^2-1]/[x+2]) is an example of a rational function (the fraction of two polynomials). For what values of is (x) continuous? Explain.

 What is the largest possible domain of the function (x) = root ([x-1]/[x+2]) ? Is this function continuous everywhere where it is defined? Explain.

 Where is the function ttp://calculus.sfsu.edu/latexrender/pictures/a7ef1e3a8802c2f50a824d8f8bc6a7e1.png discontinuous? Is this a removable discontinuity?

 Where is the function ttp://calculus.sfsu.edu/latexrender/pictures/f9b8eed0fb6c7e6cc46338491c307639.png discontinuous? Is this a removable discontinuity? Is it a jump discontinuity?

 We define the floor function to be the greatest integer not exceeding . For example, 4] = 4 , [2.37]] = 2 , [-1]] =-1 , [-1.2]] =-2 . Sketch by hand the graph of = [x] by first tabulating the values of for several numbers . Then compare your graph with the plot form the grapher. What are the discontinuities of (x) = [x] where the domain of is 2.3 leq x leq 1.5 ? Are these removable discontinuities? At the numbers where (x) is not continuous, is (x) continuous from the right? Is (x) continuous from the left?

 (a) Use the Intermediate Value Theorem to show that if part of the graph of a polynomial function = p(x) is located below the -axis and above the -axis, then it must intersect the -axis at some number = c . (This number such that (c) = 0 is called a zero of (x) ). In algebraic terms: if for some numbers , < b , (a) < 0 but (b) > 0 , then for some in the interval a,b) , (x) = 0 . (b) Then give an example of a polynomial (x) without a zero (a zero is a number such that (c) = 0 ).

(c) Give an example of a function (x) whose graph is above the -axis and below the -axis, yet (x) does not intersect the -axis.

 Is the function ttp://calculus.sfsu.edu/latexrender/pictures/a0e0412645672f16dbb7f563e6276d46.png continuous from the right at = 0 ? What is the domain of continuity of (x) ? Use the grapher for small to verify what your conclusion.