Mathematics

 
1. Sketch by hand the graph of a function f that satisfies: (a) f(2) = 3 and (b)limx?2 f(x) = 4.
Is the function f(x) continuous at x = 2? Explain.
2. Sketch by hand the graph of a function f that satisfies: (a) f(1) = 2 and (b)limx?1+ f(x)=4
and(c) limx?1- f(x) = 2. Is the function f(x) continuous from the left at x = 1? Is the
function f(x) continuous at x = 1 from the right? Is the function f(x) continuous at x = 1?
Explain.
3. Sketch by hand the graph of a function f that satisfies: (a) f(0) is not defined, (b) limx?0- f(x)
does not exist and (c) limx?0+ f(x) = 2. Is the function f(x) continuous from the left at
x = 0? Is the function f(x) continuous at x = 0 from the right? Is the function f(x)
continuous at x = 0? Explain.
4. The function f(x) = x2-1
x+2 is an example of a rational function (the fraction of two polynomials).
For what values of x is f(x) continuous? Explain.
5. What is the largest possible domain of the function f(x) = !x-1
x+2 ? Is this function continuous
everywhere where it is defined? Explain.
6. Where is the function f(x) = ” 1
x4 if x ?= 0
0 if x = 0 discontinuous? Is this a removable discontinuity?
7. Where is the function f(x) = ” ex if x < 0
x2 if x = 0 discontinuous? Is this a removable discontinuity?
Is it a jump discontinuity?
8. We define the floor function [[x]] to be the greatest integer not exceeding x. For example,
[[4]] = 4, [[2.37]] = 2, [[-1]] = -1, [[-1.2]] = -2. Sketch by hand the graph of y = [[x]] by
first tabulating the values of [[x]] for several numbers x. Then compare your graph with the
plot form the grapher. What are the discontinuities of f(x) = [[x]] where the domain of x is
-2.3 = x = 1.5? Are these removable discontinuities? At the numbers x where f(x) is not
continuous, is f(x) continuous from the right? Is f(x) continuous from the left?
9. (a) Use the Intermediate Value Theorem to show that if part of the graph of a polynomial
function y = p(x) is located below the X-axis and above the X-axis, then it must intersect
the X-axis at some number x = c. (This number c such that f(c) = 0 is called a zero of f(x)).
In algebraic terms: if for some numbers a, b, a<b, p(a) < 0 but p(b) > 0, then for some x in
the interval (a, b), p(x) = 0. (b) Then give an example of a polynomial p(x) without a zero
(a zero is a number c such that p(c) = 0 ).
(c) Give an example of a function h(x) whose graph is above the X-axis and below the X-axis,
yet h(x) does not intersect the X-axis.
10. Is the function f(x) = ” vx cos 1
x2 if x > 0
0 if x = 0 continuous from the right at x = 0? What is
the domain of continuity of f(x)? Use the grapher for small x to verify what your conclusion.
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11. Discuss but do not submit. Let f(x) = ” x sin 1
x if x ?= 0
0 if x = 0 . Show that f(x) is continuous
at x = 0, that is show that limx?0 f(x) = f(0).