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AP Calculus AB – Shubleka Hebron Academy - Fall 2006 Fiesta # 3 NAME_____________________ Read the questions carefully. Present your answers neatly and justify each step in your solutions. Show all your work for full credit. No calculators are allowed on this quiz. You may only consult with Mrs. Leyden. Problem 1 (10 points) i) Discuss the following function in terms of limits, continuity, and differentiability. Do the first and the second derivative functions exist? Justify your answer. If the derivative exists, sketch its graph. 2 x if x 0 f ( x) 2 x if x 0. Problem 2 (10 points) The signum (or sign) function, denoted by sgn, is defined by:
1 if x 0 sgn x 0 if x 0 1 if x 0. i) Sketch the graph of this function. ii) Find each of the following limits or explain why it does not exist: a) lim sgn x. b) lim sgn x. c) lim sgn x. d) lim | sgn x | . x 0
x 0
x 0
x 0
Problem 3 (10 points) i) Show by means of an example that lim ( f ( x) g ( x)) may exist even though x a
neither lim f ( x) nor lim g ( x) exists. x a
ii)
x a
Show by means of an example that lim ( f ( x) g ( x)) may exist even though neither x a
lim f ( x) nor lim g ( x) exists. x a
x a
Problem 4 (10 points)
3x 2 ax a 3 exists? If so, find the value of a and the x 2 x2 x 2
Is there a number a such that lim value of the limit. Problem 5 (10 points)
2 x x 2 if 0 x 2 2 x if 2 x 3 f ( x ) Discuss the continuities of and sketch the graph of the x 4 if 3 x 4 if x 4 function. Provide as many details as you can. Bonus Question (3 points) Show that the equation 2sin x = 3- 2x has a root in the interval (0, 1).
The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. Aristotle, Metaphysica, 3-1078b.
1 if x 0 sgn x 0 if x 0 1 if x 0. i) Sketch the graph of this function. ii) Find each of the following limits or explain why it does not exist: a) lim sgn x. b) lim sgn x. c) lim sgn x. d) lim | sgn x | . x 0
x 0
x 0
x 0
Problem 3 (10 points) i) Show by means of an example that lim ( f ( x) g ( x)) may exist even though x a
neither lim f ( x) nor lim g ( x) exists. x a
ii)
x a
Show by means of an example that lim ( f ( x) g ( x)) may exist even though neither x a
lim f ( x) nor lim g ( x) exists. x a
x a
Problem 4 (10 points)
3x 2 ax a 3 exists? If so, find the value of a and the x 2 x2 x 2
Is there a number a such that lim value of the limit. Problem 5 (10 points)
2 x x 2 if 0 x 2 2 x if 2 x 3 f ( x ) Discuss the continuities of and sketch the graph of the x 4 if 3 x 4 if x 4 function. Provide as many details as you can. Bonus Question (3 points) Show that the equation 2sin x = 3- 2x has a root in the interval (0, 1).
The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. Aristotle, Metaphysica, 3-1078b.
