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AP Calculus – Shubleka Hebron Academy – F all ‘05 Quiz Five NAME_____________________ Definition: Let f be a function defined on an interval I =[c, d] containing x = a. If lim f ( x) f (a) x a
then f is continuous at x = a. To say that f is continuous on I means that f is continuous at each point of I (including endpoints of I, if any). Problem 1 (20 points)
bx2 1 if x 2 Let f ( x) .What value of b makes f continuous at x = -2? if x 2 x Problem 2 (20 points) 1 if x Z Let f ( x) 2 if x Z . a) Draw a graph of f(x) over the interval [0, 5]. b) Evaluate lim f ( x). x 4
c) Evaluate lim f ( x). x 5 / 2
d) For which values of a does lim f ( x) exist? Why? x a
Problem 3 (20 points)
x 2 a if x 1 if x 1 Let g ( x) x a 2 a x if x 1 a) Find all values of a for which lim g ( x) exists. x 1
b) Find all values of a for which g(x) is continuous at x = -1. Problem 4 (20 points) Suppose that the function f is continuous at x = 2 and that f is defined by the rule: ax2 3 if x 2 f ( x) if x 2 3x 5 a) Find the value of a. Evaluate the following limits. b) lim f ( x) c) lim f ( x) d) lim f ( x) e) lim f ( x) x 3
x 0
x 2
x 1
Problem 5 If f and g are continuous functions with f(3)= 5 and lim [2 f ( x) g ( x)] 4 , find x 3
g(3). (20 points)
I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors. Everybody a mathematician? James Caballero
then f is continuous at x = a. To say that f is continuous on I means that f is continuous at each point of I (including endpoints of I, if any). Problem 1 (20 points)
bx2 1 if x 2 Let f ( x) .What value of b makes f continuous at x = -2? if x 2 x Problem 2 (20 points) 1 if x Z Let f ( x) 2 if x Z . a) Draw a graph of f(x) over the interval [0, 5]. b) Evaluate lim f ( x). x 4
c) Evaluate lim f ( x). x 5 / 2
d) For which values of a does lim f ( x) exist? Why? x a
Problem 3 (20 points)
x 2 a if x 1 if x 1 Let g ( x) x a 2 a x if x 1 a) Find all values of a for which lim g ( x) exists. x 1
b) Find all values of a for which g(x) is continuous at x = -1. Problem 4 (20 points) Suppose that the function f is continuous at x = 2 and that f is defined by the rule: ax2 3 if x 2 f ( x) if x 2 3x 5 a) Find the value of a. Evaluate the following limits. b) lim f ( x) c) lim f ( x) d) lim f ( x) e) lim f ( x) x 3
x 0
x 2
x 1
Problem 5 If f and g are continuous functions with f(3)= 5 and lim [2 f ( x) g ( x)] 4 , find x 3
g(3). (20 points)
I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors. Everybody a mathematician? James Caballero
