Download Calculus - maccalc

HW 1.4
Calculus
Section 2.5 Continuity
Name _____________________________
x 2  1,
-1  x  0

0x1
2x,

f(x)  1,
x 1
- 2x  4, 1  x  2

0,
2x3

USE THE GIVEN PIECE-WISE FUNCTION TO CREATE A GRAPH, THEN ANSWER THE FOLLOWING:
1. Answer each of the following based on the function above.
a. f(0) = _______
c.
b.
lim f(x) = _________
x0 
lim f(x) = ________
x0 
d. Is f continuous at x = 0? ________
2. Answer each of the following based on the function above.
a. f(1) = _______
b. lim f(x) = _________
x1
c. Does lim f(x)  f(1) ? _________
x1
d. Is f continuous at x = 1? _______
3. Is f continuous at x = 2? If not, state which condition of continuity fails.
4. In the interval (-1,3), at what values of x is f discontinuous? For each state whether the discontinuity is
removable or nonremovable.
5. a. What is the value of lim f(x) ? ______
x2
b. What value should be assigned to f(2) to make f continuous at x = 2? ______
6. To what new value should f(1) be changed to make f continuous at x = 1? ______
7. What is the domain of this function? (Interval Notation) ________________________
8. What is the range of this function? (Interval Notation) ______________________
 x2  1
, x1

9. g(x)   x  1
2
x 1

a. Is g continuous at x = 1? Justify using the three conditions of continuity.
 x2  9
, x3

10. h(x)   x  3
?,
x3

a. Define h(3) in a way that extends h(x) to be continuous at x = 3.
 x 2  3x  10
, x2

11. f(x)  
x 2
?,
x2

a. Define f(2) in a way that extends f(x) to be continuous at x = 2.
12. What value should be assigned to “a” to make the function
2

x  1,
f(x)  

2ax,
x2
x2
continuous at x = 2. Draw a complete
graph of f(x) for this value of “a”,
13. What value should be assigned to “a” to make the following function continuous everywhere?
 x2  a2
,

g(x)   x  a
8,

xa
xa