AP Calculus AB/BC
Name___________________________
Skill Builder: Topics 1.11 – 1.13
Skill Builder: Topic 1.11 – Determining Continuity at a Point
1.) In the following graphs determine if the function, f (x), is continuous at the marked value of c, and if not,
using proper notation, determine for which of the three rules of continuity the function fails and state
whether the discontinuity is removable or non-removable.
Skill Builder Topic 1.12 – Confirming Continuity on an Interval
2.) Find the value(s) of x where the function is discontinuous. Classify each discontinuity a being either
removable or nonremovable.
5
a) f ( x) = x3 + 3x
b) f ( x) = 2
x - 81
c.) f ( x) =
x
x -x
e.) f ( x) =
x 2 + 2 x - 24
x 2 - 36
2
d.) f ( x) =
x+3
x+3
f.) f ( x) = tan x
3.) State whether or not the function is continuous at the value where the rule for the function changes. If
continuous, state the three criteria of continuity. If not, state why.
ì2 x ,
ì2- x , x < -1
x<3
c.) f ( x) = í
d.) f ( x) = í
î10 - x, x ³ 3
î x + 3, x ³ -1
ì 1
ïx-2, x < 2
ï
e.) f ( x) = í3,
x=2
ï x + 1, x > 2
ï
î
ì x3 - x
ï x 2 - x , x ¹ 0, x ¹ 1
ï
ï
f.) f ( x) = í3,
x=0
ï2,
x =1
ï
ï
î
Skill Builder Topic 1.13 – Removing Discontinuities
4.) Find the value of the constant a which makes the function continuous. Be sure to justify your solution
using proper notation.
ì9 - x 2 ,
b.) f ( x) = í
îax,
x>2
x£2
ìa 2 - x 2 , x < 2
d.) f ( x) = í
î1.5ax, x ³ 2
6.) The piece-wise function for g ( x) is below. Find the values for a and b that makes g ( x) continuous
everywhere. Be sure to use the definition of continuity and demonstrate proper notation.
ì x3 + 8
x < -2
ï x+2
ïï
k ( x) = í2ax + b
- 2 £ x <1
ïax 2 - bx - 2 x ³ 1
ï
ïî