AP Calculus
2.3 Worksheet
All work must be shown in this course for full credit. Unsupported answers may receive NO credit.
1. What is the definition of continuity?
2. Sketch a possible graph for each function described.
a) f (5) exists, but lim f ( x ) does not exist.
x5
b) The lim f ( x ) exists, but f (5) does not exist.
x5
ìï x 2 - a 2 x ; x < 2
3. Let f ( x) = ïí
. Find all values of a that make f continuous at 2.
ïïî 4 - 2 x 2 ; x ³ 2
Using the definition of continuity, justify your response.
 2x  5  x  7

4. If f  x   
x2
k  3

if x  2
, and if f is continuous at x = 2, then k = ? Justify your response.
if x  2
5. Complete the following questions from your textbook: page 85 #47 and 48
6. Let f be the function defined by the following:
x0
sin x,
 x2 ,

f  x  
 2  x,
 x  3,
0  x 1
1 x  2
x2
For what values of x is f NOT continuous?
7. If the function f is continuous for all real numbers and if f  x  
8. Let f be the function given by f  x  
A) None
9. Let h ( x) =
 x  1  x 2  4 
B) 1 only
x2  a
x2  4
, when x  2 , then f (–2) =
x2
. For what positive values of a is f continuous for all real numbers?
C) 2 only
x2 + 5x + 6
.
x 2 + 7 x + 10
a) Find the domain of g (x).
b) Find the lim g ( x ) for all values of c where g (x) is not defined.
x c
c) Find any horizontal asymptotes and justify your response.
d) Find any vertical asymptotes and justify your response.
e) Write an extension to the function so that g (x) is continuous at x = –2.
D) 4 only
E) 1 and 4
10. Complete the following questions from the textbook: page 84 – 86 #1, 7, 10, 11 – 18, 28, 29, 58, 59
11. Without using a picture, give a written explanation of why the function f ( x) = x 2 - 4 x + 3 has a zero in the interval [2, 4].
ìï3 x 2 - 4, if x £ 2
12. Let h ( x ) = ïí
.
ïïî 5 + 4 x, if x > 2
a) What is h (0) ?
b) What is h (4) ?
c) Is there a value of x such that h (x) = 10? Explain why this result does not contradict the IVT.
AP Calculus
2.4 Worksheet
All work must be shown in this course for full credit. Unsupported answers may receive NO credit.
1. What is a difference quotient?
2. How do you find the slope of a curve (aka slope of the tangent line to a curve) when x = a ?
3. What is a normal line ?
4. What is the difference between the AVERAGE RATE OF CHANGE and INSTANTANEOUS RATE OF CHANGE?
5. Let f ( x) = x3 .
a) Write an expression for f (a + h) .
b) Find the slope of the curve at x = a.
c) When does the slope equal 12?
d) Write the equation of the tangent line to the curve at x = 4 ___________________________________________
e) Write the equation of the normal line to the curve at x = 4 ___________________________________________
6. Let g ( x) = x
a) Find the average rate of change from x = 4 to x = 9.
b) Find the instantaneous rate of change at x = 9.
c) Write the equation of the tangent line when x = 9
d) Write the equation of the normal line when x = 9.
7. Complete the following from the textbook: page 92 – 93 #3, 6, 7, 11, 12, 23, 25, 29, 38, and 39
8. You should also begin reviewing for your chapter 2 test: page 95 – 97 #1 – 29, 31, 33, 35, 39, 40, 43 – 49, 52.
This isn’t due until the day of your exam.