Calculus AB
Quarter 1 Test 2 Review (2009-2010)
_________
Mathematician _________________________
Date _______________ Period
50 Point Total
NO CALCULATORS
-----------------------------------------------------------------------------------------------------------------------------(4 points)
1. Write the formal definition of limit. L = lim f ( x ) if and only if . . .
x c
-----------------------------------------------------------------------------------------------------------------------------(4 points)
2. For the function f whose graph is at the right, complete the following.
(a) lim f (x) = __________
x®3
(b) lim f (x) = __________
x®0
(c) lim f (x) = __________
x®1
(d) lim- f (x) = __________
x®0
-----------------------------------------------------------------------------------------------------------------------------(2 points)
x
f(x)
3. Answer the questions using the table.
0.6 1.127017
(a) What appears to be lim f ( x) ? __________
0.8 1.055728
x 1
0.9
0.99
0.999
1.001
1.01
1.1
1.2
1.4
(b) If f(x) is kept within 0.02 unit of the limit found in
part (a), then x will be kept within __________ unit of 1.
1.026334
1.002513
1.00025
0.99975
0.997512
0.976177
0.954451
0.91608
-----------------------------------------------------------------------------------------------------------------------------(1 point each)
4. If lim f ( x)  4.5, then if lim f ( x) exists, to what value does it converge?
x 3
x3
(a) 6.5
(b) 4.5
(c) 1
(d) 2
(e) 6
(3 points)
5. The function graphed to the right has a limit of 1
as x approaches 4. If epsilon is 0.5, estimate the largest
value for delta that will satisfy the definition of limit.
Show the delta and epsilon neighborhoods on the graph.
 = __________
-----------------------------------------------------------------------------------------------------------------------------(3 points)
6. Write the definition of continuity at a point.
____________________________________________________________________________________
(2 points each)
Use the graph of f(x) to answer 7-9.
7. Use the 3-part definition of continuity to show
f(x) is continuous at x = 3.
8. What kind(s) of discontinuity are shown in the
graph of f(x)?
9. If there is removable discontinuity, assign a value
to remove it.
___________________________________________________________________________________
(2 points)
10.
Calculus AB
Quarter 1 Test 2 Review (2009-2010)
Mathematician _________________________
Date _______________ Period _________
50 Point Total
CALCULATOR ACTIVE
-----------------------------------------------------------------------------------------------------------------------------(3 points)
14. Suppose lim f ( x)   8 and lim g ( x) 10. Find the following.
x 7
(a) lim[2 f ( x)  g ( x)]
x 7
x 7
(b) lim
x 7
f ( x)
g ( x)
(c) lim[-4ig(x) + 9]
x®7
-----------------------------------------------------------------------------------------------------------------------------(2 points each)
15. Sketch a graph that illustrates the function described below.
(a) f(-2) = 1
(b) lim f ( x)  3
f(-1) = 4
lim f ( x)   3
x  2
horizontal asymptote y = 0
x 
lim f ( x)   
x 1
lim f ( x)  3
x 
-----------------------------------------------------------------------------------------------------------------------------(5 points)
16. For f(x) = 4 x  1 , f(x) = 3 when x = 2. How close must we hold x to 2 for f(x) to be within 0.6
unit of 3? Solve algebraically and calculate the answer to six decimal places.
If x is within _______________ unit of 2 (but x  2), then f(x) will be within 0.6 unit of 3.
Identify the following:
L = __________
c = __________
 = __________
 = __________
(3 points)
17. Given lim  2 x  6   4 . Write an equation that shows how delta depends on epsilon.
x 1
-----------------------------------------------------------------------------------------------------------------------------(5 points)
18. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the
tank in an hour, then Torricelli’s Law gives the volume of water remaining in the tank after t minutes as
2
t 

V(t) = 100, 000 1   , 0  t  60
 65 
(a) Find the average rate of change of the volume of water over the interval 0  t  20 ?
(b) Estimate the instantaneous rate of change at t = 20 minutes? Show your work.
-----------------------------------------------------------------------------------------------------------------------------(5 points)
19. Cy Kling coasts down the hill on his bicycle and then up the next hill. Every 4 seconds, he notes the
speed the bike is going and records the values in the table below. Use the Trapezoidal Rule with seven
trapezoids to estimate the distance he traveled between t = 0 and t = 28 seconds. Show your work.
sec
0
4
8
12
16
20
24
28
ft/sec
16
26
30
34
37
36
32
25