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Semester 1 Final Exam Review Name ________________ AP Calculus AB
1. Determine whether the graph of the function is symmetric about the y-axis, x-axis, origin or neither. 5 1 2
(a) y x (b) y x (c) x y
2
4.9
2. A water balloon dropped from a window high above the ground falls y t m t sec. Find the balloon’s… in (a) average speed during the first 3 sec. of fall.
(b) speed at the instant t = 3
3. Given the picture to the right. True or False? (a) lim ( ) f x 1 (b) lim ( ) f x DNE
x x 1 2
(c) lim ( ) f x
2 (d) lim ( ) f x
2 x 2 x 1
(e) lim ( ) f x
1 (f) lim ( ) f x DNE x 1 x 1 ( )
4. Find the limit of f x as
(a) x ___________ x
2 , x
(b) x ___________ x
f x ( )
1
x 0
1
(c) ___________
, x 2
x
(d) x 0 ___________
5. Find each point of discontinuity and what type of discontinuity each is. Which discontinuities are removable? Which are not?
Points of discontinuity __________
3 x x ,
2
Types ____________________________
f x x (b) ( ) 2,
2 x , x
2 Removable or not? ________________
2
6. Give a formula for the extended (simplified) function that is continuous at the indicated point. 2 x
9
f x ( ) , x
3
x
3 7. Find the limit (if it exists). 2 3 3 2 6 4 2 x
3 4 x 4 x 4 x
(a) lim (b) lim x x 2 2 4 5 2
5 x
7 4 x 4 x
1
1 x
sin 2
2 x
2
(c) lim (d) lim x x 0 x x
4
8. The accompanying figure shows the velocity v ( ) f t of a particle moving on a coordinate line.
(a) When does the particle move forward? Move backward? Speed up? Slow down? (b) When is the particle’s acceleration positive? Negative? Zero? (c) When does the particle move at the greatest speed? (d) When does the particle stand still for more than an instant?
9. A particle moving along a line so that its position at any time t 0 is given by the function: 2 ( )
3
2
s t t t (a) Find the displacement during the first 5 seconds.
(b) Find the average velocity during the first 5 seconds. t = 4. (c) Find the instantaneous velocity when (d) Find the acceleration of the particle when t = 4 (e) At what values of t does the particle change direction? s is a minimum? (f) Where is the particle when dy
10. Find dx x
2
1
(a) y (b) y x 2 x
1
x
2 2
1 1
(c) y x csc 5 x (d) y ln(cos x )
2 2
log ( 7) (e) y tan (3 x ) (f) y x 5
2
y ln(1 ) x y x (g) (h) 11. Find the equation of the line tangent to the curve at the given point. 2
sin x = 3 (a) y x x , at
3
3 , at x = 2
1 (b) y x x
2
12. If f '( ) x 3 x 2 x , then f x ( ) (a)
(2) ( )
(b) If f , then f x 13. Find equations for the lines that are (a) tangent, and (b) normal at the given point. 2 2
x xy y 1, at (2,3) (a) _____________________
(b) _____________________ dy x
2
1
14. If , for what values of x are the tangents to the graph 2 dx x x
4
16
(a) Vertical? (b) Horizontal? 2 1
d y 2 y x
15. Find of (2 5) 2 dx
16. Use analytic methods to find the extreme values of the function. Indicate the intervals on which the function is increasing and decreasing. 3 2
y x x
8 x
5 17. Find the absolute maximum and minimum on the closed interval [-1, 6]. 2
18. Identify all points of inflection of the given function. Then find the intervals on which the function is concave up and concave down. x 3
(a) y (b) (4 )
2 y x x
x
1
19. For the given function, find all values that satisfy the Mean Value Theorem on the given interval. 2 ( )
2
1
f x x x on [0, 3]
2
' 6( 1)( 2) 20. y x x . Find the x values for which y has:
(a) local maxima (b) local minima (c) point(s) of inflection
21. Suppose the edge lengths x, y, and z of a closed rectangular box are changing at the following rates: dx dy dz
1 m/sec 2
1
dt dt dt
Find the rates at which the box’s (a) volume and (b) surface area are changing at the instant when x = 4, y = 3, and z = 2
22. What are the dimensions of the lightest open-top right circular cylindrical can that will
3
hold a volume of 1000cm ?
23. A 13ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.
(a) How fast is the top of the ladder sliding down the wall at that moment? (b) At what rate is the area of the triangle formed by the ladder, wall and ground changing at that moment? (c) At what rate is the angle between the ladder and the ground changing at that moment?
24. An open-top rectangular box is constructed from a 10- by 16-in. piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Find analytically the dimensions of the box of largest volume and the maximum volume. x h x x h x
cot 7( ) cot(7 ) ln 2( ) ln(2 )
25. (a) lim (b) lim h h h h
Answers: 1a. origin 1b. origin 1c. x-axis 2a. 14.7m/s 2b. 29.4m/s 3a.T 3b.F 3c.F 3d.T 3e.F 3f.T 4a. 1 4b. 0 4c. 2 4d. 5a. x=2, jump, not removable 5b. x=2, removable (assign f(2)=1) 6. f x ( ) x
3
2 4
1
1
7a. 7b. 4 7c. 7d.
5
4
2
8a. forward [0, 1) and (5, 7), backward (1, 5), speed up (1, 2) and (5, 6), slow down [0, 1.3), (3, 5), and (6, 7) 8b. positive (3, 6), negative [0, 2) and (6, 7), zero (2, 3) and (7, 9] 8c. at t=0 and (2, 3) 8d. (7, 9] 23
1 9a. 10m 9b. 2m/s 9c. 5m/s 9d. 2m/s 9e. t= s 9f. s m
2
4
4
3 x
1 2 10a. 10b. 10c. 2 5 x csc 5 cot 5 x x 2 csc 5 x x x
(2 1) x
2
1
1 2 2 2
1
10d. 10e. 4 tan(3 x x ) sec (3 x ) 10f. , x
7
1 2
x ( 7) ln5 cos x
1 x
1 2 1 10g. , x 10h.
2 x x e e
11a. y=-8.063x+25.460 11b. (y-3)=9(x-2) 11c. y x ( ) 2( ) 3 2 3 2
2
2 12a. f x ( ) x x C 12b. f x ( ) x x
12
7
4
13a. ( y 3) ( x 2) 13b. ( y 3) ( x 2)4
7
3
1
y
14a. x=0 and x=4 14b. x
15. ''
52 2
2 2 x
5
4
4
4
16. max at ,15.074 , min at 2,1 , increasing , 2, , decreasing , 2
3
3
3
17. abs max (6, 33), abs min (2,1)
3 3 and
18a. 0,0 , 3, , 3, , con down , 3 0, 3 , and con up
4
4
,0 2, , con up 0, 2 3,0 3, 18b. (0, 0) and (2, 16), con down
3 19. x 20a. x = -1 20b. x = 2 20c. x = 0 and x = 2
2 1 3 2 3 21a. 2m /sec 21b. 0m /sec 22. r h cm or cm
10
6.83
AP Calc AB Semester 1 Final Review Packet Help Videos Use a “Barcode Scanning” app with your smart phone to view videos below.
No smart phone? All help videos can be found at my website Problem 8 ……………… Problem 10 ……………… Problem 14 ……………… Problem 15 ……………… Problem 21 ……………… Problem 23 ………………