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.
(a)
(a) average speed during the first 3 sec. of fall.
(b) speed at the instant t = 3
5. Find each point of discontinuity and what type of discontinuity each is. Which discontinuities are removable? Which are not?
(a)
Points of discontinuity __________
Types ____________________________
(b)
6. Give a formula for the extended (simplified) function that is continuous at the indicated point.
(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?
Points of discontinuity __________
Types ____________________________
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. (c) Find the instantaneous velocity when t = 4.
(d) Find the acceleration of the particle when t = 4
(e) At what values of t does the particle change direction? (f) Where is the particle when s is a minimum?
10. Find dy dx (a)
2 1
2 1
x y
x (b) y x 2x 1
(c) y x2csc 5x (d) y ln(cos1x)
(e) y tan (32 x2) (f) y log (5 x 7)
(g) y ln(1 x ) (h) 2
y x
11. Find the equation of the line tangent to the curve at the given point. (a) y x2sinx , at x = 3
(b) y x3 3x 1, at x = 2
12. If 2
'( ) 3 2
f x x x , then (a) f x( )
(b) If f (2)0, then f x( )
13. Find equations for the lines that are (a) tangent, and (b) normal at the given point.
2 2
1, (2,3)
x xy y at (a) _____________________
(b) _____________________
14. If
2
2 1
4 16
dy x
dx x x
, for what values of x are the tangents to the graph (a) Vertical?
(b) Horizontal?
15. Find 2
2 d y
dx of
(2 5) 12
y x
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
8 5
y x x x
17. Find the absolute maximum and minimum on the closed interval [-1, 6]. 2
2 8 9
18. Identify all points of inflection of the given function. Then find the intervals on which the function is concave up and concave down.
(a) 2
1 x y
x
(b)
3
(4 )
y x x
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]
20. y '6(x 1)(x 2)2. 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:
1 dx
dt m/sec 2
dy
dt 1
dz 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
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.
25. (a) 0
cot 7( ) cot(7 ) lim
h
x h x
h
(b)
0
ln 2( ) ln(2 ) lim
h
x h x
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.
AP Calc AB Semester 1 Final Review Packet Help Videos
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No smart phone? All help videos can be found at my
website:https://sites.google.com/a/hf233.org/dmcclain/home/ap-calculus-ab
Problem 8 ………
Problem 10 ………
Problem 14 ………
Problem 15 ………
Problem 21 ………
