Cherry Creek High School
Summer Assignment
AP Calculus AB
Please have the following worksheets completed and ready to be handed
in on the first day of class in the fall. Make sure you show your work
where appropriate. Answers are provided for you to check; however,
you will not be given credit if you don’t show work on problems that
require it. It is expected that you have a good understanding of this
material coming into these courses as teachers will not be doing an
extensive review of previously learned material.
Have a great summer and we look forward to seeing you in the fall!
The CCHS Math Department
Cherry Creek HS
AP Calc AB Summer Packet Supplement
Limit Flowchart
Cherry Creek HS
AP Calc AB Summer Packet Supplement
Derivative Rules
Power
d n
x = n•xn-1
dx
Product
d
f•g = f’•g + f•g’
dx
Quotient
d f f ′• g − f • g′
=
g2
dx g
Chain
d
f( g(x) ) = f’( g(x) )•g’(x)
dx
exponential
d f(x)
e = ef(x) • f’(x)
dx
logarithm
f ′(x )
d
ln[f(x)] =
f (x )
dx
Trig Derivative Rules
d
sinx = cosx
dx
d
cosx = -sinx
dx
d
tanx = sec2x
dx
d
cotx = -csc2x
dx
d
secx = secx•tanx
dx
d
cscx = -cscx•cotx
dx
Pythagorean Identities
cos x + sin2x = 1
tan2x + 1 = sec2x
2
Double Angle
Sin(2A) = 2 sinAcosA
Cos(2A) = cos2A – sin2A = 2cos2A – 1 = 1 – 2sin2A
2tan A
Tan(2A) =
1− tan 2 A
cot2x + 1 = csc2x
Cherry Creek HS
AP Calc AB Summer Packet
Solving Equations
Name
Solve each. Give exact solutions.
1. x2 = 7x
2. x2 – 3x – 54 = 0
4. 2x − 1 = 5
7.
5.
x2 −4 = 5
x
7
24
−
= 2
x − 3 x + 5 x + 2x −15
2x-1
9. 4
= 8x+2
6. x4 – 2x2 – 24 = 0
8.
10. 42 + 3e3x/ 2 = 965
12. . log2x + log2(x-6) = 4
3. (x-3)2 = 12
x −1 + 3 = x
11. log9x =
3
2
13. e2x - 4ex = 0
Solve by graphing on a calculator.
14. e-x = 3x - x2
15. x2 = 5•cosx
Give a general solution for each.
17. 2sinx – 1 = 0
18. 2cos2x + cosx – 1 = 0
Solve each inequality. Give answers in interval notation.
x2 −4
20. 2x2 - x - 6 ≥ 0
21. 2
≤0
x − 10x + 21
16. 2•1.02x = 4 + .5x
x: [-30,300], y: [-30,150]
19. tan(4x) =
22. 2x + 3 ≤ 7
3
Cherry Creek HS
AP Calc AB Summer Packet
Graphing
Name
Sketch the graph of each:
1. . y =
3
x–5
4
4. y = -3(x-2)2 + 6
7. y =
x +4 +1
10. y = ln(x+3)
−3
(x+4)
2
2. 2x – 5y = 10
3. y - 2 =
5. y = x2 – 6x + 8
6. y = -2•| x-4 | - 3
8. y =
4−x2
9. y =
3−x
11. y = ex+1
12. Write the equation of the line thru (2,-3) and perpendicular to y = 2x - 4.
Sketch one fundamental period for each.
13. y = 3sinx + 2
14. y = cos(2x – π)
15. y = 2secx
Given a graph of f(x) at right,
sketch a graph of each transformation.
16. f(x) + 2
17. f(x+2)
18. f (x )
19. f(-x)
Use a calculator to
(a) Find any relative maximum or minimum values of the function.
(b) Determine the intervals over which the function is increasing
(c) Find the zeros of the function
(d) Sketch the graph of the function.
20. f(x) = x3 - 4x2 + x - 4
21. f(x) = x4 - 3x2 – 4
22. Let f’(x) = -2x•(x-3)2•(x+4). Give intervals f(x) is increasing and decreasing.
23. Let g(x) = x6 - 6x5 + 12x4 - 8x3 + 4. Give intervals g(x) is concave up and down.
Show the use of the derivative and appropriate derivative chart.
x 2 + 2x + 4
on calculator. Give relative max/min as ordered pairs (nearest
x +1
hundredth). Give asymptotes as equations.
24. Graph y =
25. Let h(x) = 2x3 - 3x2 - 12x + 24. Give absolute maximum and minimum over
the interval [-3,3].
Cherry Creek HS
AP Calc AB Summer Packet
Limits and Continuity
Name
Find each limit.
1. lim 2x 3 − 3x 5
⎛ 3πx − 4 ⎞
2. lim sin⎜
⎟
x →∞
⎝ 2x + 1 ⎠
3. lim
3x
4. lim
x →∞ 4x − 1
x2 −3
5. lim
x →−∞ x + 1
2x 2 − 7x + 6
6. lim
x →2
x3 − 4x
x →∞
x 2 + 5x + 4
x →−1 2x 2 − x − 3
x
x −2
2x
x →−∞ x − 5
2
x− 3
x2 − 9
7. lim
8. lim+
x2 − 4
10. lim 2
x →2 x − 4 x + 4
11. lim
x 2 − 25
9x 2
⎛ 9x + 1⎞
12. lim log 3 ⎜
⎟
x →∞
⎝ x+2⎠
13. lim
14. lim
x+5
x+5
15. lim− x − 3
16. lim e −x
17. lim+ ln(x )
x →2
x →∞
(x + h)2 − x 2
h →0
h
x →−∞
x →∞
9. lim
x →3
x →3
18. lim +
x →0
19. Describe the continuity of f (x ) =
x →π 2
1+ sinx
cos x
x 2 + 10x + 21
.
x2 −9
20. Define f(2) so that f(x) is continuous: f (x ) =
x2 −4
x −2
⎧x 2 − 2c,x ≤ 2
21. Give ‘c’ so that f(x) is continuous: f (x ) = ⎨
⎩ cx,x > 2
22. What feature of a graph (asymptote, hole, endpoint, break, etc) is described by the each limit
statement(s):
a. lim f (x) = ∞
b. lim+ f (x) = 1 and lim− f (x) = 3
x →2
x →2
c. lim− f (x) = −∞ and lim+ f (x) = ∞
x →1
x →1
d.. lim f (x) = 1
x →−∞
23. Give one sketch of f(x) using all of the following information:
a. lim f (x) = 1
b. lim f (x) = ∞
c. f(0) = 0
x →±∞
x →2
x →2
Cherry Creek HS
AP Calc AB Summer Packet
Derivatives and Applications
Name
dy
for each.
dx
1. y = 6x4 - 5x2 + x – 2
Find
3. y =
2. y = 3x11/3 - 2x7/4 - 2x1/ 2 + 8
3
2
+ 5 + 53 x
4
x
x
4. y = 2 x •(3x - 2)
x2
cos x
5. y = 3sin2(4x)
6. y =
4 x 3 + 10x 4
8. y =
2x
⎧4x 3 − 3x 2 ,x ≥ 2
9. y = ⎨
⎩ 2x + 1,x < 2
10. y =
11. y = (3x – 1) 1+ 2x
12. y = 3tan2(x)
14. y = cos2(3x) - sin2(3x)
15. xy2 = siny
h( x )
⎛ πx ⎞
13. y = csc ⎜ ⎟
⎝3⎠
7. y =
x 2 − 6x
16. Give the instantaneous rate of change of y = 5 - 2 x at (4,1).
17. Write the equation of the tangent line to y =
18. Determine the x-value(s) at which f(x) =
3x 2 − 2 at x = 3.
x2
has a horizontal tangent.
3x − 4
19. Use implicit differentiation to find dy/dx: x2 + 4xy +4y2 – 3x = 6
20. Find the slope of the tangent to x2 + y2 = 16 at (3,
7 )
21. A ladder 25 feet long is leaning against the wall of a house.
The base of the ladder is pulled away from the wall at a rate
of 2 ft/sec. How fast is the top moving down the wall when the
base of the ladder is 7 feet from the wall?
250
+ 6 + .1x
x
How many units should you produce to result in a minimum cost per unit?
Find the minimum cost per unit.
______
22. The cost per unit for a product is given by: C (x ) =
Selected Answers
Solving Equations
1. 0,7
2. 9,-6
3. 3± 2 3
4. -2,3 5. ± 29 6. ± 6 (±2ext)
7. -1 (3 ext) 8. 5 (2 ext) 9. 8
10. 3.819
11. 27
12. 8 (-2 ext)
13. ln4
14. .278, 2.983
15. ±1.252
π
5π
π 2nπ
π nπ
16. 199.381, -4.329 17.
19.
+ 2nπ,
+ 2nπ 18. +
+
6
6
3
3
12 4
20. (-∞,-3/2] [2,∞) 21. [-2,2](3,7)
22. [-5,2]
Graphing
−1
20a. max(.131,-3.935) min (2.535,-10.879)
(x − 2)
2
20b. (-∞,.131)(2.535,∞)
20c. x = 4
21a.max (0,4) min (±1.225,-6.25)
21b. (-1.225,0)(1.225,∞)
21c. x =±2 22. decr: (-∞,-4)(0,∞)
23. conc up: (-∞,0)(.553, 1.447)(2,∞)
24. rel.max: (-2.732,-3.464) rel min (.732,3.464) 25. Abs max (-1,31) abs min(-3,-21)
12. y + 3 =
Limits and Continuity
1. -∞ 2. -1 3. 0
4.
3
4
5. -∞ 6.
1
8
7.
−3
5
8. +∞
9.
1
12 3
1
12. 2
13. 2x
14. -1
15. DNE
3
16. 0
17. -∞
18. -∞
19. Removeable discontinuity @ x=-3, essential disc. @ x=3, continuous x ≠ ±3
20. f(2) = 4 21. c = 1
22a. vert. asymptote 22b. break
22c. vert asymptote
22d. horizontal asymptote
10. DNE
11.
Derivatives and Applications
−1
−12 10
5
7 34
− 6+ 3 2
3.
x −x 2
5
x
x
2
3 x
2
2x cos x + x 2 sinx
3/2
1/2
4. y = 3x – 4x à y ‘ = 9 x −
5. 12sin(8x) 6.
cos 2 x
x
⎧12x 2 − 6x,x > 2
x −3
2
3
2
7.
8. y = 2x +5x à y’ = 4x +15x
9. y ′ = ⎨
2,x < 2
⎩
x 2 − 6x
−π ⎛ −πx ⎞ ⎛ −πx ⎞
h ′(x )
csc⎜
10.
11. (1+2x)-1/2(9x+2) 12. 6tanx•sec2x 13.
⎟ cot ⎜
⎟
3
2 h(x )
⎝ 3 ⎠ ⎝ 3 ⎠
dy
y2
=
14. y = cos(6x) à y’ = -6sin(6x)
15.
16. -1/2
dx cos y − 2xy
−3
dy 3 − 2x − 4 y
9
=
17. y – 5 = (x − 3)
18. 0, 8/3
19.
20.
dx
4x + 8 y
5
7
−7
21.
22. x = 50 à $16 per unit
ft / sec
12
1. 24x3 -10x + 1
2. 11x
8
3
−