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Summer work for Honors Calculus(349)
This packet contains problems involving skills that you should already know. Please take your
time with these problems and SHOW YOUR WORK. Do not use a calculator unless otherwise
noted.
The review should be time to practice things you’ve relearned (or learned) over the summer.
Please use your resources wisely. There are many online resources available to guide you
through any problems you may have. Ex. khanacademy.org, wolframalpha.com, mathisfun.com
You may also contact me at [email protected] if you have questions.
This packet is due August 29, 2016.
A. Simplify. Show the work that leads to your answer.
x4
1) 2
x  3x  4
3)
5 x
x 2  25
x3  8
2)
x2
4)
x 2  4 x  32
x 2  16
B. Simplify each expression in order to obtain a single fraction. Show all work.
1)
1
1

xh x
2
x2
2)
10
x5
3)
2
1
1  a2
4)
x 1 
2
x 1
C. If f ( x)  1  x 2 and g ( x)  2 x  1 , find:
1) f (3)
2) g (2)
3) g ( f (4))
4) f (g (0))
5) f ( x  h)
6) f ( g ( x))
7) g ( x)  f ( x)
8) f ( x) g ( x)
9)
g ( x  h)  g ( x )
h
D. Using point-slope form y  y1  m  x  x1  , write an equation for the line…
1) with slope –2, containing the point (3, 4)
_________________________________
2) containing the points (1, -3) and (-5, 2)
_________________________________
3) with slope 0, containing the point (4, 2)
_________________________________
4) perpendicular to the line in #1, containing
the point (3, 4)
__________________________________
E. Find the equation of all vertical ( x  ? ) and horizontal ( y  ? ) asymptotes, if they exist.
1) y 
x
x3
2) y 
x3  4
x2  1
3) y 
2x  5
x2  1
F. For each of the following, sketch the function and then determine its domain and range.
1) y 
x4
2) y  3sin 2 x
2
3) y  9  x
4) y 
G. Complete the following identities.
2
2
1) sin x  cos x  _________
2
2
3) csc x  cot x  _________
2
2) 2  2 tan x  _________
1
x 1
H. Factor the following completely.
2) t 4  13t 2  36
1) 2 x 2  13x  15
5
2
3
2
3) x  x  12x
1
2
4) ( x  3)2 (2 x  1)3  ( x  3)3 (2 x  1)2
I. Multiply and simplify your results.
1)
2x 9 y

3 y 8x
2)
2u v3

v 3u
3)
6s 2 10st

5t 3 6s 3
4)
x2  4 2 x  4

6
x2
5)
3y  9
y3
 2
14 y y  9
6)
t 2  25 2s  3

4s 2  9 5  t
J. Complete the missing sides and angles of the special right triangles.
1)
2)
X
2X
X
X
K. Determine the exact value of each expression. Remember NO CALCULATORS!
1) sin 0  ______
2) sin
3
 __________
4
3) cos   __________
4) cos
7
 _______
6
5) tan
7
 __________
4
6) tan 0  __________
7) csc
2
 _______
3
8) sec
3
 __________
2
9) cot

 ______
3
11) cos
10) sin
13) tan 1  1 =____

 __________
2
14) arccos 1 =_____
11
 __________
6
12) arcsin
3
=_________
2
 1
15) arcsin     ________
 2
L. Simplify each radical/radical expression.
1)
56
2)
4)
64
121
5)
3
108
 4  6  4  6 
6)
M. Solve the equation for x, where x is a real number.
1) x2  3x  4  14
3)
 x  5
2
9
125  80
3)
2)
x4 1
0
x3
4) 2 x2  5x  8

11  2

2
5)
5
1
e 1
2x
7) 5 ln(2 x  1)  3  6
6)
8)
x 1 
5
0
x 1
4
1
5
 
x 1 6 x  3
Solve each equation on the interval [0,2 ) .
9) sin x  1  0
10) 2 cos x  3  0
11) 4 sin 2 x  1
12) 2 sin x cos x  sin x  0
Solve for z.
13) 4 x  10 yz  0
14) h 
3
2x4
z
N. The number of elk after t years in a state park is modeled by the function P(t ) 
You may use a calculator for this problem.
1) What was the initial population?
2) When will the number of elk be 750?
3) What is the maximum number of elk possible in the park?
1216
1  73e 0.03t
O. Limits
 x 
2) lim sin  
x 5
6
1) lim 2 x  5
x 3
x 2  4 x  21
x 7
x 2  49
x2
x  x 2  5
4) lim
5) lim
3) lim
x  4
x4
x  5x  4
2
4 x2  5x  2
x   3 x 2  5 x  12
6) lim
P. Geometry
1) Find the area of the trapezoid below.
12cm
45o
16cm
2) Find the volume and surface area
of the sphere that has a radius
of r  4 cm.