Summer Review Packet
Calculus with Applications
The problems in this packet are designed to help you review topics that are important
to your success in Calculus. You must know how to do all these problems WITHOUT a
calculator.
If you need help with any of the problems, check the Poolesville web site for links to
on-line classes at Montgomery College, which are available to you free of charge. You
may also e-mail Mrs. Loomis at [email protected] with questions.
Calculus with Applications Summer Review
I.
Simplify. Show the work that leads to your answer.
1.
x4
2
x  3x  4
3.
5 x
x 2  25
II.
1.
x3  8
2.
x2
x 2  4 x  32
4.
x 2  16
Simplify each expression.
1
1

xh x
1
1

3. 3  x 3
x
2
2
2. x
10
x5
4.
2x
1
8

 2
x  6x  9 x 1 x  2x  3
2
Calculus with Applications Summer Review
III.
Complete the following identities.
1. sin2x + cos2x = __________
3. cot2x + 1 =
__________
5. sin 2x =
__________
IV.
2. 1 + tan2x = __________
4. cos 2x =
__________
Solve for z.
2. y2 + 3yz – 8z – 4x = 0
1. 4x + 10yz = 0
V. If
x 3
f(x) = {(3,5), (2,4), (1,7)}
g(x) =
h(x)= {(3,2), (4,3), (1,6)}
k(x) = x2 + 5
determine each of the
following:
1. (f + h)(1) = _______________
2. (k – g)(5) = _______________
3. (f ◦ h)(3) = _______________
4. (g ◦ k)(7) = _______________
5. f -1(x) =
_______________
6. k-1(x) =
1
=
f ( x)
_______________
7.
8. (kg)(x) =
_______________
_______________
Calculus with Applications Summer Review
VI.
Follow the directions for each problem.
1. Evaluate
f ( x  h)  f ( x )
and simplify if f(x) = x2 – 2x.
h
2. Expand (x + y)3
3. Simplify:
VII.
1.
3.
3
2
5
2
x ( x  x  x2 )
Simplify
x
x
e(1ln x )
_________________ 2.
eln3
_________________
4. ln 1
_________________
_________________
5. ln e7
_________________ 6. log3(1/3)
_________________
7. log 1/2 8
_________________
_________________
9.
e3ln x
8. ln
_________________ 10.
1
2
4 xy 2

1
3
12 x y
_________________
5
11. 272/3
_________________ 12. (5a2/3)(4a3/2)
13. (4a5/3) 3/2
_________________ 14. Blank!!
_________________
Calculus with Applications Summer Review
VIII.
Using the point-slope form y – y1 = m(x – x1), write an equation for the line.
1. with slope –2, containing the point (3, 4)
1. __________________________
2. containing the points (1, -3) and (-5, 2)
2. __________________________
3. with slope 0, containing the point (4, 2)
3. __________________________
4. perpendicular to the line in problem #1,
containing the point (3, 4)
4. __________________________
IX.
Determine the exact value of each expression.
1. sin 0
________
4. cos 
________
7. tan
7
4
________
10. cos(Sin –1
1
)
2
2. sin

2
________
3. sin
3
4
________
________
5. cos
3
4
________
6. cos

3
8. tan

6
________
9. tan
2
________
3
________
11. Sin –1 (sin
7
)
6
________
Calculus with Applications Summer Review
X.
For each function, determine its domain and range.
Function
1. y  x  4
2. y  x2  4
3. y  4  x2
4. y  x2  4
Domain
Range
_________________
_________________
_________________
_________________
_________________
_________________
_________________
_________________
Calculus with Applications Summer Review
XI.
Solve for x, where x is a real number. Show the work that leads to your solution.
1. x + 3x – 4 = 14
x4  1
0
2.
x3
3. (x – 5)2 = 9
4. 2x2 + 5x = 8
5. (x + 3)(x – 3) > 0
6. x2 – 2x - 15  0
7. 12x2 = 3x
8. sin 2x = sin x , 0  x  2
9. |x – 3| < 7
10. (x + 1)2(x – 2) + (x + 1)(x – 2)2 = 0
11. 272x = 9x-3
12. log x + log(x – 3) = 1
2
Calculus with Applications Summer Review
XII.
Graph each function. Give its domain and range.
1. y = sin x
2. y = ex
Domain_________________
Domain_________________
Range _________________
Range _________________
3. y =
4. y =
x
3
x
Domain_________________
Domain_________________
Range _________________
Range _________________
Calculus with Applications Summer Review
5. y = ln x
6. y = |x + 3| - 2
Domain_________________
Domain_________________
Range _________________
Range _________________
7.
1
y
x
8.
x 2

y = x + 2
4

if x < 0
if 0  x  3
if x > 3
Domain_________________
Domain_________________
Range _________________
Range _________________