Trigonometric Identities
Reciprocal Identities
1
1
1
=
csc x
, sec x
, cot x
sin x
cos x
tan x
Tangent/Cotangent Identities
sin x
cos x
=
tan x
, cot x
cos x
sin x
Pythagorean Identities
sin 2 x + cos
Page |1
Unit 1
Chapter 1 Precalculus Review
In this chapter, we review the essential topics of precalculus that will relate directly to many of
the calculus problems you will have to solve. This is no
Geometry Formulas
Perimeter: The perimeter of a simple closed figure is the distance once around the outside of
the figure.
Circumference of a Circle: C = 2 r
Area of Rectangle or Parallelogram: A = b
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Geometry Examples Solutions
Example 1 Solution
Note that the length is 2w 5. So, the perimeter is
2 w 2 2 w 5 2 w 4 w 10 6 w 10.
Example 2 Solution
We have 2 r 25 . Dividing both sides by 2
Unit 1
Additional Problems
No Calculator
1
1. Simplify
( x + h)
2
1
+ 5 2 + 5
x
h
.
2. Solve: sin ( 2 x ) = sin x on [ 0, 2 ) .
3. A box with a square base and height h has a volume of 25. Find its
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Unit 5
Chapter 6 Applications of the Definite Integral
6.1
Area
y
y
y f ( x)
y f ( x)
0
a
b
x
y g ( x)
0
a
0
b
x
b
area f ( x) dx
0
an Rx region
a
Rx Regions We call a region an Rx region
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Unit 4
Chapter 5 Integrals
5.1
Antiderivatives and Indefinite Integrals
Definition A function F is an antiderivative of f on an interval I if F ( x) f ( x)
for every x in I.
Example 1 Find
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Unit 3
Chapter 4 Applications of the Derivative
4.1
Extrema of Functions
Definition Let f be a function defined on an interval I , and let x1 , x2 denote numbers in I .
(i) f is increasing
Unit 3
Additional Problems
Calculator
1. Suppose f ( x) x 2 2sin x. Use the graph of f to estimate the maximum and minimum
values of f on 0, 2 correct to four decimal places.
2. Find all numbers c tha
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Unit 2
Chapter 3 The Derivative
3.1
Tangent Lines and Rates of Change
Slope The slope of a non-vertical line measures how much y changes for a unit change in x
(called the rate of change of
Derivative Rules
Definition
f ( x) = lim
h0
f ( x + h) f ( x )
h
Power Rule Dx ( x r ) = r x r 1
Constant Multiple Dx ( c f ( x) ) = c f ( x)
Sum/Difference Dx [ f ( x) g ( x) ] = f ( x) g ( x)
Produc
Unit 2
Additional Problems
No Calculator
f ( x + h) f ( x )
to find f ( x).
h0
h
7
5
(b) f ( x) =
(c) f ( x)= 6
2x + 3
x +1
1. Use the definition f ( x) = lim
(a) f ( =
x)
1
x x3
2
2. The graph of y
Unit 2
Additional Problems
Calculator
1 + x2
f ( a + h) f ( a )
with h = 0.0001 to
1. Suppose f ( x) =
. Use the slope of the secant line
2
h
sin ( x )
3
approximate the slope of the line tangent to t
Calculus IExam 1 (Ch. 1, Ch. 2) June 19th, 2012
Solutions
1.
4
2 0x
4 2x
4 2x
0 xx
0
x
0
x
22x
x
,0
2, Note that x 0 is not a solution because of division
by 0.
2.
2 1 sin2 x sinx 1 0
2sin2 x sinx 1 0
5.
x 2 x2 2x 4
x 2 x2 2x 4
f(x)
2
x 2x 4
x3 x2 x2
x2 x 3 4 x 3
x3 x2
lim f(x) and lim f(x)
.x3x2
So, x 3 and x 2 are the only vertical asymptotes.Note that x 2
causes a division by zero, but there is