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Appendix D
Trigonometric Functions
Conversion from Degrees to Radians:
Multiply by
180
Conversion from Radians to Degrees:
180
Multiply by
Ex 1: Convert 300 to radians.
5
Ex 2:
What Is Coming Up Due?
Section 2.8
Related Rates
In a related rates problem we try to compute the
rate of change of one quantity in terms of the
rate of change of another quantity, which may
be more e
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Section 2.2
The Derivative as a Function
The Derivative of a function f (x) is given by:
In the previous section we found the value of the derivative of f at a particular point
Section 2.1 Derivatives and Rates of Change
Lines and Graphs and Tangents, oh my!
How would you find the slope of the line PQ?
If
x began to move to the left and approach a , the
secant would move.
Ex
What Is Coming Up Due?
Section 2.1
Derivatives and Rates of Change
How would you find the slope of the line PQ ?
If x began to move to the left and approach a, the secant would move.
The line t is tan
Section 2.2 The Derivative of a Function
The Derivative of a Function f ( x) is given by:
In the previous section, we found the value of the derivative of f at a particular point a .
We learned that t
Calculus I Test 1 Review Problems
From Section 1.1
Be able to evaluate the difference quotient of a given function and simplify your answer:
a) () = 4 + 3 2
b) () = 5 2 3 + 8
From Appendix D
Be able t
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Section 2.7
Rates of Change in the
Natural and Social Sciences
Ex 1: A Particle moves according to a law of motion s = f (t), t 0
where t is measured in seconds and s in feet.
f
Section 1.2 More Functions
There are many different types of functions and graphs that we will explore during the course of
the semester. These include the following:
Linear: y = mx + b
n
n1
Polynomia
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Section 3.3
How Derivatives Affect
the Shape of a Graph
The First Derivative of a Function Tells Us:
Ex 1: Find the intervals of increasing and decreasing and any local
extrema.
Section 2.5 Chain Rule
The Chain Rule
Example: Find the derivative using the chain rule.
1.
2 100
f ( x )=( 4 xx )
2.
3.
f ( x )=
2
( 1+ sec x )2
f ( t )=( 3 t1 )4 ( 2 t+1 )3
4.
5.
f ( t )=
( 2t +1 )3
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Section 1.2
More Examples of Functions
Types of Graphs
Linear: y = mx + b
Polynomials: anxn + an-1xn-1 + + a1x + a0
Power Functions: xa
Rational Functions: two divided polynomia
Section 2.3 Differentiation Formulas
Example: Use the definition of the derivative to find the following.
1.
d 2
(x )
dx
2.
d n
(x )
dx
First Few Derivative Rules
Power Rule:
d n
( x )=n x n1
dx
Const
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Section 1.1
Functions
Can you define the following terms?
1. Function
2. Domain
3. Range
4. Independent Variable
5. Dependent Variable
6. Graph
Ex 1: Function f is graphed.
a. F
Appendix D Trigonometric Functions
To convert from degrees to radians: multiply by
180 .
To convert from radians to degrees: multiply by
180
.
Example: Complete the following conversions.
1. Convert
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Section 3.7
Optimization
Ex 1: Find the dimensions of a rectangle with perimeter 100 m
whose area is as large as possible.
Ex 2: If 1200 cm2 of material is available to make a b
Section 2.4 Derivatives of Trig Functions
Extremely Important Limits
These two limits are EXTREMELY important to the following proofs for the
trigonometric derivatives.
Proofs for each one can be foun
Section 1.5 The Limit of a Function
Example: Use the graph and/or table to guess the value of each function or limit.
1.
2.
3.
f ( x )=
lim
x 0
x1
x 21
sin x
x
lim sin
x 0
x
Left and Right Handed Limi
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Section 3.2
The Mean Value Theorem
Ex 1: Verify that the function satisfies the three hypotheses of Rolles
Theorem on the given interval. Then find all numbers c that
satisfy th
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Section 2.6
Implicit Differentiation
We dont need to solve an equation of y in term of x to find the
derivative of y.
Ex 1: We will solve using implicit differentiation first an
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Section 3.9
Antidifferentiation
Lets look at these two functions:
f (x) = x2 7
g(x) = x2 + 5
n +1
x
Rule: xn has an antiderivative of
n +1
Ex 1: Find the most general antideriva
Section 1.1 Functions
From prior classes you should be able to define the following terms:
1.
2.
3.
4.
5.
6.
Function
Domain
Range
Independent Variable
Dependent Variable
Graph
Example: Identify the f
Section 1.6 Properties of Limits
Limit Laws
Direct Substitution Property
If f ( x) is a polynomial or rational function and a is in the domain of f
lim f ( x ) =f ( a)
x a
so long as f ( x) is continu
Section 1.3 Operations on Functions
Example: Match the equation with its graph.
1.
a.
y=f ( x 4 )
b.
y=f ( x ) +3
c.
y=
f (x) 1
= f (x)
3
3
Example: Find f g , g f
2.
f ( x )=x +
1
x
and the domains f
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Section 3.5
Summary of Curve Sketching
Types of Asymptotes:
Horizontal
Vertical
Slant (Oblique)
[
f ( x) (mx + b)] = 0
Slant (Oblique) : lim
x
For rational functions, this occu
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Section 2.9
Linear Approximation
and Differentials
The linear approximation of a function f is given by
L(x) = f (a) + f (a)(x a)
Ex 1: Find the linearization of the function at
4.1 Harmonic Functions 247
the real part of log 2, there is no analytic function on all of the annulus whose real
part equals u.
The function u that is the imaginary part of f (that is, u + iv is anal
3.5 The Riemann Mapping Theorem and Schwarz—Christoffel Transformations 243
A: —R+i1r/2
B=R+i1r
C=R
andR—voo
Figure 3.54
15. D is pictured in Figure 3.55. (Hint: Obtain this region as the limit of t