Lecture 6 : Inverse Trigonometric Functions
Inverse Sine Function (arcsin x = sin1 x)
The trigonometric function sin x is not one-to-one
functions, hence in order to create an inverse, we must restrict its domain.
The restricted sine function is given by
Lecture 4 : General Logarithms and Exponentials.
For a > 0 and x any real number, we dene
ax = ex ln a ,
a > 0.
The function ax is called the exponential function with base a.
Note that ln(ax ) = x ln a is true for all real numbers x and all a > 0. (We sa
Lecture 3 : The Natural Exponential Function: f (x) = exp(x) = ex
Last day, we saw that the function f (x) = ln x is one-to-one, with domain (0, ) and range (, ).
We can conclude that f (x) has an inverse function f 1 (x) = exp(x) which we call the natura
Lecture 11/12 : Partial Fractions
In this section we look at integrals of rational functions.
Essential Background
A Polynomial P (x) is a linear sum of powers of x, for example 3x3 + 3x2 + x + 1 or x5 x.
The degree of a polynomial P (x) is the highest po
Lecture 10 : Trigonometric Substitution
To solve integrals containing the following expressions;
a2 x 2
x 2 + a2
x 2 a2 ,
it is sometimes useful to make the following substitutions:
Expression
Substitution
2 x2
x = a sin , or = sin1 x
2
2
a
a
a2 + x 2
x =
Lecture 9 : Trigonometric Integrals
Mixed powers of sin and cos
Strategy for integrating
sinm x cosn xdx
We use substitution:
If n is odd use substitution with u = sin x, du = cos xdx and convert the remaining factors of cosine
using cos2 x = 1 sin2 x. Th
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SOLUTIONS TO EXAM 1, MATH 10560
1
1.The function f (x) = ln x x is one-to-one. Compute (f 1 ) (1).
Solution: We have
1
(f 1 ) (1) =
,
f (f 1 (1)
1
1
1
f 1 (1) = 1 and f (x) = x + x2 . Hence f (f 1 (1) = f (1) = 2, and (f 1 ) (1) = 2 .
2. Dierentiate the f
Multiple Choice
1.(6 pts) If the scalar projection of b onto a is Comp a b = 1, what is Comp 2a 3b?
(a)
2
(b)
5
(c)
3
2
(d)
6
(e)
3
2.(6 pts) Which of the following expressions gives the length of the curve defined by
r(t) = ln(t)i tj + t2 k between the p
Name:
Instructor:
Math 20550, Practice Exam 3
April 25, 2017
The Honor Code is in effect for this examination. All work is to be your own.
No calculators.
The exam lasts for 1 hour and 15 minutes.
Be sure that your name is on every page in case pages beco
Name:
Instructor:
Math 20550, Practice Exam 1
February 21, 2017
The Honor Code is in effect for this examination. All work is to be your own.
No calculators.
The exam lasts for 1 hour and 15 minutes.
Be sure that your name is on every page in case pages b
SOLUTIONS TO EXAM III, MATH 10560
1. Determine which one of the following series is convergent.
Note: Comparison Tests may help.
P cos2 n
(1)
2n
Pn=1
ln n
(2)
n
Pn=1
21/n
(3)
n
Pn=1
1
(4)
n=1 n(cos2 n+1)
P
n
(5)
n=1
2
Solution:
(1) Notice that 0 cos2 n 1
Multiple Choice
1.(6 pts) Which of the following vectors has the same direction as v = h1, 2, 2i but has
length 6?
(a)
(d)
h2, 4, 4i
2, 2 2, 2 2
(b)
h2, 4, 4i
(e)
h0, 6, 0i
(c)
h4, 2, 4i
2.(6 pts) Compute the vector projection of the vector h1, 0, 0i o