Review Packet for Exam #1
Math 121-D. Benedetto
Derivatives: Compute the derivative for each of the following functions. Do not worry about
simplifying your answers:
1. f (x) = sinh1 (ln(cos3 x)
cosh( x)
2. f (x) =
5e2x
3. f (x) =
1
arctan(17x)
4. f (x) =
Midterm Exam #1
Math 12
February 23, 2011
Answer Key
1.
[30 Points] Evaluate each of the following limits. Please justify your answers. Be clear if the
limit equals a value, + or , or Does Not Exist.
LH
(a)
0
xex x ( 0 )
=
x sin(3x)
lim
x0
LH
0
(0)
=
xex
Extra Examples of Inverse Trigonometric Integrals-Math 121-Benedetto
1.
1
dx
16 + x2
1
=
=
1
16
=
x
4
2
factor out 16 to get 1 in lead position
dx
1
1+
1
dx = arctan x + C
1 + x2
dx
x2
16 1 +
16
4
=
16
=
not quite in the form of
rewrite as perfect square
Extra Examples of Trigonometric Integrals
Math 121 D. Benedetto
ODD POWER TECHNIQUE
1.
sin3 x cos5 x dx =
sin3 x cos4 x cos x dx
2
=
sin3 x cos2 x
=
sin3 x 1 sin2 x
cos x dx
2
nd remaining even powers of cos x
cos x dx convert cos2 x using trig. identity
Integration Reference Page
Math 121 D. Benedetto
xn+1
+C
n+1
1
dx = ln |x| + C
x
sin x dx = cos x + C
cos x dx = sin x + C
tan x dx = ln | cos x| + C = ln | sec x| + C
sec x dx = ln | sec x + tan x| + C
sec2 x dx = tan x + C
Exponentials and Logarithms
xn
Answer Key for Review Packet for Exam #1
Math 121-D. Benedetto
Derivatives: Compute the derivative for each of the following functions. Do not worry about
simplifying your answers:
1. f (x) = sinh1 (ln(cos3 x)
1
1
3 cos2 x ( sin x)
f (x) =
1 + (ln(cos3 x
Overview of ex and ln x Functions
Graphs
Function Properties
Natural Exponential Function y = ex
Natural Logarithm Function y = ln x
Domain = R
Domain = (0, )
Range = (0, )
Range = R
lim ex =
lim ln x =
x
x
lim ex = 0
lim ln x =
x
x0+
e chose
Math 121, D. Benedetto
What you need to know for Exam 1
You should know Section 4.5, Sections 6.16.8, except Section 6.5. Also know Sections 7.1-7.3. The
test will not explicitly cover material from earlier sections (or from Appendix D), but of course
it
Review Handout for Derivatives and u-Substitution-Math 121
Compute the following derivatives. Do not worry about simplications here.
1. y = sin3 (x3 )
y = 3 sin2 (x3 ) cos(x3 ) 3x2 = 9x2 sin2 (x3 ) cos(x3 )
2. y = cos2 (3x)
y = 2 cos(3x) ( sin(3x) 3 = 6 c
Math 12 Spring 2011
Exam #1
February 23, 2011
Prof. D. Benedetto
Compact Version
This is a closed-book examination. No books, notes, calculators, cell phones, communication
devices of any sort, or other aids are permitted.
You need not simplify algebrai