Name_
3/10/2003-Carlson
Derivative Exam
Compute f(x) for each of the following:
1. f(x) = e3x cos (2x)
f ' ( x) = 3e3 x cos(2 x ) 2e3 x sin 2 x
2
- 2
x3
6
f ' ( x) = 20 x 4 + 4
x
2. f(x) = 4x5 -
x3 2 x
3. f(x) =
x2
3
f ' ( x) = 1 + 2
xx
4. f(x) = tan2 (3x
Section 3.2, Items #12, 18 & 40
12)
H (t ) = e t (1 + 3t 2 + 5t 4 ) H ' (t ) = (6t + 20t 3 ) + (1 + 3t 2 + 5t 4 )e t = e t (5t 4 +20t 3 + 3t 2 + 6t + 1)
18) y =
u 2 u 2 (u 2)(u + 1)
d
=
= u 2 for u 1. y ' =
(u 2) = 1
u +1
u +1
du
x 1
( x + 1)(1) ( x 1)(1)
Section 1.2, Select Problems
80 70
10 1
=
=.
9a) Using N in place of x and T in place of y, the slope is T 2 T 1 =
N 2 N1
Then, the linear equation is
Then, T =
1
6
N+
9b) The slope is
1
6
307
6
173 113
60
6
T 80 = ( N 173) .
1
6
.
as shown in part (a). I
Section 2.1, #5, 8b
5a) At t = 2 , y = 40(2) 16(2) 2 = 16. The average velocity between times 2 and 2 + h
40(2 + h) 16(2 + h) 2 16 24h 16h 2
is:
=
= 24 16h, if h 0.
h
h
(i ) h = 0.5,32 ft sec
(ii ) h = 0.1, 25.6 ft sec
(iii ) h = 0.05, 24.8 ft sec
(iv) h
MAT 270
Fall 2016
Calculus with Analytic Geometry I
Class #:
Days/Time:
Room:
Note: The syllabus is tentative and should not be considered definitive. The instructor reserves
the right to modify it (including the dates of the tests) to meet the needs of t
MAT 270
Course Objectives
Calculus with Analytic Geometry I
Evaluate the limit of a function using numerical and algebraic techniques, the properties of
limits, and analysis techniques.
Evaluate one-sided and two-sided limits for algebraic and trigonome
Section 1.3 - 30, 54, 55, 57
30) First note that the domain of f + g is the intersection of the domains of f and g; that is
f + g is only defined where both f and g are defined. Taking the horizontal and vertical
units of length to be the distance between
Exam #2
1a) The distance s (in feet) of a car moving in a straight line is given by s = 2t t + 3 where t is
measured in seconds. Find the average velocity for the time period from t = 2 to t = 5.
2
(
)(
)
s (5) s (2) 2(5) 2 5 + 3 2(2) 2 2 + 3 48 9 39
=
=
MAT 270 - Derivative Practice I
Find the derivative of each of the following functions and simplify.
1.
f ( x) = 4 x 3 3x 2 + 2 x
2.
f ( x) =
3.
f ( x) = 3 2 x 2 5 x + 1
4.
f ( x) = x
5.
f ( x) =
x +1
x2
6.
f ( x) =
x2 2
x2
7.
f ( x) =
x2
x2 2
8.
f ( x)
Derivative Practice III
Find the derivative of each of the following functions.
1.
y = x2 2x + 2
()
2. y = arcsin x 2
3. y = 10 5 x
4. y = [arccos( x )]
3
5. y = arctan(e x )
6.
f (x ) =
4 x2 x
3
x
7. g ( x ) = 5 x + 3x 7
8.
f ( x ) = arctan( 5 x )
9. 2 y
MAT 270 - Derivative Practice I Solutions
1. f ( x) = 4 x 3 3x 2 + 2 x
f ' ( x) = 12 x 2 6 x + 2
2.
x2 3
3 x2
f ( x) =
f ' ( x) =
2
6
x+ 3
3
x
(
)
3. f ( x) = 3 2 x 2 5 x + 1
f ' ( x) = 12 x + 15
4.
f ( x) = x
f ' ( x) =
1
2x
+
1
x
1
2x x
x +1
x2
3
f '
Derivative Practice III
Find the derivative of each of the following functions.
1.
y = x2 2x + 2
(
y = 2 x 2 x + x 2 ln 2
)
()
2. y = arcsin x 2
y =
2x
1 x4
3. y = 10 5 x
y =
(
1 (5 x )
10
2
1
2
) (ln 10)( 1)
4. y = [arccos( x )]
3
y =
3(arccos x )
2
1 x
MAT 270 - Derivative Practice II
Find the derivative of the following functions.
(
)
1.
f (x ) = 3x 2 4
2.
f (x ) = 3x 2 2 3 x
3.
f ( x ) = e 2 x 1 (3x + 4)
5
()
3
2
ex
4. g (x ) =
(2 x 1)3
(
)(
5. g ( x ) = e 2 x + x + 3 x 2 2 x + x
(2 3x )
f (x ) =
25
6
MAT 270 - Derivative Practice II
Find the derivative of the following functions.
1.
(
f (x ) = 3x 2 4
(
f ( x ) = 5 3 x 2 4
2.
)
5
) (6 x ) = 30 x(3x
4
2
4
)
4
()
f (x ) = 3x 2 2 3 x
()
()
f ( x ) = 6 x 2 3 x + 9 x 2 2 3 x (ln 2)
3.
f ( x ) = e 2 x 1 (3x
EVEN MORE RELATED RATES WORKSHEET
1)
(a)
10
3
x
y
(b) We are given that
(c) We need to find
dx
= 2 ft sec .
dt
d ( x + y)
when x = 10.
dt
10 x + y
3
=
10 y = 3x + 3 y y = x .
3
y
7
d
(x + y ) = d x + 3 x = dx + 3 dx = 2 + 3 2 = 14 + 6 = 20 ft sec
(e) So,
Carlson, Spring 2003
Name_
MAT 270 Exam 3
Show work on all questions. If you fail to show work, you will not receive credit.
Good Luck!
1. The position of a particle is given by the function s = f (t ) = t 6t + 9t where t is measured in
seconds and s in m
MAT 270
Exam 3
SAMPLE
NOTE: This is only a sample to give an idea of form and length it
is not meant to indicate exact content of your exam. The actual
material covered on each exam varies from semester to semester.
MAT 270 Exam 3
NO CALCULATORS.
SAMPLE
1