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12 Questions of Calculus I - Final Examination | MTH 251, Exams of Calculus

Material Type: Exam; Class: Calculus I; Subject: Math; University: Portland Community College; Term: Winter 2009;

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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MTH 251 – Mr. Simonds class – Winter Term, 2009 – Final Exam
Page 1 of 10
MTH 251 – Final Exam – No Calculator
Given March 18, 2009 Name
Please note that this test counts for 125 points
1. Fill in each blank with the
simplified
formula. Do any necessary figuring on your scratch paper;
that is, do not show steps on this problem. (2 points each)
a.
()
x
dxe
dx =
b.
4d
d
θθ
⎛⎞
=
⎜⎟
⎝⎠
c.
()
3
dt
dt =
d.
()
()
2
sin
dx
dx
=
e.
()
3x
de
dx =
f.
()
(
)
2
cos
dw
dw
=
g.
5
1d
dx e
⎛⎞
=
⎜⎟
⎝⎠
h.
1
ln 8
d
dx x
⎛⎞
=
⎜⎟
⎝⎠
i.
()
()
sin
d
d
θθ
θ
=
j.
1
dx
dx x
⎛⎞
=
⎜⎟
+
⎝⎠
k.
()
()
sec
d
d
β
β
=
l.
()
de
dt
π
=
pf3
pf4
pf5
pf8
pf9
pfa

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MTH 251 – Final Exam – No Calculator

Given March 18, 2009 Name

Please note that this test counts for 125 points

  1. Fill in each blank with the simplified formula. Do any necessary figuring on your scratch paper;

that is, do not show steps on this problem. (2 points each)

a. ( )

d (^) x x e dx

= b.

d 4

d θ θ

c. ( )

d (^) 3 t dt

= d. ( ( ))

2 sin

d x dx

e. ( )

d (^) 3 x e dx

= f. ( ( ))

2 cos

d w dw

g. (^5)

d 1

dx e

h.

ln 8

d

dx x

⎣ ⎝^ ⎠⎦

i. ( sin( ))

d

d

j. 1

d x

dx x

k. (^ sec(^ ))

d

d

= l. ( )

d e dt

π

  1. Find the critical numbers of the function (^) ( )

3 3

f x x x

= −. Show all necessary work in a

manner consistent with that illustrated and discussed during lecture.

  1. Figure 1 shows the curve sin (^) ( )

y

x − y = π e. Find the slope of

the tangent line to this curve at the point (^) ( 0,0 (^) ). Make sure

that your work is well organized and that your conclusion is

clear. (18 points)

Figure 1

  1. Figure 2 shows the curve

sin 2 4

2

x

y x

. Find the

equation of the tangent line to this curve at the point (^) ( −2, − (^1) ).

Make sure that your work is well organized and that your

conclusion is clear. (18 points)

Figure 2

  1. Sketch onto Figure 4 a function f that has each of the stated properties. Assume that all

intercepts, points of discontinuity, and points of nondifferentiability are directly implied by the

stated properties.

  • x = − 2 is a vertical asymptote for f

• f ′^ ( x ) < 0 on ( −∞ −, 5 ) ∪ −( 5, − 2 ) ∪ ( 1,3 ) ∪ ( 5,∞)

• f ′^ ( x ) > 0 on ( −2,1 ) ∪( 3,5)

• f ′′(^ x ) < 0 on ( −5, − 2 ) ∪ ( −2,1 ) ∪ ( 1, 3) ∪( 3, 5)

• f ′′(^ x ) > 0 on ( −∞ −, 5 ) ∪ ( 5,∞)

5

lim 3 xf^ x → −

5

lim 2 x

  • f^ x → −

= − , and lim ( ) 0

x

f x → ∞

• f ( − 5 ) = − 2 , f ( 0 )= − 2 , and f ( 4 ) = 0

8. Find the first derivative formula for ( ) ( )

1 2 y x tan x ln x 1

− = − +. Show your work.

Figure 4

  1. The first derivative for the function (^) ( )

( ) 3 2 4

x f x x

is (^) ( )

( )

( )

(^2 )

x f x x x

.

a. List the critical numbers of f ; no explanation nor sentence required.

b. Build an increasing/decreasing table for f. Make sure that you include the details

illustrated and discussed during lecture.

c. State the local minimum points and local maximum points that occur on f.

  1. Several function values for f and f ′^ are given in Table 1. Use these function values to help

you find each of the following. (16 points total)

a. Find the value of (^) ( ( 4 ))

d g dx

if g (^) ( x ) (^) = f (^) ( x ).

b. Find the value of h ′^ ( 4 )if (^) ( ) ( )

2 h x = ⎡⎣ f x ⎤⎦.

c. Find the value of ( 6, 6)

dy

dx

along the curve (^) ( )

2 y = x f x.

Table 1

x f^ ( x^ ) f^ ′( x )

1 0 5

2 4 4

3 7 3

4 9 1

(^5 8) − 3

(^6 6) − 4

7 3 − 5