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MTH 251 – Final Exam – No Calculator
Given March 18, 2009 Name
Please note that this test counts for 125 points
- 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
- 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.
- 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
- 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
- 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 x − f^ x → −
5
lim 2 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
- 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.
- 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
