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lim
x
3
x
x
x
lim
x
3 x
5 x
x
f '( x )
f ( x ) 10 x
5
8 x
3
5 x 3
f
x
x
2
3
x
5
f ( x ) 2 sin x e
3
x
f ( x ) cos 3 x
4
f ( x ) ln
x
2
x
4
MAT 124 Name ______________________________
Calculus SS # ______________________________
Final Exam Instructor ________________________
Spring 1999
It is to your advantage to show all work neatly in the space provided. Partial credit will only be
awarded for substantive progress on a problem.
In problems 1-3, evaluate the limit if possible.
In problems 3-7, find.
8
0
f ( x ) dx 3
4
0
f ( x ) dx 12
4
1
f ( x ) dx 7
8
1
3 f ( x ) dx
d
dx
x 1
( x 1 )
2
x
2
xy y
2
3 y 10
y f ( x ) x 2
f ( 2 )
f ( 2 )
- Suppose , and. Find.
- Use the definition of the derivative to show that.
(Short cuts result in zero points.)
- Find the equation of the line tangent to the curve described by
at the point. (Hint: Use implicit differentiation.)
- The table below shows values of near.
x 1.000 1.750 1.900 1.995 2.000 2.002 2.
f ( x ) 1.500 6.226 7.800 8.937 9.000 9.025 13.
a. Estimate as accurately as possible. Is your estimate an over- or under-
estimate? Support your answer.
b. Estimate as accurately as possible.
x
x
dx
e
t
5 cos 2 t dt
2
1
y
3
y
dy
15
0
f ( x ) dx
In Problems 14-16, evaluate the indefinite or definite integrals.
- The table contains values of a continuous function f.
x 0 3 4 8 9 10 15
f ( x ) 1 7 0 -3 -2 8 5
Estimate using a left sum.
x
y
w ( x ) > 0 on ( , 4 ) ( 4 , )
w ( x ) < 0 on ( 4 , 4 )
w ( x ) > 0 on ( , 6 ) ( 3 , )
w
x
on
lim
x !"#
w ( x ) $ 0
- Sketch the graph of a twice differentiable function y = w(x) that satisfies all the stated
conditions:
(i) w (-4) = 6, w (-2) = 0, w (4) = -
(ii)
(iii)
(iv)
