Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
Sample Final Exam 1
MATH 1110 CALCULUS I FOR ENGINEERS
Section I: Multiple Choice Problems [20% of Total final Mark, distributed
equally]
No partial credit on this part, but show all your work anyway. It might help you if you
come close to a border.
Circle only ONE (the correct answer) for each problem.
1- If ln a = 7 and ln b = 4 , then ln(
A- 20
ea 5
b
) is:
B- -24
2- Given the graph of f ( x) =
C- 32
D- 34
x2
, locate the absolute extrema (if they exist) on
x2 −1
the interval ( −1,1) .
A) absolute max: ( 0, 0 )
B)
absolute min: ( −1, −5 ) and (1, −5 )
C) absolute min: ( 0,1)
D)
no absolute extrema
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Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
3- A rectangle has its two bottom corners lying on the x-axis and the two top corners
on the parabola y = 48 − x 2 , where y ≥ 0 . If the area of the rectangle is to be
maximized, then what are its dimensions?
A)
7 × 33
B)
8 × 32
C)
10 × 35
D)
4 × 32
4- If the concentration of a chemical changes according to the equation
x '(t ) = 2.5 x(t ) [ 6 − x(t )] , find the concentration x(t ) for which the reation rate
is a maximum. Round answer to nearest hundredth.
A)
x (t ) = 2.55
B)
x (t ) = 2.68
C)
x (t ) = 2.78
D)
x (t ) = 3.00
5- Find the general antiderivative of 10 x 3 + x 6
A)
5 4 1 7
x + x +c
2
7
B)
10 4 1 7
x + x +c
3
6
C)
5 3 1 6
x + x +c
2
7
D)
10 x 6 + x 7 + c
Ans: A
Difficulty: Easy
6- .Find the position function s (t ) from the given velocity function and initial value.
Assume that units are feet and seconds v (t ) = 7 – 3sin t , s (0) = –5
A)
s (t ) = 7t + 3cos t – 8
B)
s (t ) = 7t – 8sin t
C)
s (t ) = t – 3cos t – 3
D)
s (t ) = –7t + 3cos t
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Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
7- Identify the graph and the area bounded by the curves
y = ( x −1)
2
and
y = 1− x2
A)
area = 5/3
B)
area = 1/6
C)
area = 1/3
D)
area = 1/3
8- A ball is thrown at an angle of π / 2 with an initial speed of 20 meters/second.
What is its time of flight? [Assume the ball is thrown from ground level and lands at
ground level.]
A) 0.20 seconds
B) 2.50 seconds
C) 1.25 seconds
D) 4.08 seconds
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Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
Section II: Multiple-Step problems [60% of Total Final Mark distributed equally]
Show all work (no work – no credit). Write clearly.
1- The velocity graph of a car moving along a straight line is shown in Fig 1. Use this graph
to find an estimation of the distance traveled by the car in the 30 seconds elapsed from
time
t=0
m/sec
1.5
1
0.5
0
5
10
15
20
Figure 1
25
30
sec
3- Sketch the graph of the function f ( x) = 2 x 3 − 15 x 2 + 24 x , and show all local
extrema and inflection points
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Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
4- Find the derivatives
dy
of the following (DON’T SIMPLIFY):
dx
y ( x) = sin 2 (2 x 3 ) + cos 2 (2 x 3 )
1
b) y sin( ) = 3 + xy , using implicit differentiation
y
a)
5- Determine the real value(s) of x for which the line tangent to f ( x) = 4 x 2 – x – 6 is
horizontal.
6- a) Use Integration by parts to evaluate the following integrals. Show all your steps
to receive credit.
π
i-
∫ x sin x dx
ii-
0
b) Find
∫ x ln x dx
(3x 2 + 2)
∫ cos 2 ( x 3 + 2 x + 3) dx
7- Sand is being dumped from a truck at a rate of 4 cubic meters per minute. The
resulting sand pile is conical, and the height is one third times the diameter of the
base. How fast is the height increasing when the pile is 2 meters high?
[The volume of a cone is V =
1
π r 2 h where r is the radius of the base and h is the
3
height.]
8- Compute the volume of a solid formed by revolving the region bounded by
y = 2 − x 2 and y = − 3, about y = 6.
5/8
Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
Section III: Multiple-Step problems [20% of Total Final Mark distributed equally]
1- Find the 100th derivative of f ( x) = x 92 + 5 sin x
2- For each of the graphs (a)-(e) below, determine which of the graphs (I)-(X) is the
graph of its derivative. Write your answer in the space provided.
6/8
Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
3- Find an antiderivative by reversing the chain rule, product rule or quotient rule.
∫(x
7
)
sin 8 x + x8 cos8 x dx
A)
x8 cos8 x
B)
x8 sin 7 x
C)
x7
cos 7 x
8
D)
x8
sin 8 x
8
4- Write the expression as a single integral:
6
A)
∫–5
C)
− ∫ 2 f ( x) dx
2 f ( x) dx
5- Use the graph to determine whether
5
6
∫1
f ( x) dx
∫–5 2 f ( x) dx
6
∫–5 f ( x) dx
D)
1
∫–5
f ( x) dx +
1
B)
6
1
3
∫0 f ( x) dx is positive or negative.
y
4
3
2
1
-5 -4 -3
-2 -1 0
-1
x
1
2
3
4
5
-2
-3
-4
-5
6- .Make the indicated substitution for an unspecified function f ( x) .
u = x5 for
6
∫–3 x
4
f ( x5 ) dx
A)
1 6
f (u ) du
5 ∫–3
B)
∫–3 f ( x) du
C)
1 7776
f (u ) du
5 ∫–243
D)
1 7776 4
x f (u ) du
4 ∫–243
7/8
6
Dept. of Math. Sciences, UAEU
Sample Final Exam 1 Fall 2006
MATH 1110 Calculus I for Engineers
BONUS (OPTIONAL 3 % of the Total Mark)
SOLVE ONLY ONE OF THE FOLLOWING
OR
Suppose that a car engine exerts a force of 800 x(1 − x) pounds when the car
is at position x miles, 0 ≤ x ≤ 1 . Compute the work done.
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