Midterm Review: Example 1
Which one of the definite integrals below equals the limit
lim
n→∞
n
X
k
n2
k=1
1+
A.
Z
Z
1
dx
1 + x2
1
x
dx
1 + x2
1
1
dx
1 + x2
2
x2
dx
1 + x2
0
C.
Z
0
D.
Z
1
Math 105 (Section 204)
?
2
1
B.
k2
n2
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Midterm Review: Example 2
You are trying to create a topographical map of a terrain whose height
function is given by
h(x, y ) = x 2 y 2 ,
2x 2 + y 2 ≤ 1.
Where should the highest and lowest points on your map be?
A. (0, 0), (0, ±1)
B. (0, 0), (± 12 , ± √12 )
C. (0, 0), (± √12 , 0)
D. (0, ±1), (± √12 , 0)
Math 105 (Section 204)
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Midterm Review: Example 3
What is the xy -trace of the surface
z = 4x 2 + y 2 − 2
at height z = 2?
A. A circle centered at (0, 0) of radius 2
B. A parabola with vertex at (0, 2)
C. An ellipse that is longer in the x-direction than in the y -direction
D. An ellipse that is longer in the y -direction than in the x-direction
Math 105 (Section 204)
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Midterm Review: Example 3
What is the xy -trace of the surface
z = 4x 2 + y 2 − 2
at height z = 2?
A. A circle centered at (0, 0) of radius 2
B. A parabola with vertex at (0, 2)
C. An ellipse that is longer in the x-direction than in the y -direction
D. An ellipse that is longer in the y -direction than in the x-direction
This question could also have been phrased as follows: What is the
level curve of the surface z = 4x 2 + y 2 − 2 at z = 2?
Math 105 (Section 204)
Integral Calculus – Definite integrals
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Midterm Review: Example 4
√
Suppose you are standing at the point (1/2, 3, 2) on the surface in the
2
previous example, z = 4x 2 +
√y − 2. As you start walking on the surface
directly above the√
line y = 3, estimate the change in height as you move
to the point (.51, 3).
A. 0.4
B. 0.04
C. 0.08
√
D. 3/5
Math 105 (Section 204)
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Midterm Review: Example 5
Find the left Riemann sum of the function
f (x) =
1
x
on the interval [1, 4], with n = 3.
A. 19/12
B. 11/6
C. 5/2
Math 105 (Section 204)
Integral Calculus – Definite integrals
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Midterm Review: Example 5
Find the left Riemann sum of the function
f (x) =
1
x
on the interval [1, 4], with n = 3.
A. 19/12
B. 11/6
C. 5/2
Is the correct answer above an overestimate or underestimate of the value
of
Z 4
1
dx?
1 x
Math 105 (Section 204)
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Midterm Review: Example 6
Suppose that a function f has continuous partial derivatives of all orders,
and that
fx = sin x cos y .
Then the value of fyxy must be
A. fxxy
B. fyxx
C. fyx
D. −fx
Math 105 (Section 204)
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Midterm Review: Example 7
For some unknown function f , you are given that
Z 1
Z 1
f (x) dx = 4,
(6 − 2f (x)) dx = −24.
0
3
The signed area of f in [0, 3] is
A. 10
B. −2
C. −6
D. 3.
Math 105 (Section 204)
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