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1
MCQ
오답
\(f\) continuous on \([a,b]\) has rel max at \(c\), \(a < c < b\). Which true? I. \(f'(c)\) exists II. If \(f'(c)\) exists, \(f'(c)=0\) III. If \(f''(c)\) exists, \(f''(c) \leq 0\)
A
II only
B
III only
C
I and II only
D
I and III only
내 답
II and III only
정답
해설
I: not necessarily (corner). II: yes (Fermat). III: yes (concavity at max).
2
Series - Graphing
오답
Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
\(\displaystyle\sum_{n=1}^{\infty} \dfrac{12}{(-5)^n}\)
내 답안
(미작성)
정답
해설 없음
3
MCQ
오답
\(\sum (x+2)^n/\sqrt{n}\) converges for
A
\(-3 < x < -1\)
\(-3 \leq x < -1\)
정답
C
\(-3 \leq x \leq -1\)
D
\(-1 \leq x < 1\)
E
\(-1 \leq x \leq 1\)
해설
Center \(-2\), radius 1. At \(x = -3\): \(\sum (-1)^n/\sqrt{n}\), alternating, converges. At \(x = -1\): \(\sum 1/\sqrt{n}\), p=1/2, diverges. So \([-3, -1)\).
4
Polar Conics - Equation Writing
오답
Write a polar equation of a conic with the focus at the origin and the given data.
Ellipse, eccentricity \(0.6\), directrix \(r = 4 \csc \theta\)
내 답안
(미작성)
정답
해설 없음
5
Parametric Equations - Eliminate Parameter
오답
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
\(x = e^t\), \(y = e^{-2t}\)
내 답안
(미작성)
정답
해설 없음
6
Vectors - Applications
오답
A quarterback throws a football with angle of elevation \(40^{\circ}\) and speed 60 ft/s. Find the horizontal and vertical components of the velocity vector.
내 답안
(미작성)
정답
해설 없음
7
MCQ
오답
The equation of the tangent line to the curve with parametric equations \(x(t) = 2 t + 1\), \(y(t) = 3 - t^3\) at \(t = 1\) is:
A
\(2 x + 3 y = 12\)
\(3 x + 2 y = 13\)
정답
C
\(6 x + y = 20\)
D
\(3 x - 2 y = 5\)
E
None of the above.
해설 없음
8
FRQ
오답
Let \(y = f(x)\) be the particular solution to the differential equation \(\dfrac{d y}{d x} = y \cdot (x \ln x)\) with initial condition \(f(1) = 4\). It can be shown that \(f''(1) = 4\).
(a) Write the second-degree Taylor polynomial for \(f\) about \(x = 1\). Use the Taylor polynomial to approximate \(f(2)\).
(b) Use Euler's method, starting at \(x = 1\) with two steps of equal size, to approximate \(f(2)\). Show the work that leads to your answer.
(c) Find the particular solution \(y = f(x)\) to the differential equation \(\dfrac{d y}{d x} = y \cdot (x \ln x)\) with initial condition \(f(1) = 4\).
내 답안
(미작성)
정답
해설 없음
9
Logistic Growth - Fish Population
오답
Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,000. The number of fish tripled in the first year.
(a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after \(t\) years.
(b) How long will it take for the population to increase to 5000?
내 답안
(미작성)
정답
해설 없음
10
MCQ
오답
Identify the false statement.
A
\(\dfrac{d \sinh(x)}{d x} = \cosh(x)\)
B
\(\dfrac{d \cosh(x)}{d x} = \sinh(x)\)
C
\(\displaystyle\int_{a}^{t} sech^2(x) d x = \tanh(t) - \tanh(a)\)
D
\(\cosh^2(x) - \sinh^2(x) = 1\)
All four statements are true.
정답
해설 없음
11
MCQ
오답
The series \(\displaystyle\sum_{n=0}^{\infty} n! (x - 3)^n\) converges if and only if
A
\(x = 0\)
B
\(2 < x < 4\)
\(x = 3\)
정답
D
\(2 \leq x \leq 4\)
E
\(x < 2\) or \(x > 4\)
해설 없음
12
Spheres - Midpoint and Medians
오답
(a) Prove that the midpoint of the line segment from \(P_1(x_1, y_1, z_1)\) to \(P_2(x_2, y_2, z_2)\) is
\( \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}, \dfrac{z_1 + z_2}{2}\right) \)
(b) Find the lengths of the medians of the triangle with vertices \(A(1, 2, 3)\), \(B(-2, 0, 5)\), and \(C(4, 1, 5)\). (A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.)
내 답안
(미작성)
정답
해설 없음
13
Integral Test - Concepts
오답
Draw a picture to show that
\(\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n^{1.3}} < \displaystyle\int_{1}^{\infty} \dfrac{1}{x^{1.3}} d x\)
What can you conclude about the series?
내 답안
(미작성)
정답
해설 없음
14
MCQ
오답
Evaluate \(\displaystyle\int_{0}^{6} \sqrt{6 x - x^2} d x\)
A
\(\pi\)
B
\(2 \pi\)
C
\(\dfrac{5 \pi}{2}\)
\(\dfrac{9 \pi}{2}\)
정답
E
\(3 \pi\)
해설 없음
15
Polar Curves - Intersection Points
오답
Find all points of intersection of the given curves.
\(r = \sin \theta\), \(r = 1 - \sin \theta\)
내 답안
(미작성)
정답
해설 없음
16
Polar Curves - Horizontal/Vertical Tangents
오답
Find the points on the given curve where the tangent line is horizontal or vertical.
\(r = e^\theta\)
내 답안
(미작성)
정답
해설 없음
17
Cylinders and Quadric Surfaces - Reduce and Classify
오답
Reduce the equation to one of the standard forms, classify the surface, and sketch it.
\( 4x^2 - y + 2z^2 = 0 \)
내 답안
(미작성)
정답
Elliptic paraboloid
해설 없음
18
Polar Curves - Sketching
오답
Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates.
\(r = 2(1 + \cos \theta)\)
내 답안
(미작성)
정답
해설 없음
19
Integration
오답
Evaluate the integral: \(\int x^3 \sqrt{x^2 + 1} d x\)
내 답안
(미작성)
정답
\(\dfrac{1}{5}(x^2 + 1)^{\dfrac{5}{2}} - \dfrac{1}{3}(x^2 + 1)^{\dfrac{3}{2}} + C\)
해설
Let \(u = x^2 + 1\), then \(x^2 = u - 1\). Rewrite as \(\dfrac{1}{2} \int (u - 1) \sqrt{u} d u\) and integrate.
20
Spheres - Completing the Square
오답
Show that the equation represents a sphere, and find its center and radius.
\(3x^2 + 3y^2 + 3z^2 = 10 + 6y + 12z\)
내 답안
(미작성)
정답
Center \((0, 1, 2)\), radius \(\sqrt{\dfrac{25}{3}} = \dfrac{5 \sqrt{3}}{3}\).
해설
Divide by 3: \(x^2 + y^2 + z^2 - 2y - 4z = \dfrac{10}{3}\). Complete squares: \(x^2 + (y - 1)^2 + (z - 2)^2 = \dfrac{10}{3} + 1 + 4 = \dfrac{25}{3}\).
21
Series - Convergence
오답
Determine whether the series is convergent or divergent.
\(\displaystyle\sum_{k=1}^{\infty} k e^{-k^2}\)
내 답안
(미작성)
정답
Convergent
해설 없음
22
Polar Curves - Lemniscate
오답
Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates.
\(r^2 = 9 \sin 2 \theta\)
내 답안
(미작성)
정답
해설 없음
23
Integration
오답
Evaluate the definite integral: \(\displaystyle\int_{0}^{2 \dfrac{\pi}{3}} \dfrac{3}{5 + 4 \cos \theta} d \theta\)
내 답안
(미작성)
정답
\(2 \tan^{-1}(3) - \dfrac{\pi}{2}\) or approximately \(0.964\)
해설
Use \(t = \tan\left(\dfrac{\theta}{2}\right)\). Then \(\cos \theta = \dfrac{1-t^2}{1+t^2}\). The integrand simplifies to a form involving \(\dfrac{1}{9 + t^2}\).
24
Absolute Convergence
오답
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
\(\displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} \dfrac{n}{n^2 + 4}\)
내 답안
(미작성)
정답
Conditionally convergent
해설 없음
25
Comparison Test
오답
Determine whether the series converges or diverges.
\(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{1 + \dfrac{1}{n}}}\)
내 답안
(미작성)
정답
Divergent
해설 없음
26
Polar Coordinates - Polar Equation
오답
Find a polar equation for the curve represented by the given Cartesian equation.
\(y = 1 + 3x\)
내 답안
(미작성)
정답
해설 없음
27
MCQ
오답
\(\int \dfrac{d x}{(x - 1)(x + 2)} =\)
| \(\dfrac{1}{3} \ln | \dfrac{x-1}{x+2} | + C\) |
|---|
B
| \(\dfrac{1}{3} \ln | \dfrac{x+2}{x-1} | + C\) |
|---|
C
| \(\dfrac{1}{3} \ln | (x-1)(x+2) | + C\) |
|---|
D
| \((\ln | x-1 | )(\ln | x+2 | ) + C\) |
|---|
E
| \(\ln | (x-1)(x+2)^2 | + C\) |
|---|
해설
Partial fractions: \(1/((x-1)(x+2)) = \left(\dfrac{1}{3}\right)[1/(x-1) - 1/(x+2)]\). Integral: \(\left(\dfrac{1}{3}\right) \ln|\dfrac{x-1}{x+2}|\).
28
Vectors - Dot Product
오답
Find \(\mathbf{a} \cdot \mathbf{b}\).
\(\mathbf{a} = \langle 1.5, 0.4 \rangle\), \(\mathbf{b} = \langle -4, 6 \rangle\)
내 답안
(미작성)
정답
-3.6
해설 없음
29
MCQ
오답
MVT for integrals: \(\displaystyle\int_{a}^{b} f =\)
A
\(f\dfrac{c}{b-a}\)
B
\(\dfrac{f(b)-f(a)}{b-a}\)
C
\(f(b) - f(a)\)
D
\(f'(c)(b-a)\)
\(f(c)(b-a)\)
정답
해설
Mean value theorem for integrals.
30
Vectors - Calculus Connection
오답
(a) Find the unit vectors that are parallel to the tangent line to the curve \(y = 2 \sin x\) at the point \(\left(\dfrac{\pi}{6}, 1\right)\).
(b) Find the unit vectors that are perpendicular to the tangent line.
(c) Sketch the curve \(y = 2 \sin x\) and the vectors in parts (a) and (b), all starting at \(\left(\dfrac{\pi}{6}, 1\right)\).
내 답안
(미작성)
정답
해설 없음