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1
MCQ
Wrong
\(f\) 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
My Answer
II and III only
Correct Answer
Explanation
I: not necessarily (corner). II: yes (Fermat). III: yes (concavity at max).
2
Series - Graphing
Wrong
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.
\(\displaystyle\sum_{n=1}^{\infty} \dfrac{12}{(-5)^n}\)
My Answer
(No answer)
Correct Answer
No explanation
3
MCQ
Wrong
\(\sum (x+2)^n/\sqrt{n}\) converges for
A
\(-3 < x < -1\)
\(-3 \leq x < -1\)
Correct Answer
C
\(-3 \leq x \leq -1\)
D
\(-1 \leq x < 1\)
E
\(-1 \leq x \leq 1\)
Explanation
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
Wrong
Write a polar equation of a conic with the focus at the origin and the given data.
Ellipse, eccentricity \(0.6\), directrix \(r = 4 \csc \theta\)
My Answer
(No answer)
Correct Answer
No explanation
5
Parametric Equations - Eliminate Parameter
Wrong
(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.
\(x = e^t\), \(y = e^{-2t}\)
My Answer
(No answer)
Correct Answer
No explanation
6
Vectors - Applications
Wrong
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.
My Answer
(No answer)
Correct Answer
No explanation
7
MCQ
Wrong
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
\(2 x + 3 y = 12\)
\(3 x + 2 y = 13\)
Correct Answer
C
\(6 x + y = 20\)
D
\(3 x - 2 y = 5\)
E
None of the above.
No explanation
8
FRQ
Wrong
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 \(f(1) = 4\).
My Answer
(No answer)
Correct Answer
No explanation
9
Logistic Growth - Fish Population
Wrong
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?
My Answer
(No answer)
Correct Answer
No explanation
10
MCQ
Wrong
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.
Correct Answer
No explanation
11
MCQ
Wrong
The series \(\displaystyle\sum_{n=0}^{\infty} n! (x - 3)^n\) converges if and only if
A
\(x = 0\)
B
\(2 < x < 4\)
\(x = 3\)
Correct Answer
D
\(2 \leq x \leq 4\)
E
\(x < 2\) or \(x > 4\)
No explanation
12
Spheres - Midpoint and Medians
Wrong
(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.)
My Answer
(No answer)
Correct Answer
No explanation
13
Integral Test - Concepts
Wrong
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?
My Answer
(No answer)
Correct Answer
No explanation
14
MCQ
Wrong
Evaluate \(\displaystyle\int_{0}^{6} \sqrt{6 x - x^2} d x\)
A
\(\pi\)
B
\(2 \pi\)
C
\(\dfrac{5 \pi}{2}\)
\(\dfrac{9 \pi}{2}\)
Correct Answer
E
\(3 \pi\)
No explanation
15
Polar Curves - Intersection Points
Wrong
Find all points of intersection of the given curves.
\(r = \sin \theta\), \(r = 1 - \sin \theta\)
My Answer
(No answer)
Correct Answer
No explanation
16
Polar Curves - Horizontal/Vertical Tangents
Wrong
Find the points on the given curve where the tangent line is horizontal or vertical.
\(r = e^\theta\)
My Answer
(No answer)
Correct Answer
No explanation
17
Cylinders and Quadric Surfaces - Reduce and Classify
Wrong
Reduce the equation to one of the standard forms, classify the surface, and sketch it.
\( 4x^2 - y + 2z^2 = 0 \)
My Answer
(No answer)
Correct Answer
Elliptic paraboloid
No explanation
18
Polar Curves - Sketching
Wrong
Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates.
\(r = 2(1 + \cos \theta)\)
My Answer
(No answer)
Correct Answer
No explanation
19
Integration
Wrong
Evaluate the integral: \(\int x^3 \sqrt{x^2 + 1} d x\)
My Answer
(No answer)
Correct Answer
\(\dfrac{1}{5}(x^2 + 1)^{\dfrac{5}{2}} - \dfrac{1}{3}(x^2 + 1)^{\dfrac{3}{2}} + C\)
Explanation
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
Wrong
Show that the equation represents a sphere, and find its center and radius.
\(3x^2 + 3y^2 + 3z^2 = 10 + 6y + 12z\)
My Answer
(No answer)
Correct Answer
Center \((0, 1, 2)\), radius \(\sqrt{\dfrac{25}{3}} = \dfrac{5 \sqrt{3}}{3}\).
Explanation
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
Wrong
Determine whether the series is convergent or divergent.
\(\displaystyle\sum_{k=1}^{\infty} k e^{-k^2}\)
My Answer
(No answer)
Correct Answer
Convergent
No explanation
22
Polar Curves - Lemniscate
Wrong
Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates.
\(r^2 = 9 \sin 2 \theta\)
My Answer
(No answer)
Correct Answer
No explanation
23
Integration
Wrong
Evaluate the definite integral: \(\displaystyle\int_{0}^{2 \dfrac{\pi}{3}} \dfrac{3}{5 + 4 \cos \theta} d \theta\)
My Answer
(No answer)
Correct Answer
\(2 \tan^{-1}(3) - \dfrac{\pi}{2}\) or approximately \(0.964\)
Explanation
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
Wrong
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
\(\displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} \dfrac{n}{n^2 + 4}\)
My Answer
(No answer)
Correct Answer
Conditionally convergent
No explanation
25
Comparison Test
Wrong
Determine whether the series converges or diverges.
\(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{1 + \dfrac{1}{n}}}\)
My Answer
(No answer)
Correct Answer
Divergent
No explanation
26
Polar Coordinates - Polar Equation
Wrong
Find a polar equation for the curve represented by the given Cartesian equation.
\(y = 1 + 3x\)
My Answer
(No answer)
Correct Answer
No explanation
27
MCQ
Wrong
\(\int \dfrac{d x}{(x - 1)(x + 2)} =\)
| \(\dfrac{1}{3} \ln | \dfrac{x-1}{x+2} | + C\) |
|---|
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
| \(\ln | (x-1)(x+2)^2 | + C\) |
|---|
Explanation
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
Wrong
Find \(\mathbf{a} \cdot \mathbf{b}\).
\(\mathbf{a} = \langle 1.5, 0.4 \rangle\), \(\mathbf{b} = \langle -4, 6 \rangle\)
My Answer
(No answer)
Correct Answer
-3.6
No explanation
29
MCQ
Wrong
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
\(f'(c)(b-a)\)
\(f(c)(b-a)\)
Correct Answer
Explanation
Mean value theorem for integrals.
30
Vectors - Calculus Connection
Wrong
(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 \(\left(\dfrac{\pi}{6}, 1\right)\).
My Answer
(No answer)
Correct Answer
No explanation