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1
MCQ
Wrong
\(\int x \sqrt{4 - x^2} d x =\)
A
\(\dfrac{(4 - x^2)^{\dfrac{3}{2}}}{3} + C\)
B
\(-(4 - x^2)^{\dfrac{3}{2}} + C\)
C
\(\dfrac{x^2 (4 - x^2)^{\dfrac{3}{2}}}{3} + C\)
D
\(-\dfrac{x^2 (4 - x^2)^{\dfrac{3}{2}}}{3} + C\)
\(-\dfrac{(4 - x^2)^{\dfrac{3}{2}}}{3} + C\)
Correct Answer
Explanation
\(u = 4 - x^2\), \(du = -2x dx\). \(-\left(\dfrac{1}{2}\right) \int u^{\dfrac{1}{2}} du = -u^{\dfrac{3}{2}}/3\).
2
Series - Proof
Wrong
If \(\sum a_n\) is convergent and \(\sum b_n\) is divergent, show that the series \(\sum (a_n + b_n)\) is divergent. [Hint: Argue by contradiction.]
My Answer
(No answer)
Correct Answer
No explanation
3
MCQ
Wrong
If \(3 x^2 + 2 x y + y^2 = 1\), then \(\dfrac{d y}{d x} =\)
A
\(-\dfrac{3 x + y}{y^2}\)
\(-\dfrac{3 x + y}{x + y}\)
Correct Answer
C
\(\dfrac{1 - 3 x - y}{x + y}\)
D
\(-\dfrac{3 x}{1 + y}\)
E
\(-\dfrac{3 x}{x + y}\)
No explanation
4
MCQ
Wrong
\(\displaystyle\int_{0}^{1} x^3 e^{x^4} dx =\)
\(\dfrac{1}{4}(e - 1)\)
Correct Answer
B
\(\dfrac{1}{4} e\)
C
\(e - 1\)
D
\(e\)
E
\(4(e - 1)\)
Explanation
\(u = x^4\), \(du = 4 x^3 dx\). \(\left(\dfrac{1}{4}\right) \displaystyle\int_{0}^{1} e^u du = (e - 1)/4\).
5
Vectors - Orthogonality
Wrong
Find a unit vector that is orthogonal to both \(\mathbf{i} + \mathbf{j}\) and \(\mathbf{i} + \mathbf{k}\).
My Answer
(No answer)
Correct Answer
No explanation
6
Regions in R^3
Wrong
Describe in words the region of \(RR^3\) represented by the equation or inequality.
\(z \geq -1\)
My Answer
(No answer)
Correct Answer
The half-space consisting of all points on or above the plane \(z = -1\).
No explanation
7
Polar Conics - Equation Writing
Wrong
Write a polar equation of a conic with the focus at the origin and the given data.
Ellipse, eccentricity \(\dfrac{1}{2}\), directrix \(x = 4\)
My Answer
(No answer)
Correct Answer
No explanation
8
Differential Equations
Wrong
Solve the initial value problem: \(y \dfrac{d y}{d x} = x^2\), \(y(0) = 1\)
My Answer
(No answer)
Correct Answer
\(y = \sqrt{\dfrac{2x^3}{3} + 1}\) or \(y^2 = \dfrac{2x^3}{3} + 1\)
Explanation
Separate: \(y d y = x^2 d x\). Integrate: \(\dfrac{y^2}{2} = \dfrac{x^3}{3} + C\). Apply \(y(0) = 1\) to find \(C = \dfrac{1}{2}\).
9
Lines and Planes - Plane Equations
Wrong
Find an equation of the plane through the points \((2, 1, 2)\), \((3, -8, 6)\), and \((-2, -3, 1)\).
My Answer
(No answer)
Correct Answer
No explanation
10
Vectors - Projections
Wrong
Find the scalar and vector projections of \(\mathbf{b}\) onto \(\mathbf{a}\).
\(\mathbf{a} = 3 \mathbf{i} - 3 \mathbf{j} + \mathbf{k}\), \(\mathbf{b} = 2 \mathbf{i} + 4 \mathbf{j} - \mathbf{k}\)
My Answer
(No answer)
Correct Answer
No explanation