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1
Separable Equations - Solve
Wrong
Solve the differential equation.
\( \dfrac{d y}{d x} = x \sqrt{y} \)
My Answer
(No answer)
Correct Answer
No explanation
2
Differentiation - Finding Points
Wrong
Find the points on the curve \(y = x^3 + 3x^2 - 9x + 10\) where the tangent is horizontal.
My Answer
(No answer)
Correct Answer
\(y' = 3x^2 + 6x - 9 = 3(x + 3)(x - 1) = 0\) gives \(x = -3\) and \(x = 1\). Points: \((-3, 37)\) and \((1, 5)\).
No explanation
3
MCQ
Wrong
Which of the following equations has a graph that is symmetric with respect to the origin?
A
\(y = \dfrac{x + 1}{x}\)
\(y = -x^5 + 3 x\)
Correct Answer
C
\(y = x^4 - 2 x^2 + 6\)
D
\(y = (x - 1)^3 + 1\)
E
\(y = (x^2 + 1)^2 - 1\)
Explanation
Origin symmetric ⟺ odd function. Only \(-x^5 + 3x\) is odd.
4
Chain Rule - Tangent Lines
Wrong
Find an equation of the tangent line to the curve at the given point.
\(y = \sqrt{1 + x^3}\), \((2, 3)\)
My Answer
(No answer)
Correct Answer
\(y' = \dfrac{3x^2}{2 \sqrt{1 + x^3}}\)
At \((2, 3)\): \(y'(2) = \dfrac{12}{6} = 2\)
Tangent line: \(y - 3 = 2(x - 2)\), or \(y = 2x - 1\)
No explanation
5
Polar Area
Wrong
Find the area of the shaded region.
My Answer
(No answer)
Correct Answer
No explanation
6
MCQ
Wrong
Which of the following gives the area between the curves \(y = x^2\) and \(y = 2 x\) over the interval \([-2, 2]\)?
A
\(\displaystyle\int_{-2}^2 (x^2 - 2 x) d x\)
B
\(\displaystyle\int_{-2}^2 (2 x - x^2) d x\)
\(\displaystyle\int_{-2}^0 (x^2 - 2 x) d x + \displaystyle\int_{0}^{2} (2 x - x^2) d x\)
Correct Answer
D
| \( | \displaystyle\int_{-2}^2 (x^2 - 2 x) d x |
\) |
|---|
E
none of the above
No explanation
7
Text Structure and Purpose
Wrong
The Bayeux Tapestry, from eleventh-century France, depicts 75 scenes over 250 feet of fabric. It was likely produced by workers embroidering in sections and then joining the resulting panels together. It's plausible that the workshop that produced the tapestry had never produced one so large, and some researchers claim that a close examination of the joins—the places where the panels are stitched together—suggests that the workers developed and refined their joining process over the course of production. For example, the first join the workers completed exhibits a clear misalignment of the borders of the two panels, whereas the later joins are virtually invisible.
Which choice best describes the function of the underlined sentence in the text as a whole?
A
It identifies the people and events depicted in the Bayeux Tapestry.
It supports an argument about the workers who produced the Bayeux Tapestry.
Correct Answer
C
It compares the Bayeux Tapestry with other tapestries from eleventh-century France.
D
It describes how researchers determined where the Bayeux Tapestry was produced.
No explanation
8
Related Rates - Basic Geometry
Wrong
The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 80 mm?
My Answer
(No answer)
Correct Answer
\(\dfrac{d V}{d t} = 25600 \pi\) mm³/s
Explanation
\(V = \dfrac{4}{3} \pi r^3\), so \(\dfrac{d V}{d t} = 4 \pi r^2 \dfrac{d r}{d t}\). When diameter \(= 80\), \(r = 40\). Thus \(\dfrac{d V}{d t} = 4 \pi (40)^2 (4) = 25600 \pi\) mm³/s.
9
Vectors - Dot Product
Wrong
Find \(\mathbf{a} \cdot \mathbf{b}\).
\(\mathbf{a} = \langle 4, 1, \dfrac{1}{4} \rangle\), \(\mathbf{b} = \langle 6, -3, -8 \rangle\)
My Answer
(No answer)
Correct Answer
19
No explanation
10
Solution Sets of Linear Systems
Wrong
Suppose \(A\) is the \(3 \times 3\) zero matrix (all entries are zero). Describe the solution set of \(A \mathbf{x} = \mathbf{0}\).
My Answer
(No answer)
Correct Answer
Every vector in R³ is a solution. The solution set is all of R³, since 0·x = 0 for every x.
No explanation