Choose the letter that is the best antonym of the vocabulary word prodigious.

a. varied

b. meager

c. quiet

d. permanent

e. kind

f. verified

g. obscurity

h . brief

Right, or wrong? Give a brief reason for your answer.

$$\int \sqrt{2 x+1} d x=\frac{1}{3}(\sqrt{2 x+1})^3+C$$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. The appropriate trial solution for the 5th-order differential equation

$$D ^ { 3 } \left( D ^ { 2 } + 1 \right) y = x ^ { 4 } \text { is } y _ { p } ( x ) =A _ { 0 } + A _ { 1 } x + A _ { 2 } x ^ { 2 } + A _ { 3 } x ^ { 3 } + A _ { 4 } x ^ { 4 }.$$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. If

$$p ( x ) = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } \text { and } q ( x ) = b _ { 0 } + b _ { 1 } x + b _ { 2 } x ^ { 2 },$$

then we can define an inner product on

$$P _ { 2 } ( \mathbb { R } )$$

via

$$\langle p , q \rangle = a _ { 0 } b _ { 0 }.$$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. If 0 is a root of multiplicity four of the auxiliary polynomial of an nth-order constant coefficient homogeneous linear differential equation, then any polynomial of degree three or less is a solution to the differential equation.

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. The appropriate trial solution for the 5th-order differential equation

$$D ^ { 3 } \left( D ^ { 2 } + 1 \right) y = \cos x \text { is } y _ { p } ( x ) =x \left( A _ { 0 } \cos x + B _ { 0 } \sin x \right).$$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. With the inner product

$$\langle f , g \rangle = \int _ { 0 } ^ { \pi } f ( t ) g ( t ) d t,$$

the functions f(x) = cos x and g(x) = sin x are an orthogonal basis for span{cos x, sin x}.

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. Any two differential operators

$$L _ { 1 } ( D ) \text { and } L _ { 2 } ( D )$$

commute; that is,

$$L _ { 1 } ( D ) L _ { 2 } ( D ) = L _ { 2 } ( D ) L _ { 1 } ( D ).$$

Consider the following parametric equations. Make a brief table of values of t, x, and y.

$$x = t ^ { 3 } - 1 , y = 5 t + 1 ; - 3 \leq t \leq 3$$

A 3.46 kg briefcase is sitting at rest on a level floor. What is its weight in newtons?

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. Each pair of complex eigenvalues

$$\lambda = a \pm i b ( b \neq 0 )$$

of a matrix A gives rise to a pair of real-valued solutions to the vector differential equation

$$\mathbf { x } ^ { \prime } = A \mathbf { x }.$$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. A matrix of the form

$$\left[ \begin{array} { l l l } { a } & { b } & { c } \\ { d } & { e } & { f } \\ { g } & { h } & { i } \end{array} \right]$$

such that ae - bd = 0 cannot be invertible.

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. If S is a spanning set for a vector space V, then any proper subset S' of S

$$\left( \mathrm { i } . \mathrm { e } , S ^ { \prime } \neq S \right)$$

not a spanning set for V.

Using information from this section, write a brief newspaper story with a sensational headline in the same style that was used in William Randolph Hearst's newspapers around the turn of the century.