Explain whether the given equation defines an exponential function. Give reasons for your answers. Write the base for each exponential function. $y = x ^ { x }$

Prove that a conjugate of a glide reflection in M is a glide reflection, and that the glide vectors have the same length.

Rank the following compounds in order of increasing strength of intermolecular forces:

$$\begin{array} { l } { \text { a. } \mathrm { CH } _ { 3 } \mathrm { NH } _ { 2 } , \mathrm { CH } _ { 3 } \mathrm { CH } _ { 3 } , \mathrm { CH } _ { 3 } \mathrm { Cl } } \\ { \mathrm { b } . \mathrm { CH } _ { 3 } \mathrm { Br } , \mathrm { CH } _ { 3 } \mathrm { I } , \mathrm { CH } _ { 3 } \mathrm { Cl } } \\ \mathrm { c } . \left( \mathrm { CH } _ { 3 } \right) _ { 2 } \mathrm { C } = \mathrm { C } \left( \mathrm { CH } _ { 3 } \right) _ { 2 },\left( \mathrm { CH } _ { 3 } \right) _ { 2 } \mathrm { CHCOOH } , \left( \mathrm { CH } _ { 3 } \right) _ { 2 } \mathrm { CHCOCH } _ { 3 } \\ \mathrm { d } . \mathrm { NaCl } , \mathrm { CH } _ { 3 } \mathrm { OH } , \mathrm { CH } _ { 3 } \mathrm { Cl } \end{array}$$

Determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. The set of all real-valued functions f defined everywhere on the real line and such that f(1) = 0.

Sketch the region bounded between the given curves and then find the area of the region.

$$y=\sin x, y=\sin 2 x, x=0, x=\pi$$

Find each of the following limits: a) $\lim _ { z \rightarrow 0 } e ^ { z }$ b) $\lim _ { z \rightarrow 2 \pi i } \left( e ^ { z } - e ^ { - z } \right)$ c) $\lim _ { z \rightarrow \pi i / 2 } ( z + 1 ) e ^ { z }$ d) $\lim _ { z \rightarrow - \pi i } \exp \left( \frac { z ^ { 2 } + \pi ^ { 2 } } { z + \pi i } \right)$

Draw a voltage-amplifier model. Is the gain parameter measured under open-circuit or short-circuit conditions? Repeat for a current amplifier model, a transresistance-amplifier model, and a transconductance-amplifier model.

What effect would an increase in the cost of productive resources have on the aggregate supply curve and on the equilibrium prices of products?

Find the exact value of the expression. $\sin \left(\frac{3 \pi}{4}+\frac{2 \pi}{3}\right)$

find (if possible) the complement and the supplement of each angle. 11π/12

Prove that if $A$ and $B$ are stochastic matrices, then $AB$ is a stochastic matrix.

Consider the vectors

$$\mathbf { v } _ { 1 } = ( 2 , - 1,5 ) , \mathbf { v } _ { 2 } = ( 1,3 , - 4 ), \mathbf { v } _ { 3 } = ( - 3 , - 9,12 ) \text { in } \mathbb { R } ^ { 3 }.$$

(a) Show that

$$\left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$$

is linearly dependent. (b) Is

$$\mathbf { v } _ { 1 } \in \operatorname { span } \left\{ \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\} ?$$

Draw a picture illustrating your answer.

A magazine line plot shows results of a survey. Explain how to use the line plot to find each of the following: the number of people who answered the survey.

Find the area of the following regions. In each case, graph the curve(s) and shade the region in question. The area that is inside the cardioid r=1+cos

$$\theta$$

and outside the cardioid r=1-cos

$$\theta$$