Sketch the curve in polar coordinates.

$$\theta = - \frac { 3 \pi } { 4 }$$

Evaluate the integral.

$$\int \frac { d x } { \left( 3 + x ^ { 2 } \right) ^ { 3 / 2 } }$$

Write each of the following polynomials over $\mathbb{Q}$ as a product of factors that are irreducible over $\mathbb{Q}.$ $(a) x^{3}-x^{2}-5 x+5$ $(b) 3 x^{3}-2 x^{2}+3 x-2$ $(c) x^{3}-2 x^{2}+2 x$

Evaluate the double integral

$$\int_R \int f(x,y) \,dA$$

for the function f(x,y) and the region R. f(x,y) = xy; R is the bounded by the lines x=1, y=0 and y=x.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I multiplied an

$$m \times n$$

matrix and an

$$n \times p$$

matrix by multiplying corresponding elements.

So far this baseball season, Philip has gotten a hit 8 times out of 40 at-bats. He wants to increase his batting average to 0.333. Calculate the number of consecutive hits, h, he would need in order to achieve this goal. Round your answer to the nearest whole number.

Evaluate each function at the given values of the independent variable and simplify.

$$h ( x ) = x ^ { 4 } - x ^ { 2 } + 1$$

$$h ( - x )$$

Find the relation between the efficiency of a reversible ideal heat engine and the coefficient of performance of the reversible refrigerator obtained by running the engine backwards.

If the statement is true for all sets A and B, write "true." If it is not true for all sets A and B, write "false." Assume that A≠∅, U≠∅, and A⊂U. U⊂∅

If A is an

$$n \times n$$

matrix, then the matrices B and C defined by

$$B = \frac { 1 } { 2 } \left( A + A ^ { T } \right) , \quad C = \frac { 1 } { 2 } \left( A - A ^ { T } \right)$$

are referred to as the symmetric and skew-symmetric parts of A respectively. Use the properties of the transpose to show that B and C are symmetric and skew-symmetric, respectively.

Describe a real-world object that shows the Hinge Theorem or its converse.

Let v and w denote two nonzero vectors. Show that the vectors ||w|| v + ||v||w and ||w||v - ||v||w are orthogonal.

Find the exact values of the cosine and sine of each angle. Then find the decimal values. Round your answers to the nearest hundredth. 720°

Determine whether the statement is true or false. Explain your answer. If f'(2)=5, then

$$\frac { d } { d x } \left. \left[ 4 f ( x ) + x ^ { 3 } \right] \right| _ { x = 2 } = \frac { d } { d x } \left. [ 4 f ( x ) + 8 ] \right| _ { x = 2 } = 4 f ^ { \prime } ( 2 ) = 20$$