The set $D_1$ consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set $D_2$ consists of four students: Dale, who is 5 feet 11 inches tall; Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Mary, who is 6 feet tall. The set $D_3$ consists of one student: Dale, who is 6 feet tall. The set $D_4$ consists of three students: Part, Sandy, and Gal, each of whom is 5 feet 11 inches tall. $T_1(x,y)$ is the propositional function "x is taller than y." Tell whether each proposition is true or false if the domain of discourse is $D_i\times D_j$ for the given values of i and j.

1. $\forall x\forall yT_1(x,y)$
2. $\forall x\exists yT_1(x,y)$
3. $\exists x\forall yT_1(x,y)$
4. $\exists x\exists yT_1(x,y)$

i=4, j=1

Evaluate the definite integral, if it exists. $\int_0^{-4} \sqrt{1-2 x} d x$

If

$$f \in \mathcal{R}[a, b]$$

and

$$|f(x)| \le M$$

for all

$$x \in [a,b]$$

, show that

$$\left|\int_a^bf\right|\le M\left(b-a\right).$$

Evaluate the given definite integrals.

$$\displaystyle\int_1^4 x^{5/2}dx$$

Use the properties of logarithms to evaluate the expression.

$$\log _{8} 4+\log _{8} 16$$

Find the sum of the given series, or show that the series diverges (possibly to infinity or negative infinity). $\sum_{n=1}^{\infty} \frac{2}{n+1}$

Evaluate the definite integral $\int_2^4\left(1+x^{-2}\right) d x$ .

Simplify:

$$7 \left( 4 x ^ { 2 } + 3 x \right) + 2 \left( 5 x ^ { 2 } + x \right).$$

Consider the polynomial

$$f=1+x^2+x^3 \in \mathbb{Z}_2(\alpha)[x],$$

where $\mathbb{Z}_2(\alpha)$ is the field considered in the previous exercise. (a) Use the Root Theorem to show that this polynomial is irreducible. (b) By Kronecker's the previous Theorem we can construct an extension field of $\mathbb{Z}_2(\alpha)$ so that $f$ has a root $\beta$. How many elements does this field have? (c) If you're not doing anything this weekend, construct a multiplication table for this field.

Find the sum. .

$$\left(5 x^2-2 x+3\right)+\left(x^2+x+2\right)$$

Geography Students at the University of Minnesota in Minneapolis built a model globe 42 ft in diameter using a scale of 1: 1,000,000. About how tall is Mount Everest on the model? (Mount Everest is about 29,000 ft tall.)

Simplify.

$$- 3 \left( x ^ { 2 } - 2 \right) + 2 \left( 3 - 2 x ^ { 2 } \right)$$

Rewrite the following paragraph, adding, deleting, or changing any end marks and commas to correct the numbered word groups that are incorrectly punctuated.

EXAMPLE: In looking through a United States atlas have you cver been tempted to use it, as a menu to create a meal of places named for foods. In looking through a United States atlas, have you ever been tempted to use it as a menu to create a meal of places named for foods?*

For an appetizer, that will take the edge off your family's hunger, without filling them up why not serve a relish tray assembled from Pickleville in Utah. Olive in Montana, and Pepperton in Georgia, along with Rolls from Arizona and Indiana and Butters from North Carolina

Evaluate the definite integral $\int_0^1 \cos(\pi t/2)~dt$.