Simplify each difference.

$$\left( - 4 x ^ { 2 } + x - 1 \right) - \left( x ^ { 2 } - x + 8 \right)$$

Identify each series as arithmetic, geometric, or neither. Then rewrite the series using sigma notation. 20+17+14+

$$\cdots$$

+2

Suppose that a function f is differentiable at $x_0$ and that $f^{\prime}\left(x_0\right)>0$. Prove that there exists an open interval containing $x_0$ such that if $x_1$ and $x_2$ are any two points in this interval with $x_1<x_0<x_2$, then $f\left(x_1\right)<f\left(x_0\right)<f\left(x_2\right)$.

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. $-1290^{\circ}$

Use the properties of logarithms to evaluate the expression.

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

Evaluate the definite integral. $\int_{0}^{2} x \sqrt{x^{2}+1} d x$

Simplify.

$$8 \left( 3 x ^ { 5 } \right) - ( 2 x ) ^ { 3 } \left( 5 x ^ { 2 } \right)$$

A finite number of terms are deleted from a divergent series. Determine whether the new series will still diverge. Explain your reasoning.

If

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

and

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

for all

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

, show that

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

A plan view of a field :
a) Using a scale of $1 \mathrm{~cm}$ for every $5 \mathrm{~m}$, construct a scale drawing of the field. b) A farmer divides the field by running a fence from $\mathrm{X}$ in such a way that it bisects angle WXY. By construction, show the position of the fence on your diagram. c) Work out the length of fencing used.

Find exact values of the six trigonometric functions forthe angle. Rationalize denominators when applicable. $1020^{\circ}$

Determine exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable.

$$570^{\circ}$$

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