The vectors $\mathbf { u }$ and $\mathbf { v}$ are given. Calculate the dot product $\mathbf { u } \cdot \mathbf { v }$. $\mathbf { u } = \langle 4,5 , - 6 \rangle$, $\mathbf { v } = \langle 0 , - 2 , - 3 \rangle$

Insert necessary commas. In the space provided, indicate briefly the reason for the given comma. For example, write independent adjectives, the adverb too, introductory verbal phrase, independent clauses, and so forth. Write C if the sentence is correct.

Example:

Nadia read several enlightening, business-related articles. $\underline{\text{independent adjectives}}$

If you want to be successful too learn from any mistakes you make along the way. ______________

For the following exercises, the vectors $\mathbf{u}$ and $\mathbf{v}$ are given. Calculate the dot product $\mathbf { u } \cdot \mathbf { v }$.

$\mathbf { u } = \langle 3,0 \rangle$, $\mathbf { v } = \langle 2,2 \rangle$

Evaluate the following integrals. $\int{\frac{x^9}{\sqrt{1-x^{20}}} ~ dx}$

Determine the following integrals: $\int e^{-3 x} \cos 2 x d x$

Evaluate the integrals using appropriate substitutions.

$$\int e^{-\frac{x}{2}}\ dx$$

Evaluate the following integrals. $\int \frac { d x } { x ^ { 2 } - 6 x + 34 }$

Maximize $z=3 x+2 y$ subject to

$$\begin{array}{r} 2 x-3 y \leq 6 \\ x+y \leq 4 \\ x \geq 0 \\ y \geq 0 . \end{array}$$

Evaluate the integrals.

$$\int \frac{x^5}{e^{2-3 x^6}} d x$$

Find the dot product of $\mathbf{U}$ and $\mathbf{V}.$ Then determine if $\mathbf{U}$ and $\mathbf{v}$ are orthogonal. $\mathbf{U}=\langle 2,5,2\rangle, \mathbf{V}=\langle 8,2,-13\rangle$

Solve the integrals. $\int \frac{d x}{x^2+2 x+3}$ (Complete the square.)

Evaluate the integrals

$$\displaystyle\int \frac{d x}{2 \sqrt{x}+2 x}$$

Evaluate the integrals using the indicated substitutions. $\int e^{-5 x} d x ; u=-5 x$

The velocity of a mouse running alongside the baseboard of a room is modeled by the function

$$v(t)=3 \cos (\pi t / 12)-0.5 t, \quad 0 \leq t \leq 8$$

where $t$ is in seconds and $v$ is in meters per second. (Assume that positive values of $v$ indicate motion to the right.) Estimate the time(s) at which the mouse is $2 \mathrm{~m}$ from its starting position.