I have a passion for mathematics and plan to purse a career as a professor of mathematics.
Evaluate the definite integral. Use a graphing utility to verify your result.$$\int_{0}^{2}\left(2^{x}+6\right) d x$$
Find the directional derivative of $ f(x, y, z) = xy^2z^3 $ at $ P(2, 1, 1) $ in the direction of $ Q(0, -3, 5) $.
A 5 -kg pendant chandelier is designed such that the alabaster bowl is held by four chains of equal length, as shown in the following figure. a. Find the magnitude of the force of gravity acting on the chandelier.b. Find the magnitudes of the forces of tension for each of the four chains (assume chains areessentially vertical).
Volume of a Fish Tank $A$ fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is 300 $\mathrm{cm}$ long and 120 $\mathrm{cm}$ wide. The front and back faces are vertical, but the left and right faces areslanted at $30^{\circ}$ from the vertical and measure 120 $\mathrm{cm}$ by150 $\mathrm{cm} .$ (See the figure.)(a) Let $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ be the three vectors shown in thefigure. Find $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .$ [Hint: Recall that $\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta$ and $|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}| \sin \theta . ]$(b) What is the capacity of the tank in liters?[Note: $1 \mathrm{L}=1000 \mathrm{cm}^{3} . ]$
Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? (a) from $n=4$ to $n=2$ , (b) from an orbit of radius 2.12 A to one of radius $8.46 \hat{A},(\mathbf{c})$ anelectron adds to the $\mathrm{H}^{+}$ ion and ends up in the $n=3$ shell?
For the following exercises, the vectors $\mathbf{u}$ and $\mathbf{v}$ are given.a. Find the vector projection $\mathrm{w}=\mathrm{proj}_{\mathrm{u}} \mathrm{v}$ of vector $\mathrm{v}$ onto vector u. Express your answer in component form.b. Find the scalar projection comp_ $\mathrm{v}$ of vector $\mathbf{v}$ onto vector $\mathbf{u} .$$$\mathbf{u}=\langle 4,4,0\rangle, \quad \mathbf{v}=\langle 0,4,1\rangle$$
A crane suspends a 500-lb steel beam horizontally by support cables (with negligible weight) attached from a hook to each end of the beam. The support cables each make an angle of $ 60^\circ $ with the beam. Find the tension vector in each support cable and the magnitude of each tension.
