Use the spinner to find each probability. If the spinner lands on a line it is spun again.

P (pointer landing on orange)

Given that sec b = csc 0.64 and that b lies in the first quadrant, use a cofunction identity to determine the measure of angle b, to two decimal places.

Evaluate the integrals by making appropriate u-substitutions and applying the formulas.

$$\int \frac{x^{1 / 5}}{9+x^{12 / 5}} d x$$

W is a subset of $R^3$ consisting of vectors of the form

$$\mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]$$

. In each case, determine whether W is a subspace of $R^3$. If W is a subspace, then give a geometric description of W. $W=\left\{\mathbf{x}: x_{3}=x_{2}=2 x_{1}\right\}$

Plot the asteroid, which is the hypocycloid that results with r = 1 and k = 4. How many cusps does the curve have?

Evaluate the integrals by making appropriate u-substitutions and applying the formulas.

$$\int \frac{e^{\tan ^{-1} 3 x}}{1+9 x^2} d x$$

The asteroid Ceres has a mass of 7 x 10²⁰ kg and a radius of 500 km. What is g on the surface of Ceres?

W is a subset of $R^3$ consisting of vectors of the form

$$\mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]$$

. In each case, determine whether W is a subspace of $R^3$. If W is a subspace, then give a geometric description of W. $W=\left\{\mathbf{x}: x_{2}=x_{3}+x_{1}\right\}$

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $\mathbf{a}=[-8,6], \quad \mathbf{b}=[\sqrt{7}, 3]$

Find the directional derivative of f at P in the direction of a vector making the counterclockwise angle $\theta$ with the positive $x$-axis.

$$f(x, y)=\frac{x-2 y}{2 x-y} ; P(2,-1) ; \theta=\pi / 6$$

Given that $\cos \frac{5 \pi}{9}=-\sin y$, first express $\frac{5 \pi}{9}$ as a sum of $\frac{\pi}{2}$ and an angle, and then apply a trigonometric identity to determine the measure of angle y.

Graph the given functions. In the previous Exercise, first rewrite the function with a positive angle, and then graph the resulting function.
$y=-5 \cos (-4 \pi x)$

Evaluate the integral by making an appropriate trigonometric substitution.

$$\int \frac{\sqrt{1+9 x^2}}{x} d x$$

Graph the given functions. In the previous Exercise, first rewrite the function with a positive angle, and then graph the resulting function. \

$$y=3 \sin (-2 x)$$