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Approximate f by a Taylor polynomial with degree n at the number a.
${\displaystyle f\left(x\right)=\ln (1+2x),a=1,n=3,0.5\le x\le 1.5}$

Suppose that A and B are independent events such that the probability that neither occurs is a and the probability of B is b. Show that
${\displaystyle P\left(A\right)=\frac{1-b-a}{1-b}}$

Find the real numbers x and y that make the equation true.
${\displaystyle 2+3i=x+yi}$

Find all values of k for which the given augmented matrix corresponds to a consistent linear system.
(a) [3 -4 k, -6 8 5]

Write the solution of the given homogeneous system in parametric vector form
${\displaystyle 3x_1+3x_2+6x_3=0}$
${\displaystyle -9x_1-9x_2-18x_3=0}$
${\displaystyle -6x_2-6x_3=0}$
where the solution set is $$x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$$

Write the column matrix b as a linear combination of the columns of A.
$$A=\left[\begin{array}{cc}-3& 5\\ 3& 4\\ 4& -8\end{array}\right],b=\left[\begin{array}{c}-22\\ 4\\ 32\end{array}\right]$$

You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. (a) What is the probability that you will have to wait longer than 10 minutes? (b) If, at 10:15, the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

To find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. Find a rational number halfway between the two fractions in each pair. 1/100 and 1/10

Find the angle ${\displaystyle \theta }$ between the given vectors to the nearest tenth of a degree.
${\displaystyle U=-5i+7j,V=8i+3j}$

A box contains 8 red marbles and 3 green ones. Six draws are made at random without replacement. True or false: the chance that the 3 green marbles are drawn equals
${\displaystyle \frac{6!}{3!3!}{\left(\frac{8}{11}\right)}^3{\left(\frac{3}{11}\right)}^3}$

Use the dot product to find an angle between ${\displaystyle u=5i+2j}$ and ${\displaystyle v=8i-3j}$ to the nearest tenth of a degree.

Find an equation of the plane passing through the point perpendicular to the given vector or line. Point: (3, 2, 2) Perpendicular to: n = 2i + 3j - k

Find the curl and of the vector field.
${\displaystyle F(x,y,z)=(x+yz)i+(y+xz)j+(z+xy)k}$

A student is getting ready to take an important oral examination and is concerned about the possibility of having an “on” day or an “off” day. He figures that if he has an on day, then each of his examiners will pass him, independently of each other, with probability .8, whereas if he has an off day, this probability will be reduced to .4. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student feels that he is twice as likely to have an off day as he is to have an on day, should he request an examination with 3 examiners or with 5 examiners?

In how many ways can a set of five letters be selected from the English alphabet?

Find a polynomial of degree n that has the given zero(s). (There are many correct answers.) Zero(s) x = -3 Degree n = 2

Let P and Q be polynomials. Find ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{P\left(x\right)}{Q\left(x\right)}}$ if the degree of P is greater than the degree of Q.