Modular arithmetic. Find the following numbers modulo the given number $n$. Three examples to do.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Previous throws don't affect the probability distribution of subsequent throws. Believing otherwise is the gambler's fallacy.

Questions on powers, the laws of indices, and exponential growth.

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$I$ compact interval. $\displaystyle g: I \rightarrow I, g(x)=\frac{ax}{x^2+b^2}$. Find stationary points and local maxima, minima. Using limits, has $g$ a global max, min? 

$I$ compact interval, $g:I\rightarrow I,\;g(x)=ax^3+bx^2+cx+d$. Find stationary points, local and global maxima and minima of $g$ on $I$

$I$ compact interval, $g:I\rightarrow I$, $g(x)=(x-a)(x-b)^2$. Stationary points in interval. Find local and global maxima and minima of $g$ on $I$. 

$g: \mathbb{R} \rightarrow \mathbb{R}, g(x)=\frac{ax}{x^2+b^2}$. Find stationary points and local maxima, minima. Using limits, has $g$ a global max, min? 

$I$ compact interval. $\displaystyle g: I\rightarrow I, g(x)=\frac{x^2}{(x-c)^{a/b}}$. Are there stationary points and local maxima, minima? Has $g$ a global max, global min? 

$I$ compact interval. $\displaystyle g: I \rightarrow I, g(x)=\frac{ax}{x^2+b^2}$. Find stationary points and local maxima, minima. Using limits, has $g$ a global max, min? 

Previous throws don't affect the probability distribution of subsequent throws. Believing otherwise is the gambler's fallacy.

Compute the experimental probability of a particular score on a die given a sample of throws, and compare it with the theoretical probability.

The last part asks what you expect to happen to the experimental probability as the sample size increases.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Previous throws don't affect the probability distribution of subsequent throws. Believing otherwise is the gambler's fallacy.

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.

Questions on powers, the laws of indices, and exponential growth.