Bill Foster's content - Numbas at mathcentre.ac.uk

204 results authored by Bill Foster - search across all users.

Max and Min 5
Ready to use
Question in Bill's workspace by Bill Foster

$I$ 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?
Max and Min 6
Ready to use
Question in Bill's workspace by Bill Foster

$I$ 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?
Matrix manipulation 2
Ready to use
Question in Bill's workspace by Bill Foster and 1 other

Exercises in multiplying matrices.
Matrix Multiplication 1
Ready to use
Question in Bill's workspace by Bill Foster and 1 other

Multiplication of $2 \times 2$ matrices.
Linear combinations of 2 x 2 matrices
Ready to use
Question in Bill's workspace by Bill Foster and 1 other

Linear combinations of $2 \times 2$ matrices. Three examples.
IS2.3
Ready to use
Question in Bill's workspace by Bill Foster and 1 other

Given the pdf $f(x)=\frac{a-bx}{c},\;r \leq x \leq s,\;f(x)=0$ else, find $P(X \gt p)$ , $P(X \gt q | X \gt t)$ .
Integration by parts
Ready to use
Question in Bill's workspace by Bill Foster

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$ , find $g(x)$ . Find $h(x)$ , $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$ .
IP3.1
Ready to use
Question in Bill's workspace by Bill Foster

Given a probability mass function $P(X=i)$ with outcomes $i \in \{0,1,2,\ldots 8\}$ , find the expectation $E$ and $P(X \gt E)$ .
IP3.2
Ready to use
Question in Bill's workspace by Bill Foster

Determine if the following describes a probability mass function.

$P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$ .
IP3.3
Ready to use
Question in Bill's workspace by Bill Foster

Two numbers from a set of $5$ numbers are chosen at random, without replacement. Find the distribution $X$ of their sum and $E[X]$ .
IS2.1
Ready to use
Question in Bill's workspace by Bill Foster

The random variable $X$ has a PDF which involves a parameter $k$ . Find the value of $k$ . Find the distribution function $F_X(x)$ and $P(X \lt a)$ .
IS2.2
Ready to use
Question in Bill's workspace by Bill Foster

The random variable $X$ has a PDF which involves a parameter $c$ . Find the value of $c$ . Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$ .
Integration by partial fractions
Ready to use
Question in Bill's workspace by Bill Foster

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d$ .
Integration by partial fractions
Ready to use
Question in Bill's workspace by Bill Foster

Find $\displaystyle\int \frac{a}{(x+b)(x+c)}\;dx,\;b \neq c$ .
Integration by partial fractions. (Video)
Ready to use
Question in Bill's workspace by Bill Foster

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Video in Show steps.
Integration by parts
Ready to use
Question in Bill's workspace by Bill Foster

Find $\displaystyle \int (ax+b)\cos(cx+d)\; dx$ and hence find $\displaystyle \int (ax+b)^2\sin(cx+d)\; dx$
Integration by parts
Ready to use
Question in Bill's workspace by Bill Foster

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx$ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$
Integration by parts
Ready to use
Question in Bill's workspace by Bill Foster

Given that $\displaystyle \int x({ax+b)^{m}} dx=\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$ , find $g(x)$ .
Functions of two variables: Stationary points 1
Ready to use
Question in Bill's workspace by Bill Foster

Find the stationary point $(p,q)$ of the function: $f(x,y)=ax^2+bxy+cy^2+dx+gy$ . Calculate $f(p,q)$ .
Functions of two variables: Stationary points 2
Ready to use
Question in Bill's workspace by Bill Foster

Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.
Functions of two variables: Stationary points 3
Ready to use
Question in Bill's workspace by Bill Foster

Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$ : $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$ , $D$ is a disk centre $(c,d)$ .
Find the stationary point of a function of two variables (2)
Ready to use
Question in Bill's workspace by Bill Foster

Find the critical point $(0,a)$ of the function: $f(x,y)=ax^3+bx^2y+cy^2+dy+f$ and find its type using the test given by the Hessian matrix.
Complex Numbers: Modulus, Argument
Ready to use
Question in Bill's workspace by Bill Foster

Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\pi$ and $\pi$ and careful with quadrants!
Bill's copy of Rearranging equations by multiplying or dividing: One step
Ready to use
Question in Bill's workspace by Bill Foster

No description given

Bill Foster

Show results for Show results for

Refine by Refine by

Status

Tags

Usage rights

Ability Level

Topics