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

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  • Testing and False Positives
    Has some problems
    Question in Bill's workspace by Bill Foster

    Given a population size and data on a test for a condition on that population, use a tabular approach to find the increase in risk after a positive test in which there is a small chance of a false positive i.e. the probability that the test is positive even though  the condition is not present. See Advice for a graphical version.

  • Exam (6 questions) in Newcastle University Biomedical Sciences by Bill Foster and 1 other

    A test of basic concepts to do with SI units and concentrations of solutions.

  • Question in Bill's workspace by Bill Foster

    Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.

    Included in the Advice of this question are:

    6 iterations of the method.

    Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.

  • Question in Bill's workspace by Bill Foster

    Asking users to input coefficients of a system of diff equations so that the phase space is a stable spiral. All systems input by the user are graphed together with immediate feedback. Also included in the Steps are the graphs of the solutions for $x(t),\; y(t);\; x(0)=-5,\;y(0)=5.$

  • Question in Bill's workspace by Bill Foster

    Using Jsxgraph to draw the vector field for a differential equation of the form $\frac{dy}{dx}=f(x,y)=1-\sin(y)$, and also by moving the point $(x_0,y_0)$ you can see the solution curves going through that point. 

    If you want to modify $f(x,y)$ simply change the definition of  $f(x,y)$ and that of the variable  str in the user defined function testfield in Extensions and scripts. You have to use javascript notation for functions and powers in the definition of $f(x,y)$.

  • Question in Bill's workspace by Bill Foster

    Using Jsxgraph to draw the vector field for a differential equation of the form $\frac{dy}{dx}=f(x,y)=\sin(x)-\sin(y)$, and also by moving the point $(x_0,y_0)$ you can see the solution curves going through that point. 

    If you want to modify $f(x,y)$ simply change the definition of  $f(x,y)$ and that of the variable  str in the user defined function testfield in Extensions and scripts. You have to use javascript notation for functions and powers in the definition of $f(x,y)$.

  • Question in Bill's workspace by Bill Foster

    Using Jsxgraph to draw the vector field for a differential equation of the form $\frac{dy}{dx}=f(x,y)=\sin(x-y)$, and also by moving the point $(x_0,y_0)$ you can see the solution curves going through that point. 

    If you want to modify $f(x,y)$ simply change the definition of  $f(x,y)$ and that of the variable  str in the user defined function testfield in Extensions and scripts. You have to use javascript notation for functions and powers in the definition of $f(x,y)$.

  • Question in Bill's workspace by Bill Foster

    Newton-Raphson numerical method question to solve $g(x)=0$

    Includes a graph of the function $g(x)$ in Advice using Jsxgraph.

  • Question in Bill's workspace by Bill Foster

    Using Jsxgraph to draw the vector field for a differential equation of the form $\frac{dy}{dx}=f(x,y)=x^3-y^3$, and also by moving the point $(x_0,y_0)$ you can see the solution curves going through that point.

    If you want to modify $f(x,y)$ simply change the definition of  $f(x,y)$ and that of the variable  str in the user defined function testfield in Extensions and scripts. You have to use javascript notation for functions and powers in the definition of $f(x,y)$.

  • Question in Bill's workspace by Bill Foster

    Using Jsxgraph to draw the vector field for a differential equation of the form $\frac{dy}{dx}=f(x,y)=x^2-y^2$, and also by moving the point $(x_0,y_0)$ you can see the solution curves going through that point.

    If you want to modify $f(x,y)$ simply change the definition of  $f(x,y)$ and that of the variable  str in the user defined function testfield in Extensions and scripts. You have to use javascript notation for functions and powers in the definition of $f(x,y)$.

  • Question in Bill's workspace by Bill Foster

    Asking users to input coefficients of a system of diff equations so that the phase space is a centre. All systems input by the user are graphed together with immediate feedback. Also included in the Steps are the graphs of the solutions for $x(t),\; y(t);\; x(0)=-5,\;y(0)=5.$

  • Dynamical system 4
    Ready to use
    Question in Bill's workspace by Bill Foster

    Nature of fixed points of a 2D dynamical system.

    These examples are either centres or spirals.

  • Question in Bill's workspace by Bill Foster

    Asking users to input coefficients of a system of diff equations so that the phase space is a saddle. All systems input by the user are graphed together with immediate feedback. Also included in the Steps are the graphs of the solutions for $x(t),\; y(t);\; x(0)=-5,\;y(0)=5.$

  • Question in Bill's workspace by Bill Foster

    As the title says

  • Question in Bill's workspace by Bill Foster

    Questions of the form $f(x) = g(x)/h(x) $

    Optimised for display of quotients using \displaystyle - see first part - otherwise difficult to read.

  • Question in Bill's workspace by Bill Foster

    This question uses the technique of allocating zero marks to gaps so that they act as holders of values input by the user but are not assessed. The values so input are then gathered using the code found in the preamble and suitable combinations are assigned as answers to gap fills which are then assessed to see if they meet the requirements of the question. Note that is a gap fill is created, but not addressed in the prompt of the part then it is hidden from the user.

  • Question in Bill's workspace by Bill Foster

    Solve $p - t < \text{or}> q$

  • Question in Bill's workspace by Bill Foster

    Compute a table of values for a quadratic function. The student input is now disconnected from the graph so that they slide the points on the graph after they input the values and the answer fields are not updated. Now includes a graph in advice.

  • Truth tables 0 (v2)
    Ready to use
    Question in Bill's workspace by Bill Foster

    Create a truth table for a logical expression of the form $a \operatorname{op} b$ where $a, \;b$ can be the Boolean variables $p,\;q,\;\neg p,\;\neg q$ and $\operatorname{op}$ one of $\lor,\;\land,\;\to$.

    For example $\neg q \to \neg p$.

  • Truth tables 1(v2)
    Ready to use
    Question in Bill's workspace by Bill Foster

    Create a truth table for a logical expression of the form $(a \operatorname{op1} b) \operatorname{op2}(c \operatorname{op3} d)$ where $a, \;b,\;c,\;d$ can be the Boolean variables $p,\;q,\;\neg p,\;\neg q$ and each of $\operatorname{op1},\;\operatorname{op2},\;\operatorname{op3}$ one of $\lor,\;\land,\;\to$.

    For example: $(p \lor \neg q) \land(q \to \neg p)$.

  • Truth tables 2 (v2)
    Ready to use
    Question in Bill's workspace by Bill Foster

    Create a truth table for a logical expression of the form $((a \operatorname{op1} b) \operatorname{op2}(c \operatorname{op3} d))\operatorname{op4}e $ where each of $a, \;b,\;c,\;d,\;e$ can be one the Boolean variables $p,\;q,\;\neg p,\;\neg q$ and each of $\operatorname{op1},\;\operatorname{op2},\;\operatorname{op3},\;\operatorname{op4}$ one of $\lor,\;\land,\;\to$.

    For example: $((q \lor \neg p) \to (p \land \neg q)) \lor \neg q$

  • Truth tables 3 (v2)
    Ready to use
    Question in Bill's workspace by Bill Foster

    Create a truth table for a logical expression of the form $((a \operatorname{op1} b) \operatorname{op2}(c \operatorname{op3} d))\operatorname{op4}(e \operatorname{op5} f) $ where each of $a, \;b,\;c,\;d,\;e,\;f$ can be one the Boolean variables $p,\;q,\;\neg p,\;\neg q$ and each of $\operatorname{op1},\;\operatorname{op2},\;\operatorname{op3},\;\operatorname{op4},\;\operatorname{op5}$ one of $\lor,\;\land,\;\to$.

    For example: $((q \lor \neg p) \to (p \land \neg q)) \to (p \lor q)$

  • Truth tables 4 (v2)
    Ready to use
    Question in Bill's workspace by Bill Foster

    Create a truth table for a logical expression of the form $((a \operatorname{op1} b) \operatorname{op2}(c \operatorname{op3} d))\operatorname{op4}e $ where each of $a, \;b,\;c,\;d,\;e$ can be one the Boolean variables $p,\;q,\;r,\;\neg p,\;\neg q,\;\neg r$ and each of $\operatorname{op1},\;\operatorname{op2},\;\operatorname{op3},\;\operatorname{op4}$ one of $\lor,\;\land,\;\to$.

    For example: $((q \lor \neg r) \to (p \land \neg q)) \land \neg r$

  • Question in Bill's workspace by Bill Foster

    Solve for $x$ and $y$:  \[ \begin{eqnarray} a_1x+b_1y&=&c_1\\   a_2x+b_2y&=&c_2 \end{eqnarray} \]

    The included video describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.

  • Question in Bill's workspace by Bill Foster and 1 other

    Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

  • Question in Bill's workspace by Bill Foster

    A simultaneous equations question with integers only

  • Question in Bill's workspace by Bill Foster

    Shows how to define variables to stop degenerate examples.

  • Question in Bill's workspace by Bill Foster

    Seven standard elementary limits of sequences. 

  • Question in Bill's workspace by Bill Foster

    Let $x_n=\frac{an+b}{cn+d},\;\;n=1,\;2\ldots$. Find  $\lim_{x \to\infty} x_n=L$ and find least $N$ such that $|x_n-L| \lt 10^{-r},\;n \geq N,\;r \in \{2,\;3,\;4\}$.

  • Question in Bill's workspace by Bill Foster

    $x_n=\frac{an+b}{cn+d}$. Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| \lt 10 ^{-r},\;n\geq N$, $2\leq r \leq 6$.