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📗 [2 points] (Select one or multiple answers) In an \(n\) dimensional space, consider \(2^{n}\) n-spheres (gray) with radius 1 centered at \(\left(x_{1}, x_{2}, ..., x_{n}\right), x_{i} \in \left\{-1, 1\right\}\). Consider the largest n-sphere (blue) centered at the origin that can fit inside the gray n-spheres and the smallest n-cube (red) centered at the origin that can contain all gray n-spheres. For what values of \(n\) is part of the blue sphere outside of the red cube? Below are examples in 1, 2, and 3 dimensions.
In 1D, a unit 1-sphere is just the endpoints \(\left\{x_{1} - 1, x_{1} + 1\right\}\). Note that the blue sphere is only a point and has radius 0.
In 2D, a unit 2-sphere is the circle with radius 1 centered at \(\left(x_{1}, x_{2}\right)\). Note that the blue sphere is a circle with radius \(\sqrt{2} - 1\).
In 3D, a unit 3-sphere is the sphere with radius 1 centered at \(\left(x_{1}, x_{2}, x_{3}\right)\). Note that the blue sphere has radius \(\sqrt{3} - 1\). Use mouse or touch to rotate the view in the diagram above and note that the radius of the blue sphere is larger than the one in 2D. In higher dimensions, the radius of the blue sphere will get larger and eventually larger than the side length of the red cube, which is always \(2\).
📗 You can check one or multiple answers.
📗 You can use this textbox as a calculator: .
📗 Choices:
None of the above
📗 [3 points] (Enter an expression) Enter the derivative of as a function of \(w\).
📗 You can enter expressions of w. Use "exp(w)" for \(e^{w}\). Do not change the variable name "w". You do not need to simplify the expression.
📗 You can plot your expression (red curve) against my answer (green curve) using from to . w
📗 Answer: .
📗 [4 points] (Enter a matrix) Compute the Hessian matrix of evaluated at \(\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\) = .
📗 You can enter two numbers or two expressions separated by commas in one line and another two numbers or two expressions in the next line.
📗 Answer: .
📗 [4 points] (Select grid elements) Highlight squares in the grid such that the number of highlighted squares in the rows sums up to and in the columns sum up to .
📗 Click inside a square to highlight (or dehighlight it). Dragging from one square to another flips the highlighting of all squares along the path.
grid
📗 [3 points] (Draw a graph) Draw a digraph given the following adjacency matrix: .
📗 Add a node by mouse click or touch, add a directed edge by mouse drag or touch "drag" from one node to another (or a loop that goes back to itself).
📗 Remove a node by changing to "Eraser" mode and mouse click or touch the node, remove a directed edge by mouse drag or touch "drag" from one node to another.
graph
📗 [3 points] (Enter a vector) Find the unit vector that is perpendicular (orthogonal) to the plane = \(0\) with a non-negative \(x_{3}\)-value. It's the red vector on the green plane in the diagram below (may have to rotate to see).
📗 You can enter three numbers or three expressions separated by commas.
📗 Answer: .
📗 [3 points] (Move the sliders) Move the sliders below to change the green plane normal so that all the blue points are above the plane and all the red points are below the plane.
📗 You can use mouse, touch or keyboard arrow keys to control the sliders. If you use mouse or touch, the diagram will update only when you release the slider.
📗 Rotate the 3D diagrams using the left mouse button (or one-finger touch "drag"), zoom in and out using the mouse scroll wheel (or two-finger touch "pinch"), and pan (move without rotation) using the right mouse button (or two-finger touch "drag").
📗 [1 points] (Draw a line) Draw the linear decision boundary that classifies all points correctly, i.e. all red points are on the left side of the line and all blue points are on the right side of the line. (Note: the line you draw can be viewed as a vector from your mouse-down (or touch-start) position to your mouse-up (or touch-end) position, and "left side" (and "right side") is based on the direction of that vector.)
📗 Draw a line using mouse drag or touch (drag). The line connects the point when the mouse is pressed (touch is started) and the point when the mouse is released (touch is ended). The existing line will be replaced when a new line is drawn.
linetrue
📗 [2 points] (Select objects) Highlight a spanning tree of the following directed graph by selecting the nodes and edges in the spanning tree. Use the convention that parents point to their children.
📗 Select (or deselect) a node by mouse click or touch, select (or deselect) a directed edge by mouse drag or touch "drag" from one node to another. The selected nodes and edges should appear red.
select
📗 [2 points] (Enter a number) Evaluate the following expression: \(\displaystyle\sum_{i=1}^{n}\) for \(n\) = .
📗 You can enter a number or an expression (evaluated using math.js).
📗 Use the "Calculate" button or press the "Enter" key to make sure the expression you entered can be evaluated correctly by the auto-grader.
📗 You could save the text in the above text box to a file using the button or copy and paste it into a file yourself .
📗 You could load your answers from the text (or txt file) in the text box below using the button . The first two lines should be "##m: 1" and "##id: your id", and the format of the remaining lines should be "##1: your answer to question 1" newline "##2: your answer to question 2", etc. Please make sure that your answers are loaded correctly before submitting them.
📗 You can find videos going through the questions on Link.