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📗 (Enter a number) Evaluate the following expression: \(\displaystyle\sum_{i=1}^{n}\) for \(n\) = . Hint: use the formula: \(1 + x + x^{2} + ... + x^{t-1} = \dfrac{1 - x^{t}}{1 - x}\).
📗 You can enter a number or an expression.
📗 Answer: .
📗 (Enter a vector) Find the unit vector that is perpendicular to the plane = \(0\) with a non-negative z-value (\(x_{3}\) value here). Hint: the normal vector of the plane \(a x_{1} + b x_{2} + c x_{3} + d = 0\) is \(\begin{bmatrix} a \\ b \\ c \end{bmatrix}\) and the unit normal vector is \(\begin{bmatrix} \dfrac{a}{\sqrt{a^{2} + b^{2} + c^{2}}} \\ \dfrac{b}{\sqrt{a^{2} + b^{2} + c^{2}}} \\ \dfrac{c}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{bmatrix}\).
📗 You can enter three numbers or three expressions separated by commas.
📗 Answer: .
📗 (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: .
📗 (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.
📗 Answer: .
📗 (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. Hint: the length of the line segment from the origin to the center of any of the spheres is \(\sqrt{n}\).
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 \(\left(x_{1}, x_{2}, x_{3}\right)\). Use your 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.
📗 Choices:
📗 (Select objects on canvas) Highlight a spanning tree 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 grid elements on canvas) Highlight the grid such that the number of highlighted elements in the rows sums up to (the numbers are ordered from the top row to the bottom row) and in the columns sum up to (the numbers are ordered from leftmost column to the rightmost column).
📗 Click inside a cell to highlight (or dehighlight it). Dragging from (i1, j1) to (i2, j2) flips the highlighting of all cells in the rectangle with corners (i1, j1) and (i2, j2).
📗 (Draw a line on canvas) 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. Hint: the direction you draw the line does matter: if one direction does not work, try the other direction.
📗 Draw a line using mouse drag or touch (drag).
📗 (Draw a graph on canvas) Draw a digraph given the following adjacency matrix: . Hint: there is an edge from i to j iff the entry on row i column j in the adjacency matrix is 1.
📗 Add a node by mouse click or touch, add a directed edge by mouse drag or touch "drag" from one node to another.
📗 Please copy and paste the text between the *****s (not including the *****s) and submit it on Canvas, M1.
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