Exercises * on Functions

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1 Exercises * on Functions Laurenz Wiskott Institut für Neuroinformatik Ruhr-Universität Bochum, Germany, EU 2 February 2017 Contents 1 Scalar functions in one variable Elementary transformations Exercise: Stepwise transformation of an elementary function Exercise: Stepwise construction of a given function Exercise: Stepwise construction of a given function Combination of functions Exercise: Combinations of functions Composition of functions Exercise: Composite functions Exercise: Composite functions Exercise: Analytical expressions for given function graphs Logarithmic plots Scalar functions in two variables Laurenz Wiskott (homepage This work (except for all figures from other sources, if present) is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit Figures from other sources have their own copyright, which is generally indicated. Do not distribute parts of these lecture notes showing figures with non-free copyrights (here usually figures I have the rights to publish but you don t, like my own published figures). Several of my exercises (not necessarily on this topic) were inspired by papers and textbooks by other authors. Unfortunately, I did not document that well, because initially I did not intend to make the exercises publicly available, and now I cannot trace it back anymore. So I cannot give as much credit as I would like to. The concrete versions of the exercises are certainly my own work, though. In cases where I reuse an exercise in different variants, references may be wrong for technical reasons. * These exercises complement my corresponding lecture notes available at Teaching/Material/, where you can also find other teaching material such as programming exercises. The table of contents of the lecture notes is reproduced here to give an orientation when the exercises can be reasonably solved. For best learning effect I recommend to first seriously try to solve the exercises yourself before looking into the solutions. 1

2 2.1 Visualization Set of curves Contour graphics D-graphics / surface plot Elementary transformations Linear coordinate transformations Exercise: Stepwise transformation of a simple function Exercise: Stepwise transformation of a simple function Exercise: Simplify a function with stepwise transformations Vectorial functions in one variable Visualization Set of curves Sequence of a vector Trajectory in phase space Exercise: Trajectories in phase space Exercise: Trajectories in phase space Exercise: Analytical expressions for given trajectories Vectorial functions in two variables Visualization as a vector field Exercise: Vector field Exercise: Analytic expression for a given vector field Exercise: Analytic expression for a given vector field Differential operators Partial derivatives and the Nabla operator Gradient Exercise: Contour plot and gradient field Exercise: Gradient field Divergence

3 1 Scalar functions in one variable 1.1 Elementary transformations Exercise: Stepwise transformation of an elementary function Construct the following functions step by step, for each intermediate step write down an equation and sketch the graph. (a) Begin with y = x 3, i) Shift the function 7 units to the left, ii) Stretch the function vertically by a factor of 8, iii) Flip the function at the y-axis. (b) Begin with y = cos(x), i) Stretch the function horizontally by a factor of 2; ii) Shift the function 3 units up; iii) Flip the function at the x-axis. (c) Begin with y = e x, i) Flip the function at the x-axis, ii) Shift the function 5 units to the right and 12 units down, iii) Flip the function at the diagonal y = x Exercise: Stepwise construction of a given function Construct and draw the following functions by stepwise transformation of the underlying elementary function. Illustrate each step. (a) 3 sin(x + π/4) (b) 1/[(x 2) 3 1] Exercise: Stepwise construction of a given function Construct and draw the following functions by stepwise transformation of the underlying elementary function. Illustrate each step. (a) e 2( x 3) + 7 (b) x 3 x Combination of functions Exercise: Combinations of functions Discuss the following functions intuitively, like in the lecture, and sketch them. (a) f(x) = cos (x) + x2 20 (sum of functions) (b) g(x) = x 2 sin(x) (product of functions) 3

4 1.3 Composition of functions Exercise: Composite functions Discuss the following functions intuitively, like in the lecture, and sketch them. (a) ln(x 2 ) (b) e sin x (c) sin(1/x) Exercise: Composite functions Discuss the following functions intuitively, like in the lecture, and sketch them. (a) e ( x2 +2) 2 (b) ln(x) Exercise: Analytical expressions for given function graphs Find a formula for each of the following sketched functions: (a) 3 (b) 1 (c) 3 c 1 1 (d) c (e) unclear (f) 1 Hint: You can use AnalyticalExpressionsForGivenGraphs-Exercise.ipynb and add the missing code to verify your guesses. 4

5 1.4 Logarithmic plots 2 Scalar functions in two variables 2.1 Visualization Set of curves Contour graphics D-graphics / surface plot 2.2 Elementary transformations 2.3 Linear coordinate transformations Exercise: Stepwise transformation of a simple function Transform the function z = x 2 + y 2 step by step as follows: 1. Compress the function by a factor 1/2 in the x-direction and stretch it by a factor 3 in the y-direction. 2. Shift the function horizontally 3 units to the left in x-direction and 1 unit up in y-direction. 3. Shift the function vertically 2 units down in z-direction. For each step write a corresponding equation and sketch the final result if you combine all three steps Exercise: Stepwise transformation of a simple function Transform the function as follows: f(x) = x T ( ) ( x x, x = y ) (1) 1. Rotate the function clockwise by 30 degrees. Hint: sin(π/6) = 1/2, cos(π/6) = 3/2. 2. Shift the function horizontally by a = (1, 2) T. Write a corresponding equation for each step and sketch the final result. possible. Use matrix notation as far as Exercise: Simplify a function with stepwise transformations Consider the function f 0 (x, y) := 1/2 x + 5y + 3/2x 2 + xy + 3/2y 2. (1) 5

6 1. Write the function in matrix notation like f 0 (x) = c + f T x + x T Hx, (2) with a scalar c, vectors f and x = (x, y), and a symmetric 2 2-matrix H. 2. Find a horizontal shift a and a vertical shift b such that f 1 (x) := x T Vx := b + f 0 (x a) (3) for a symmetric matrix V. 3. Find a rotation ( cos(φ) sin(φ) R := sin(φ) cos(φ) of the coordinates such that matrix V of the function ) (4) f 2 (x) := x T V x := f 1 (Rx) (5) becomes diagonal. 4. Sketch the function f 0 (x, y) without using a calculator. 3 Vectorial functions in one variable 3.1 Visualization Set of curves Sequence of a vector Trajectory in phase space Exercise: Trajectories in phase space Sketch the trajectories of the following functions without using a computer or pocket calculator. (a) f(a) = (cos (a), sin (2a)) (b) f(t) = (sin 2 (6t) sin(t), sin 2 (6t) cos(t)) Exercise: Trajectories in phase space Sketch the trajectories of the following functions without using a computer or pocket calculator. (a) f(t) = ((1 1/t) sin t, (1 1/t) cos t) T, 1 t (b) f(a) = (cos (2πa) 2 + a, cos (2πa)) T 6

7 3.1.6 Exercise: Analytical expressions for given trajectories Invent functions f(t) = (x(t), y(t)) that generate trajectory plots like those below. (a) (b) (c) CC BY-SA 4.0 Hint: You can use AnalyticalExpressionsForGivenTrajectories-Exercise.ipynb and add the missing code to verify your guesses. 4 Vectorial functions in two variables 4.1 Visualization as a vector field Exercise: Vector field Draw the vector field (y, x) Exercise: Analytic expression for a given vector field Find an analytic expression for the following vector field CC BY-SA Exercise: Analytic expression for a given vector field Find an analytic expression for the following vector field. 7

8 1 0.5 y x CC BY-SA Differential operators 5.1 Partial derivatives and the Nabla operator 5.2 Gradient Exercise: Contour plot and gradient field Consider the function X(u, v) = u 2 (u 1) 2 + v 2. (1) 1. Calculate the gradient field of the function. 2. Sketch the function with a contour plot and draw the gradient field, both without the help of a calculator Exercise: Gradient field Calculate the gradient of the function f(x, y) = xy 1 2 y2. Sketch the gradient field and the level lines. 5.3 Divergence 8

Exercises * on Principal Component Analysis

Exercises * on Principal Component Analysis Laurenz Wiskott Institut für Neuroinformatik Ruhr-Universität Bochum, Germany, EU 4 February 207 Contents Intuition 3. Problem statement..........................................