Answer Happy

please help me with the part 1 of this homework thanks.
General Information: Geogebra is a free digital tool for mathematical learning. If you go to their website https://www.geogebra.org/, and click on "App Downloads" on the left column, you will notice that you can either download the app, or just click on "Start" if you want to use a particular feature of Geogebra online. The solutions for this assignment should consist of a document (for exam- ple a word file) where you include all the images you generated, plus any steps in the solution of the problem that were used for arriving at these images. Part 1 Parametrizing Curves: Viviani's curve is defined as the intersection of the sphere x² + y² + z² = 4 with the cylinder (x - 1)2 + y2 = 1. Observe that x = 1+ cos t y = sin t z = ±2 sin (²) 1-cost gives a parametrization of this curve [this uses the identity sin () = ± To plot Viviani's curve go to "3D Calculator" and type "curve": choose the option for three entries Curve (Expression, Expression, Expression, Parameter Variable, Start Value, End Value) and enter the formulas for x, y, z in terms of t here (choose the positive sign for z and t from 0 to 7, which in Geogebra you can write as "pi". You must also write sin(t) instead of sint for example) Exercise 1. Save the image you get as a pdf or jpeg file.

Exercise 2. Now plot the curve which is obtained as the intersection of the paraboloid z = x² + y² with the cylinder (x − 1)² + y² = 1. Use the interval [0 2*pi] instead of [0 pi]. Hint: notice that this is the same cylinder as before so x and y are given by the same functions of t. Only z requires a different formula in terms of t. Part 2 Graphing Functions: Consider the function f(x, y) = y (1-²²) (x² + y²), Exercise 3. What is the domain of this function? To plot it go to 3D calculator of Geogebra and type f(x,y)= y (1-1/(x^2+y^2)). Exercise 4. Save the image you get as a pdf or jpeg file. Notice that when you plot the graph of this function, away from the origin it looks very similar to a plane, can you think of why? To graph level curves go to https://www.geogebra.org/ and enter "Level Curves" on the search bar. There is a nice animation created by Kristen Beck (https://www.geogebra.org/m/J3kDCzjz). Use this to see some level curves of this function. Exercise 5. Save the image you get as a pdf or jpeg file (take a screen- shot of the level curves shown on the ry plane) Partial Derivatives and the Gradient To plot 2d vector fields on Geogebra enter "two dimensional vector field" on the Geogebra.com search bar. There is an useful one with the link here https://www.geogebra.org/m/kdw2vf9p. Exercise 6. Find the gradient of h(x, y) = (x² + y²) and show the image you get. For 3d vector fields go to the Geogebra.com search bar and enter "vector fields". An useful one is this animation https://www.geogebra.org/m/u3xregNW. Exercise 7. Plot the vector field F = (x+y)i + (z −y)j + (x+y+z)k and show the image you get. Exercise 8. Use the 3D calculator option to plot the ellipsoid f(x, y, z) = r²+2y² +3z² = 1 together with the tangent plane at the point (1,0,0). 2