Partial Derivatives and the Gradient using Geogebra digital tool

Math 251, Spring 2024 Geogebra 2 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 example a word or pdf 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. 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 a useful one with the link here https://www.geogebra.org/m/kdw2vf9p Exercise 1 Find the gradient of h(x, y) = 21 (x2 + y 2 ) and show the image you get. For 3d vector fields go to the Geogebra.com search bar and enter “vector fields”. A useful one is this animation https://www.geogebra.org/m/u3xregNW Exercise 2 Plot the vector field F = (x + y)i + (z − y)j + (x + y + z)k. Include the image in your assignment. Exercise 3 Use the 3D calculator option to plot the ellipsoid f (x, y, z) = x2 + 2y 2 + 3z 2 = 1 together with the tangent plane at the point (1, 0, 0). Include the image in your assignment. Lagrange Multipliers In Calculus 1 you learned how to solve the following optimization problem: “Suppose that you have two numbers x, y and want to make x2 + y 2 as large or small as possible, subject to the condition that the point (x, y) must belong the parabola x = y 2 − 5. In other words, y 2 = x + 5.” Exercise 4 (a) Use Calculus 1 techniques to find a function S(x) you need to optimize and find the value(s) of x that works, together with the corresponding value(s) of y. (b) Now we will reinterpret this problem as a Lagrange multiplier problem. This means that we have a function of two variables f (x, y) = x2 + y 2 and a constraint condition g(x, y) = y 2 − x = 5. Find the values of x, y that work using the Lagrange Multiplier method. Are these the same points as those of part (a)? (c) To understand what could be going on, suppose we had solved part (a) differently. Namely, that someone writes the function to optimize in terms of y, not x. Do this using Calculus 1 techniques and see what values of x, y you get. (d) Use the animation https://www.geogebra.org/m/PSzG4pe6 to show that at these three points the gradients ∇f and ∇g are parallel. Generate an image for each point. In the animation the blue vector represents ±∇g and the red vector represents ±∇f . Include the image(s) in your assignment.