Lines, curves, and planes in Space (40 points) a. Find the equation of the line of intersection between x+y+2=3 and 2x-y+2-10 b. Derive the formula for a plane, wrote the vector equation first and then derive the equation involving x, y, and z. c. Write the equation of a line in 3D, explain the idea behind this equation (2-3 sentences) d. Calculate the curvature of y = x^3 at x=1. Graph the curve and the osculating circle using GeoGebra 4. Multivariate functions (40 points): a. For the function f(x,y) = 100 – x^2 - y^2 i. Sketch the domain using GeoGebra ii. Sketch f(x,y) using GeoGebra
iii. Find the first partial derivative with respect to x and with respect to y iv. Explain what the first partial derivative with respect to x represents geometrically at x = 3