1. (In this problem, you'll derive the equation for the tangent plane for a particular example problem, comparing your answer to the formula. This problem s very similar to the derivation we did together in class.) Consider the hemisphere, z = /9 22 2. a. (6 pts) Write the equation of the space curve ry() that lies on the hemisphere in the vertical plane y = 2. Then, write the equation of the tangent line to this curve at the point (1,2,2). b. (6 pts) Write the equation of the space curve ra(t) that lies on the hemisphere in the vertical plane x = 1. Then, write the equation of the tangent line to this curve at the point (1,2,2). c. (6 pts) Using methods from 2.5, compute the equation of the plane that contains the tangent lines from (a) and (b). d. (6 pts) Now compute the tangent plane to z = /9 2? y? at the point (1,2, 2) using the formula, z = 29 + fo(o, y0)(z 20) + fy(xo,y0)(y yo). You should get the same answer you found in (c). 2. Let f(z,y) = (y/z) + sin(zy). a. (6 pts) Find the linearization L(z,y) to f(z,y) at the point (2, 7). b. (3 pts) Use the linearization to approximate f(2,3) (i.e., compute L(2, 3)). 3. The pressure, volume, and temperature of a mole of an ideal gas are related by the equation PV = 831T, where P is measured in kilopascals, V' in liters, and T in kelvins. You may round any number in your formula (a) or answer in (b) to the nearest 0.1. a. (6 pts) Write the pressure P as a function of 7" and V. Then, compute a linear equation for approximating the change in pressure were the volume of the gas to increase from 12 L to (12+ AV) L and the temperature of the gas were to increase from 310 K to (3104 AT) K. b. (3 pts) Use your formula in (a) to approximate the change in pressure if the volume increases from 12 L to 12.2 L and the temperature decreases from 310 K to 308 K