Problem 1. Calculate the derivatives of the following implicit functions: (a) 3:2 cos y + sin 2y = my (b) 3x2y5 + tany = 5 sins: Problem 2. Find the equation of the tangent line to the curve 31:2 + 43:11; + y2 = 13 at the point (2,1). Problem 3. An airplane is ying towards a radar station at a constant height of 6 km above the ground. If the distance 3 between the airplane and the radar station is decreasing at a rate 400 km per hour when .9 = 10 km, what is the horizontal speed of the plane? Problem 4. A lter lled with liquid is in the shape of a vertex-down cone with a height of 15 in and a diameter of 10 in at its open (upper) end. If the liquid drips out the bottom of the lter at a constant rate of 4 in3 /sec, how fast is the level of the liquid dropping when the liquid is 1in deep? Problem 5. A 10ft ladder resting against a wall begins to slip. How fast is the top of the ladder sliding down the wall when the angle between the ladder and the wall is 0 = 7r/4 and the angle is increasing at a rate of rad/sec? Problem 6. The kinetic energy of an object is K mv2 where m is the mass and v is the velocity. Let's say you're driving a 1000kg vehicle and are accelerating at 3m/s2. (a) HOW much fast is the kinetic increasing when you are traveling at 30m/s ( 67mph)? (b) How about 50m/s ( 89mph)? (e) What does this calculation tell you about kinetic energy and its relation to velocity? Problem 7. A slider crank linkage is a bar that connects a piston to a rotating wheel. They are a way of converting horizontal oscillation into rotation. More complicated versions of these linkages are what converts the motion of pistons in steam (https://en.wikipedia.org/wiki/Slider-crank_linkage#/media/File:Steam_engine_in_action.gif) and internal combustion engines into rotation. They can also be used to convert the linear motion of hydraulic pistons into rotational motion (https: //en.wikipedia. org/wiki/ File:Hydraulikzylinder01 . jpg). You can and should make use of this Desmos graph to help your thinking: https: //www.desmos . com/calculator/uge22lglvc 1. As in the drawing above, assume that the wheel is centered at (0,0). Let l he the length of the bar connecting the piston to the wheel, 7" the radius of the wheel, as the horizontal position of the piston head and A the angle between the point on the Wheel holding the bar and the horizontal axis. Using the law of cosines nd a formula relating l, r, A and w. Hint: Recall that the law of Cosines is a generalization of the Pythagorean theorem that includes the cos of an angle measure. 2. Suppose l = 3 meters and 1' = 1 meter. Compute using implicit differentiation. Your answer will depend on m, l, 'I' and A. What is a physical meaning of (ti:1? 3. Solve for x in terms of A,l and r. 4. Find using the explicit formula for a: and see that it is the same as the derivative you found using implicit dierentiation. 5. Assuming x(t) = cos(10t) + 3 (measured in meters) nd % at A = '5