Problem 1. Calculate the derivatives of the following implicit functions: (a) a' cosy + sin 2y = ry (b) 3x2y' + tany = 5 sinc Problem 2. Find the equation of the tangent line to the curve x2 + 4ry + y? = 13 at the point (2, 1).Problem 3. An airplane is flying towards a radar station at a constant height of 6 km above the ground. If the distance s between the airplane and the radar station is decreasing at a rate 400 km per hour when s = 10 km., what is the horizontal speed of the plane? Problem 4. A filter filled 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 filter at a constant rate of 4 in/ sec, how fast is the level of the liquid dropping when the liquid is lin 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 e = #/4 and the angle is increasing at a rate of rad/sec? Problem 6. The kinetic energy of an object is K = my where m is the mass and v is the velocity. Let's say you're driving a 1000kg vehicle and are accelerating at 3m/s. (a) How much fast is the kinetic increasing when you are traveling at 30m/s ( 67mph)? (b) How about 50m/s ( 89mph)? (c) 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 : Hydrauliczylinder01 . jpg). You can and should make use of this Desmos graph to help your thinking: https://www.desmos. com/calculator/uge221glvc A bar Piston (X,0 ) wheel 1. As in the drawing above, assume that the wheel is centered at (0, 0). Let / be the length of the bar connecting the piston to the wheel, r the radius of the wheel, > 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 find a formula relating I, r, A and c. Hint: Recall that the law of cosines is a generalization of the Pythagorean theorem that includes the cos of an angle measure. 2. Suppose / = 3 meters and r = 1 meter. Compute #5 using implicit differentiation. Your answer will depend on c, l, r and A. What is a physical meaning of da? 3. Solve for r in terms of A, I and r. 4. Find #5 using the explicit formula for a and see that it is the same as the derivative you found using implicit differentiation. 5. Assuming r(t) = cos(10t) + 3 (measured in meters) find S at A = ]