Please provide a detailed explanation of your solutions (show all your work) for the following questions:

1) For a - c consider the differential equation given by: a = a) On the axes provided, sketch a slope field for the given differential equation at the points indicated. b) Find the particular solution y =f(x) to the given differential equation with the initial condition f (0) = 2. Use your solution to find f (1). c) For the particular solution y =f (x) described in part (b), find limf(x). X-+0"2. Find the solution to the differential equation 6% : \"'73 with the initial condition y(3) :7 2 . 3. A bacteria culture starts With 200 bacteria and triples in size every half hour. After 3 hours, how many bacteria are there? 4. Calculator Active According to Newton's Law of Heating and Cooling, the temperature of an object decreases at a rate proportional to the difference between its temperature and that of the surrounding air. Suppose a yam is removed from a 250 degree F oven and placed on the counter in a room that is a constant 72 degreesF . After one hour, the yam is 200 degrees F . What is the temperature of the yam at time = 90 minutes ? 5. Calculator Active An object is thrown vertically upward with a velocity of 13 In] sec from a height of 4 meters. What is the highest point it reaches? (Hint: g 2 9.8 I'll/8802) 6. Match the Diff Eq to the solution equation. Each differential equation may have 0, 1 or 2 answers. 2x dy 7 302) a- a r T I. y : e 11. y = ln(x) 111. y : x3 + 2 IV. y 2* sm(x) V. y = cos(x) 7. Match the following differential equations to the corresponding slope field: 1 ) dy =- dx -2 A B dx ii) = r- - 3 111) dy : 72 4 dx 1V ) dy dx v) de vi) dy = sin (x) +y C D E F