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Find the particular solution to y ‘ = sin(x) given the general solution is y = C – cos(x) and the initial condition . (5 points)
The slope of the tangent to a curve at any point (x, y) on the curve is . Find the equation of the curve if the point (2, -2) is on the curve. (5 points)
The rate of decay in the mass, M, of a radioactive substance is given by the differential equation , where k is a positive constant. If the initial mass was 100g, then find the expression for the mass, M, at any time t. (5 points)
The temperature of a pot of coffee varies according to Newton’s Law of Cooling: , where T is the temperature of the coffee, A is the room temperature, and k is a positive constant. If the water cools from 90°C to 85°C in 1 minute at a room temperature of 30°C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes. (5 points)
The differential equation (5 points)
I. produces a slope field with horizontal tangents at y = 2
II. produces a slope field with vertical tangents at y = -1
III. produces a slope field with columns of parallel segments
Which of the following differential equations is consistent with the following slope field?
The general solution of the differential equation dy – 0.2x dx = 0 is a family of curves. These curves are all (5 points)
Estimate the value of by using the Trapezoidal Rule with n = 4. (5 points)
The table below gives selected values for the function f(x). With 5 rectangles, using the midpoint of each rectangle to evaluate the height of each rectangle, estimate the value of . (5 points)
Given f(x) > 0 with f ′(x) < 0, and f ′′(x) < 0 for all x in the interval [0, 1] with f(0) = 1 and f(1) = 0.3, the left, right, trapezoidal, and midpoint rule approximations were used to estimate . The estimates were 0.7915, 0.8405, 0.8410, 0.8421 and 0.8895, and the same number of subintervals were used in each case. Match the rule to its estimate. (5 points)