1. Verify that the following functions are solutions to the differential equation. a. b. c. d. e. 2. Create a slope field for the following differential equations. Draw a slope for every integer...

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1. Verify that the following functions are solutions to the differential equation. a. b. c. d. e. 2. Create a slope field for the following differential equations. Draw a slope for every integer point, (-4,-4) to (4,4). a. b. 3. Euler’s Method a. Create a table with through 6 rows for the differential equation with initial value and to estimate a solution. b. Repeat part a. with initial condition . Use the same differential equation and c. Sketch a slope field and your two solutions here. 1. Solve the following IVP using separation of variables. Show all your work. a. b. c. d. 2. Newton’s Law of Cooling. On your day off from investigating murders, you are baking some delicious cookies. You take the 100∘C cookies out of the oven into the 23∘C room. The cookies cool according to Newton’s Law of Cooling which says that an object heats up or cools down at a rate that is proportional to the difference between the temperature of the object and the surrounding temperature. a. Write down a differential equation that describes the current situation. b. What is the initial condition? c. It turns out that the proportionality constant for cooling cookies is approximately ? = 2.1. Solve your initial value problem to find an explicit formula for the temperature of the cookies as a function of time. d. How long will it take before the cookies cool to 50° C? e. How long until the cookies cool to 25°? NCompute the following definite integrals. Be sure to show the antiderivative. Give exact answers (do not round off decmials). a. b. c. d. 2. Determine the exact amount of area in the shaded region of the graph. 3. Let f(t) be given by the multi-colored graph below. Let . Answer the following questions about A(x) based on the graph and what you know about accumlation functions. a. Fill in the table for the whole values of x. x 0 1 2 3 -1 -2 A(x) Now fill out the table with the half values of x. Be sure they are reasonable based on the first table. x 0.5 1.5 2.5 3 -0.5 -1.5 A(x) b. Plot the points you found in the previous part on the following graph. Decide how you think the graph might fill in between the points you added. Use more calculations from the above graph if you like. 1. Solve the following improper integrals. a. b. c. d. e. 2. For each of the following, list the terms through . Then decide whether the sequence will converge or diverge as . If it converges, to what value? a. b. c. d.