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Q1: Find the general solution to the exact differential equation. A1: 5^x + tan^-1 + c Q2: Solve the initial value problem explicitly A2: u= x^7- x^3 + 5x + c Find C 1= 1-1 +5 + c -5 -5 c= -4 u = x^7 – x^3 + 5x -4 Q3: Euler’s Method with increments of x =0.1 to approximate the value of y when x = 1.3 A3: (x,y) dy/dx = x-1 x (1,2) (1.1, 2.3) ( 1.2, 3.32) 3 3.2 3.4 .1 .1 .1 2.03 Q4: Find the indefinant integral A4: t^3/3 + 1/t + c y= dy/dx ( x) .3 .32 .34 (x + x, y + y) (1.1, 2.3) (1.2, 2.32) (1.3, 2.03) Q5: use the indicated substitution to evaluate the integral. A5: du = 3dx sin u du/3 1/3 –cos 3x + c Q6: use substitution to evaluate the integral A6: u = 1-x du= -1 -du=dx -du/u^2 u^-2 du u^-1 + c 1/1-x +c Find the indefinite integral. 1. Q: A: 2. Q: A: 3. Q: A: Q1: Find the amount of time required for a $2000 investment to double if the annual interest rate r is compounded a) annually b) monthly, c) Quarterly and d) continuously Q2: An isotope of neptunium has a half-life of 65 minutes. If the decay of this is modeled by the differential equation dy/dt= -ky, where t is measured in minutes, what is the decay constant k? A2: Q3: The temperature of silver is 60 degrees above room temperature right now. Twenty minutes ago, it was 70 degrees above room temperature. When will the silver be 10 degrees above room temp? Q4: The charcoal from a tree killed in the volcanic eruption that formed crater lake in Oregon contained 44.5% of carbon-14 found in living matter. About how old is crater lake? A4: Question 1: A 2000 gallon tank can support no more than 150 guppies. Six guppies are introduced into the tank. Assume that the rate of growth of the population is: 𝑑𝑃 = 0.0015𝑃 (150 − 𝑃) 𝑑𝑡 a) Find the formula for the guppy population in terms of t. b) How long will it take for the guppy population to be 125? Part a: Final Answer (a): Part b: Final Solution (b): Question 2: Bismuth-210 has a half-life of 5.0 days. a. Suppose a sample originally has a mass of 800 mg. Find a formula for the mass remaining after t days. b. Find the mass remaining after 30 days. c. When is the mass reduced to 1 mg. Part a.) Question 3. Question 4: