1. Solve the following differential equations with the initial conditions: a) y=4e2+6x; y(0) = 4 b)...

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1. Solve the following differential equations with the initial conditions: a) y=4e2+6x; y(0) = 4 b) y' 4 sin 2x+2x; y(0) = 2022 c) y' = ln x; y(1) = 2023 2. The rate at which the risk of Down syndrome is changing is approximated by the function r'(x) = 0.004641x2 -0.3012x + 4.9 (20 x 45) where r'(x) is measured in percentage of all births/year and x is the maternal age at delivery. a) Find a function f giving the risk as a percentage of all births when the maternal age at delivery is x years, given that the risk of Down syndrome at age 30 is 0.14% of all births. b) Based on this model, what is the risk of Down syndrome when the maternal age at delivery is 25 years? c) What is the estimated average of the risk of Down syndrome when the maternal age at delivery is between 25 and 35 years? 1 Hint: The average value (AV) of f over [a, b] is AV b f(x) dx 3. Find the area of the region enclosed by the following graphs. a) y = x-4x-8, y = = x +8 b) xy=6, x+y=7 c) y=x+1, y=x+7, y = 1 x -x- 4 3 1. Solve the following differential equations with the initial conditions: a) y=4e2+6x; y(0) = 4 b) y' 4 sin 2x+2x; y(0) = 2022 c) y' = ln x; y(1) = 2023 2. The rate at which the risk of Down syndrome is changing is approximated by the function r'(x) = 0.004641x2 -0.3012x + 4.9 (20 x 45) where r'(x) is measured in percentage of all births/year and x is the maternal age at delivery. a) Find a function f giving the risk as a percentage of all births when the maternal age at delivery is x years, given that the risk of Down syndrome at age 30 is 0.14% of all births. b) Based on this model, what is the risk of Down syndrome when the maternal age at delivery is 25 years? c) What is the estimated average of the risk of Down syndrome when the maternal age at delivery is between 25 and 35 years? 1 Hint: The average value (AV) of f over [a, b] is AV b f(x) dx 3. Find the area of the region enclosed by the following graphs. a) y = x-4x-8, y = = x +8 b) xy=6, x+y=7 c) y=x+1, y=x+7, y = 1 x -x- 4 3