Assignment:
Q1. A bacteria population grows at a rate proportional to its size. Initially the population is 10,000 and after 5 days it’s 30,000.
- What is the population after 10 days?
- How long will it take for the population to double?
Q2. A solid S is generated by revolving the finite region bounded by the y-axis, the line y = 8 and the curve y = x^3 about the y-axis. Compute the volume of the solid S.
Q3. Suppose that the “trapezoidal rule” is used to estimate the definite integral ∫10ex dx.
Q4. Write down but do not evaluate the approximation to this integral given by the “trapezoidal rule” with n = 4.
Q5. Recall that the error ET,n satisfies ¦ET,n¦≤ K (b-a)^3 / 12n^2 , where K satisfies ¦f^'' (x)¦ ≤ K for all a ≤x ≤ b. How large do you need to choose n so that the approximation to the above integral by the “trapezoidal rule” is accurate to within 10^-6? [MUST use error formula to do this and show how to obtain the K that you use.]
Q6. Find the general solution for the differential equation y’ = x^4 + x^2 + 1 / y^2
Q7. Solve the initial value problem y’ = x^4 + x^2 + 1 / y^2 , y(0) = 2.
Provide complete and step by step solution for the question and show calculations and use formulas.