# Trapezoidal Rule

Please show ALL work!

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?

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.

Suppose that the "trapezoidal rule" is used to estimate the definite integral ∫_0^1▒〖e^x〗 dx.

Write down but do not evaluate the approximation to this integral given by the "trapezoidal rule" with n = 4.

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.]

Find the general solution for the differential equation y' = x^4 + x^2 + 1 / y^2

Solve the initial value problem y' = x^4 + x^2 + 1 / y^2 , y(0) = 2.

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#### Solution Preview

Please show ALL work!

1. 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.

a. What is the population after 10 days?

Solution: We will use the exponential growth formula:

After getting the constant rate, we will get the population value after 10 days:

b. How long will it take for the population to double?

For the population to the ...

#### Solution Summary

The expert examines trapezoidal rules for approximation.