course

You made scores of 73 and 80, respectively, on the two tests.

With your homework taken into account you have a B average at this point and are doing good solid work. If you want to raise your scores you have the option of taking another test on one or both chapters.

Chapter 4 test:

A radioactive decay model is of form y = A e^(k t). Given the t = 0 amount of 100 grams and the fact that the half-life is 30 years you obtain two equations 100 grams = A 3^(k * 0) and 1/2 A = A e^(k * 30 yrs). The former gives you A = 100 grams; the latter gives you e^(k * 30 yrs) = 1/2 so that k = -.7 / (30 yrs) = -.023, approx.. (Alternatively you could start with model y = A e^(-kt) and obtain k = .7, approx.. Either way you end up with y = 100 grams * e^(-.023 t) ).

In an interest rate problem you assumed form A = P e^(r t), then used .10/t as the exponent rt. There is no division in the quantity (r t), and it takes about 7 years to double, not .14 years.

The derivative of e^(7 - 3/x) is (7 - 3/x)' * e^(7 - 3/x) = 3 / x^2 e^(7 - 3/x). You had -3 ln(x) e^(7 - 3/x).

To determine whether a graph has inflection points, consider points where the second derivative is zero, and see whether the first derivative changes sign.

Critical numbers are found by setting the first derivative equal to zero and solving the resulting equation.

Chapter 5 test:

The average value of a function on an interval is equal to its integral divided by the length of the interval. Average profit is therefore obtained by integrating the profit function and dividing by the length of the interval, not by evaluating the function at a series of points and averaging the values (if endpoints are correctly weighted then this does give an approximation, which can be compared to the accurate result obtained using integration).

To find the volume of a solid of revolution you integrate the cross-sectional area from one point on the axis to another.