How This Page Works CLICK ON AN IMAGE TO SEE AN ENLARGEMENT. In each colored computation box, enter the required info, press/click the gray button, see the resultant computation. Enter negatives as "x" rather than " x" on this page. Do NOT write input x values as fractions. Write fractions as good decimal approximations. Sometimes questions are given. If possible, answer the question then, with the mouse, swipe the page between the two stars to see the answer to the question. To find the "area under a curve," from the point (a,0), draw a perpendicular to the point (a, f(a)) on the curve. One may need to draw it "up" to the curve or "down" to the curve. Repeat this procedure for b. Shade the area between x=a and x=b which is between the curve and the xaxis. This is the area needed. If instead the area between two curves (say f(x) and g(x)), from a to b, is needed, draw the perpendiculars x=a and x=b. Shade the area between the curves. For additional information see links at the bottom of this page. Enjoy. 
Linear Functions f(x) = mx + b,
(y  y_{1}) = m(x  x_{1}), ...

Write the Equation for a Tangent to a Curve Use the notes and problems and calculators of Linear Functions, on this page, if you need a review or help. A definition of tangent is provided and an example is shown below.

Exponential Functions
f(x) = P_{0}b^{kx}
1. The equation P(x) = 10(2)^{4x} is an exponential model.
2. The equation P(x) = 100(3)^{2x} is an exponential model.

Compounding  Growing Discretely, P(t) = P_{0}(1 + r/n)^{nt}
Until now there has been no emphasis on the difference between discrete and continuous. In the formula P_{0}(1 + r/n)^{nt}, some variables are discrete and some are continuous variables. Their classification depends on their use. It is discussed here because some people find it diffcult to choose which formula to choose  P_{0}b^{kx} or P_{0}(1 + r/n)^{nt}. Perhaps this is because they chose to memorize the formulas rather than understand them. In short, continuous, as in P_{0}b^{kx}, means all numbers are used. Discrete means only certain numbers are used. P_{0}b^{kx} is used for continuous growth. Trees, fish, populations of bacteria or humans grow all the time, unless some external influence exists. A classic example of this is the way rabbit populations continue to multiply (continuously) unless the fox population begins eating them and prohibits their continuous growth. This is described by a logistic model, but that is another story. P_{0}(1 + r/n)^{nt} is used for discrete growth. Money left in a bank account to grow larger is an example. Interest is paid every once and a while. The more times a year interest is compounded, the more money is earned, up to a point. This is true even though the interest rate each period is smaller. Semiannualy (n=2, twice a year), monthly (n=12, 12 times a year), quarterly (n=4, every 3 months, every quarter) are often used payment intervals. Notice, not every number is used. One can't be paid 1.5 times a year. Use the spread sheet exp.xls to compute or solve all sorts of stuff. Even cash loans on most credit cards only chage interest on a daily (n=365) basis. I think most credit cards charge interest on purchase charge interest on a monthly basis. Rate, r, is continuous. It is the percent of the principal that is paid each time interest is paid, each period.
Sample rates if n>1, interest compounded during the year.

Growth, Decay, Loans, Savings
1. In 2000, P(x) = 7,000(1 + 1.5/100)^{x}, was first used to describe the population growth, in years, of Keyport, NJ.
2. The number e is the limit, as n goes to infinity, of (1 + 1/n)^{n}  as n gets infinitely large. It is approximately 2.718281828459045...
3. Halflife means the length of time for half of the substance to decay/be changed.
Radioactive materials decay. Each has its own halflife. For example, Nobelium has a really short halflife of 23 seconds.
4. ^{226}Ra has a halflife of 1,620 years and becomes ^{222}Ra upon decay. Algebraically find its rate of decay. See source at: Isotopes of radium. *k is about .00043. * See work. 5. Which is the better interest? Rate A is 5% nominal interest compounded monthly or
Rate B which is 4% nominal interest compounded monthly? Why is it better?

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