2009 Daley Log
Page 16
I am still trying to cement my ideas about exponential and logarithmic functions.
Let me try to make a few notes from memory.
The exponential function is easy to visualize. It is a concave curve that is always positive. This makes sense. Regardless of the base (which is always a positive number), raising the power of that base to any value will still give a positive value. The curve passes through the point (0, 1). This also makes sense. Any number raised to the power of 0 is 1. The curve rises rapidly as x increases to the right of 0. If x is negative then this is equivalent to 1/x^n and the curve rapidly approaches 0, but never actually reaches it.
Note that this curve satisfies the horizontal-line test. That is, any horizontal line only passes through this curve once. That means that the exponential function has an inverse function. This function is called the logarithmic function.
Any inverse function is symmetric about the line y = x to the original function. Therefore the logarithmic function is a curve that is negative and asymptotic to the y-axis as x approaches 0. The curve passes through the point (1, 0) and then increases slowly as x increases.
Now for a few comments about trigonometric functions.
The sine function begins at 0, reaches a maximum of 1 when x is p/2. It then decreases, reaching a value of 0 when x is p. It continues to decrease, reaching a minimum of -1 when x is 3p/2. It then increases to a value of 0 when x is 2p. The cosine has the same pattern but it begins with a value of 1. The tangent is more comlicated. The tangent begins at 0 and rises quickly, reaching an asymptote at x = p/2. If x is negative then the curve decreases quickly reaching another asymptote at -p/2.
The secant is 1/sine, the cosecant is 1/cosine, and the cotangent is 1/tangent. These curves are even more complicated but can be carefully thought through.
The most important trigonometric equality is the Pythagorean Equation: sin^2 (x) + cos^2 (x) = 1.