Transcendental Calculus Chapter 2
Explain what a Continuous Function is. (2.5)
A function "f" is continuous at a number "a" if lim x -> a f(x) = a Tip: Draw the function line without removing pen from paper. The limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a.
What coordinate is the domain and range referring to? x-coordinate y-coordinate
Domain = x-coordinate or x values Range = y-coordinate
What are the requirements for f to be discontinuous at an particular value e.g. "a". (2.5)
If f is defined near "a" (in other words, f is defined on an open interval containing "a", except perhaps at "a"), we say that f is discontinuous at "a" (or f has a discontinuity at "a") if f is not continuous at a.
How do you solve the slope of a tangent line? (objective & two steps) (2.5)
The slope of a tangent is the limit of the slope of the secant line. First solve the secant line to solve for the limit of the line aka tangent slope. Step 1: Plot a point (Q) on the curve near the tangent point (P). These two points creates a secant line. Use the slope formula to solve for the slope of the secant line. m(PQ) = (Qy-Py)/(Qx-Px) Step 2: Find the limit of the secant line as point Q approaches point P around the curve. If Q is 1, find values Qx from both the left and right approaching 1. x‐ values: 0, 0.5, 0.9, 0.99, 0.999 mQP‐: 1-1.999 x+ values: 2, 1.5, 1.01, 1.001, 1.0001 mPQ+: 3-2.0001 Approaches 2. Slope of Tangent Line is 2.
What is a tangent? (2.5)
The word tangent is derived from the Latin word tangens, which means "touching." Thus a tangent to a curve is a line that touches the curve.
What are the three requirements for f to be continuous at an particular value e.g. "a". (2.5)
lim f(x) = a x -> a Requires three things if f is continuous at "a": 1. f(a) is defined (that is, "a" is in the domain of f). Essentially, "a" cannot be a point discontinuity or open point 2. The lim f(x) exists. x -> a 3. The lim f(x) = a x -> a