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TQS 124 Autumn 2007 Quinn
Calculus & Analytic Geometry I
Speed, Tangent Lines, and Successive Approximation
Average vs. Instantaneous Speed. According to MapQuest, it should take 3 hours and 42 minutes to travel the 226 mi between the University of Washington, Tacoma and Oregon State University. What would be your average speed on this trip? How does that differ from the information on your speedometer?
speed vs. velocity? Estimating the rate of change.
The average rate of change of a function y = f (x) on an interval [a, b] is given by the change in the output divided by the change in the input:
โy โx
change in output change in input
f (b) โ f (a) b โ a
f (a + h) โ f (a) h if h = b โ a.
What is the average rate of change of the function f (x) = 3x + 2 on the interval [4, 10].
For a linear function, the rate of change (slope of the line) equals the average rate of change. For other functions, an average rate of change on an interval is used to estimate the rate of change of the function at a point inside that interval. The instantaneous rate of change gives the slope of the tangent line to the curve at a point.
Problem. Approximate the average rate of change for the function f (x) = x^2 โ x at the point P (1, 0).
[1, 2] [1, 1 .5] [1, 1 .1]
As the interval of consideration shrinks, the distance between the points of intersection of the curve and the secant line ...
Definition. The slope of a graph at a point is the limit of the slopes seen in a microscope at that point, as the field of view shrinks to zero. (Also called the rate of change of a function at a point.)
Problem. Given the function
f (x) = 2 + x^3 cos(x) + 1. 5 x x + x^2
estimate the rate of change of the function at x = 2.
Given h, calculate (f (2 + h) โ f (2))/h.
Example. h = 0. 1.
f (2.1) โ f (2))
- 1
So what is a LIMIT?
Problem. Find lim zโ 3 +
bzc z
and lim zโ 3 โ
bzc z
Vertical Asymptotes. Can a limit ever be infinite?
Letโs see what happens as we approach the holes in the domain from the left and the right for
g(x) = ex x^2
h(x) = 2 x^2 โ x โ 1 3 x^3 + 2x^2 โ 5 x
Writing lim xโc+^
f (x) = โ does not mean that โ is a number. It does not mean that the limit
exists. It means that the value of f (x) becomes arbitrarily large as x approaches c from the right.
If lim xโc+^
f (x) = โ and lim xโcโ^
f (x) = โ, we will go so far as to say
lim xโc f (x) = โ.
Similar meaning for lim xโc f (x) = โโ
