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C hapter 3. Limits and Their Properties. Section 3.1. A Preview of Calculus. What is Calculus. Calculus is the mathematics of change---velocities and accelerations - PowerPoint PPT Presentation
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Chapter 3Limits and Their Properties
Section 3.1
A Preview of Calculus
What is CalculusCalculus is the mathematics of change---velocities and accelerations
Calculus is also the mathematics of tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variety of other concepts that have enabled scientists, engineers, and economists to model real-life situations
The Difference Between Precalculus and Calculus
Precalculus deals with velocities, acceleration, tangent lines, slopes, and so on, there is a fundamental difference between precalculus.
Precalculus is more statics, whereas calculus is more dynamic
Precalculus vs. Calculus
Precalculus Calculus An object traveling at a
constant velocity can be analyzed with precalculus
The slope of a line can be analyzed with precalculus
A tangent line to a circle can be analyzed with precalculus
The area of a rectangle can be analyzed with precalculus
To analyze the velocity of an accelerating object, you need calculus
To analyze the slope of a curve you need calculus
To analyze a tangent line to a general graph, you need calculus
To analyze the area under a general curve, you need calculus
The Tangent Line ProblemGiven a function f and a point P on its
graph, you are asked to find an equation of the tangent line to the graph at point P
Excluding a vertical tangent line, the problem is equivalent to finding the slope of the tangent line at P
You can approximate this slope by using a line through the point of tangency and a second point on the curve. Such a line is called a secant line
The Tangent Line Problem Cont’dIf P(c, f(c)) is the point of tangency and
Q(c + ∆x, f(c + ∆x) is a second point on the graph of f
The slope of the secant line through these two points ismsec = (f(c + ∆x) - f(c)) / (c + ∆x – c) or
msec = (f(c + ∆x) - f(c)) / (∆x)
The Tangent Line Problem Cont’dAs point Q approaches point P, the slope of
the secant line approaches the slope of the tangent line. When such a “limiting position” exists, the slope of the tangent line is said to be the limit of the slope of the secant line.
EXAMPLEThe following points lie on the graph of f(x) =
x2
Q1(1.5, f(1.5), Q2(1.1,f(1.1), Q3,(1.01,f(1.01),Q4(1.001, f(1.001) and Q5(1.0001, f(1.0001)
Estimate the slope of the tangent line of f at point P
The Area ProblemFind the area of a plane region bounded by the graphs of
functions. This problem can be solved with a limit process. In this case, the limit process is applied to the area of a rectangle to find the area of a general region.
Consider the region bounded by the graph of the function y = f(x), the x-axis, and the vertical lines x = a and x= b
You can approximate the area of the region with several rectangular regions, the more rectangles, the better the approximation
Your goal is to determine the limit of the sum of the areas of the rectangles as the number of rectangles increase without bound.
The Area Problem - Example
Consider the region bounded by the graphs of f(x) = x2, y = 0, and x = 1
First inscribe a set of rectangles, then circumscribe a set of rectangles.
Find the sum of the areas of each set of rectangles
Use your results to approximate the area of the region
Section 3.2
Finding Limits Graphically and Numerically
An Introduction to Limits
Let f(x) = (x3 – 1)/(x – 1), however x ≠ 1Since you don’t know the behavior of the graph
at x = 1, use two sets of values, one set that approaches 1 from the left and one set that approaches 1 from the right. Make a table.
The graph of f is a parabola with a gapAlthough x cannot equal 1, you can move x
arbitrarily close to 1, and as a result f(x) moves arbitrarily close to ????
An Introduction to Limits – Cont’d
Using limit notation you can write:
lim f(x) = 3x→1
Therefore, if f(x) becomes arbitrarily close to a single L as x approaches c from either side, the limit of f(x), as x approaches c, is L and is writtenlim f(x) = L
x→c
Example
Estimate the limit of each function
1. f(x) = x/((x + 1)½ - 1) at 0
2. f(x) = 1/x2 at 0
3. f(x) = x2 at 2
Section 3.3
Evaluating Limits Analytically
Some Basic LimitsLet b and c be real numbers and let n be a
positive integer lim b = b
x→c
lim x = c x→c
lim xn = cn
x→c
Properties of Limits
Let b and c be real numbers and let n be a positive integer, and let f and g be function with the following limits
lim f(x) = L and lim g(x) = K
x→c x→c
1. Scalar multiple: lim [bf(x) = bL x→c
2. Sum or difference: lim [f(x) ± g(x)] = L ± K x→c
3. Product: lim [f(x) g(x)] = LK x→c
4. Quotient: lim [f(x)/g(x)] = L/K provided K ≠ 0 x→c
5. Power: lim [f(x) n)] = Ln
x→c
Example
Find the limit of each polynomial
1. lim (4x2 + 3) x→2
2. lim (x2 + x + 2) x→1
Limits of Polynomial and Rational FunctionsIf p is a polynomial function and c is a
real number, then
lim p(x) = p(c) x→c
If r is a rational function given by r(x) = p(x)/q(x) and c is a real number such that q(c) ≠ 0, then
lim r(x) = r(c) = p(c)/q(c) x→c
Example
Find the limit of the polynomial
lim (x2 + x + 2)/(x + 1)
x→1
The Limit of a Function Involving a Radical
Let n be a positive integer. The following limit is valid for c if n is odd and is valid for c>0 if n is even.
lim n x = n c
x→c
The Limit of a Composite Function
If f and g are functions such that lim g(x) = L and lim f(x) = f(L), x→c
x→L
then lim f(g(x)) = f(lim g(x) = f(L) x→c x→c
Example
Find the limit each function lim (x2 + 4)½ x→0
lim (2x2 -10)3/2
x→3
Functions That Agree at All But One Point
Let c be a real number and let f(x) = g(x) for all x ≠ c in an open interval containing c. If the lim of g(x) as x approaches c exists, then the limit of f(x) also exists and
then lim f(x) = lim g(x) x→c x→c
Example
Find the limit: lim (x3 - 1)/(x -1) Hint: factor x→1
lim [(x + 1)½ - 1]/x Hint: rationalize
x→0
The Squeeze Theorem
If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and if
lim h(x) = L = lim g(x) x→c x→c
then lim f(x) exists and is equal to L. x→c
Section 3.4
Continuity and One-Sided Limits
Definition of ContinuityContinuity at a Point: A function f is continuous
at c if the following three conditions are met.1. f(c) is defined2. lim f(x) exists x→c
3. lim f(x) = f(c) x→c
Continuity on an Open Interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line (-∞, ∞) is everywhere continuous.
Discontinuity
Nonremovable Removable
A discontinuity at c is called nonremovable if f cannot be made continuous by appropriately defining (or redefining f(c)
Example f(x) = 1/x
A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining f(c)
Example f(x) = (x2 -1)/(x -1)
One-Sided Limits and Continuity on a Closed Interval
The limit from the right means that x approaches c from values greater than c
lim f(x) = L x→c +
The limit from the left means that x approaches c from values less than c
lim f(x) = L x→c -
One-sided limits are useful in taking limits of functions involving radicals. For instance, if n is an even integer, lim n x = 0
x→0 +
The Existence of a Limit
Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if
lim f(x) = L and lim f(x) = L x→c - x→c +
Definition of Continuity on a Closed Interval
A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and
lim f(x) = f(a) and lim f(x) = f(b) x→a + x→b-
The function f is continuous from the right at a and continuous from the left at b
Example
Is the function f(x) = (1-x2)½ continuous?
First, determine the domain of f Next use the definition of
Continuity on a Closed Interval
Properties of ContinuityIf b is a real number and f and g are
continuous at x = c, then the following functions are also continuous at c.
1. Scalar multiple: bf2. Sum and difference: f ± g3. Product: fg4. Quotient: f/g, if g(c) ≠ 0
Types of Continuous Function at Every Point in their Domain
If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c.
1. Polynomial functions: p(x) = anxn + an-1xn-1 +….
2. Rational functions: r(x) = p(x)/q(x), q(x) ≠ 0
3. Radical functions: f(x) = n x
Continuity of a Composite Function
If g is continuous at c and f is continuous at g(c), then the composite function given by (f ° g)(x) = f(g(x)) is continuous at c.
Intermediate Value Theorem
If f is continuous on the closed interval [a,b] and k is any number between f(a), and f(b), then there is at least one number c in [a,b] such that f(c) = k
Example
Use the intermediate value theorem to show the polynomial has a zero in the interval [0,1]
f(x) = x3 + 2x - 1
1. First, is the function continuous on the interval
2. Find f(0) and f(1) then compare relationship to determine if there is a zero in the interval or not
Section 3.5
Infinite Limits
Definition of Vertical Asymptote
If f(x) approaches infinity ( or negative infinity) as x approaches c from the right or the left, then the line x =c is a vertical asymptote of the graph of f
Vertical Asymptote
Let f and g be continuous on an open interval containing c. If f(c) ≠ 0, g(c) = 0 and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of the function given by
h(x) = f(x)/g(x)
Has a vertical asymptote at x = c
Example
Find the vertical asymptotes for each function
f(x) = 1/[2(x+1)]
f(x) = (x2 +1)/(x2 – 1)
f(x) = (x2 – 1)/(x – 2)
f(x) = (x2 + 2x – 8)/ (x2 – 4)
Properties of Infinite Limits
Let c and L be real numbers and let f and g be functions such that
lim f(x) = ∞ and lim g(x) = L x→c x→c
1. Sum or difference: lim[f(x) ± g(x)] = ∞ x→c
2. Product: lim[f(x) g(x)] = ∞, L > 0 x→c
lim[f(x) g(x)] = -∞, L < 0 x→c
3. Quotient: lim g(x)/f(x) = 0 x→c
Example
Find the limit of each function
lim (1 + 1/x2) x→0
lim (x2 + 1)/1/(x-1)) x→1-
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