- •Functions
- •The Concept of a Function
- •Trigonometric Functions
- •Inverse Trigonometric Functions
- •Logarithmic, Exponential and Hyperbolic Functions
- •Limits and Continuity
- •Introductory Examples
- •Continuity Examples
- •Linear Function Approximations
- •Limits and Sequences
- •Properties of Continuous Functions
- •The Derivative
- •The Chain Rule
- •Higher Order Derivatives
- •Mathematical Applications
- •Antidifferentiation
- •Linear Second Order Homogeneous Differential Equations
- •Linear Non-Homogeneous Second Order Differential Equations
- •Area Approximation
- •Integration by Substitution
- •Integration by Parts
- •Logarithmic, Exponential and Hyperbolic Functions
- •The Riemann Integral
- •Volumes of Revolution
- •Arc Length and Surface Area
- •Techniques of Integration
- •Integration by formulae
- •Integration by Substitution
- •Integration by Parts
- •Trigonometric Integrals
- •Trigonometric Substitutions
- •Integration by Partial Fractions
- •Fractional Power Substitutions
- •Numerical Integration
- •Integrals over Unbounded Intervals
- •Discontinuities at End Points
- •Improper Integrals
- •Sequences
- •Monotone Sequences
- •Infinite Series
- •Series with Positive Terms
- •Alternating Series
- •Power Series
- •Taylor Polynomials and Series
- •Applications
- •Parabola
- •Ellipse
- •Hyperbola
- •Polar Coordinates
- •Graphs in Polar Coordinates
- •Areas in Polar Coordinates
- •Parametric Equations
364CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES
through rotating the xy-coordinate system through the angle θ. Then the given second degree equation
ax2 + bxy + cy2 + dx + ey + f = 0
becomes
a0x02 + c0y02 + d0x + e0y + f0 = 0
where
a0 = a cos2 θ + b cos θ sin θ + c sin2 θ c0 = a sin2 θ − b sin θ cos θ + c cos2 θ d0 = d cos θ + e sin θ
e0 = −d sin θ + e cos θ f0 = f
Furthermore, the given second degree equation represents
(i)an ellipse, a circle, a point or no graph if b2 − 4ac < 0;
(ii)a hyperbolic or a pair of intersecting lines if b2 − 4ac > 0;
(iii)a parabola, a line, a pair of parallel lines, or else no graph if b2 −4ac = 0.
9.5Polar Coordinates
Definition 9.5.1 Each point P (x, y) in the xy-coordinate plane is assigned the polar coordinates (r, θ) that satisfy the following relations:
x2 + y2 = r2, y = r cos θ, y = r sin θ.
The origin is called the pole and the positive x-axis is called the polar axis. The number r is called the radial coordinate and the angle θ is called the angular coordinates. The polar coordinates of a point are not unique as the rectangular coordinates are. In particular,
(r, θ) ≡ (r, θ + 2nπ) ≡ (−r, θ + (2m + 1)π)
where n and m are any integers. There does exist a unique polar representation (r, θ) if r ≥ 0 and 0 ≤ θ < 2π.
9.6. GRAPHS IN POLAR COORDINATES |
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9.6Graphs in Polar Coordinates
Theorem 9.6.1 A curve in polar coordinates is symmetric about the
(a)x-axis if (r, θ) and (r, −θ) both lie on the curve;
(b)y-axis if (r, θ) and (r, π − θ) both lie on the curve;
(c)origin if (r, θ), (r, θ + π) and (−r, θ) all lie on the curve.
Theorem 9.6.2 Let e be a positive number. Let a fixed point F be called the focus and a fixed line, not passing through the focus, be called a directrix. If P is a point in the plane, let P F stand for the distance between P and the focus F and let P D stand for the distance between P and the directrix. Then the locus of all points P such that P F = eP D is a conic section representing
(a)an ellipse if 0 < e < 1;
(b)a parabola if e = 1;
(c)a hyperbola if e > 1;
The number e is called the eccentricity of the conic.
In particular an equation of the form
ek
r = 1 ± e cos θ
represents a conic with eccentricity e, a focus at the pole (origin), and a directrix perpendicular to the polar axis and k units to the right of the pole, in the case of + sign, and k units to the left of the pole, in the case of − sign.
Also, an equation of the form
ek
r = 1 ± e sin θ
represents a conic with eccentricity e, a focus at the pole, and a directrix parallel to the polar axis and k units above the pole, in the case of + sign, and k units below the pole, in the case of − sign.
366CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES
9.7Areas in Polar Coordinates
Theorem 9.7.1 Let r = f(θ) be a curve in polar coordinates such that f is continuous and nonnegative for all α ≤ θ ≤ β where α ≤ β ≤ 2π + α. Then the area A bounded by the curves r = f(θ), θ = α and θ = β is given by
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Theorem 9.7.2 Let r = f(θ) be a curve in polar coordinates such that f and f0 are continuous for α ≤ θ ≤ β, and there is no overlapping, the arc length L of the curve from θ = α to θ = β is given by
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9.8 Parametric Equations
Definition 9.8.1 A parametrized curve C in the xy-plane has the form
C = {(x, y) : x = f(t), y = g(t), t I}
for some interval I, finite or infinite.
The functions f and g are called the coordinate functions and the variable t is called the parameter.
Theorem 9.8.1 Suppose that x = f(t), y = g(t) are the parametric equations of a curve C. If f0(t) and g0(t) both exist and f0(t) 6= 0, then
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At a point P0(f(t0), g(t0)), the equation of
9.8. PARAMETRIC EQUATIONS |
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(a) the tangent line is
g0(t0)
y − g(t0) = f0(t0) (x − f(t0))
(b) the normal line is
f0(t0)
y − g(t0) = −g0(t0) (x − f(t0))
provided g0(t0) 6= 0 and f0(t0) 6= 0.
Theorem 9.8.2 Let C = {(x, y) : x = f(t), y = g(t), a ≤ t ≤ b} where f0(t) and g0(t) are continuous on [a, b]. Then the arc length L of C is given by
Z b
L = [(f0(t))2 + (g0(t))2]1/2dt
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Theorem 9.8.3 Let C = {(x, y) : x = f(t), y = g(t), a ≤ t ≤ b}, where f0(t) and g0(t) are continuous on [a, b].
(a)If C lies in the upper half plane or the lower half plane and there is no overlapping, then the surface area generated by revolving C around the x-axis is given by
Z b
p
2πg(t) (f0(t))2 + (g0(t))2 dt.
a
(b)If 0 ≤ f(t) on [a, b], (or f(t) ≤ 0 on [a, b]) and there is no overlapping, then the surface area generated by revolving C around the y-axis is
Z b
p
2πf(t) (f0(t))2 + (g0(t))2 dt.
a
368CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES
Definition 9.8.2 Let C = {(x(t), y(t)) : a ≤ t ≤ b} for some interval I. Suppose that x0(t), y0(t), x00(t) and y00(t) are continuous on I.
(a) The arc length s(t) is defined by
Z t
s(t) = [(x0(t))2 + (y0(t))2]1/2dt.
a
(b) The angle of inclination, φ, of the tangent line to the curve C is defined
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(d) The radius of curvature, R, is defined by
R(t) = κ(1t).