- •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
360 |
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CHAPTER 8. INFINITE SERIES |
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Part 9. For each |x| ≤ 1, we perform term-by-term integration to get |
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arctan x = Z0 |
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0 k=0
∞
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=(−1)k
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x2k+1
(2k + 1).
Part 10. By performing term-by-term integration of the binomial series, we get
arcsin x = Z0x |
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Zx
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This series converges for all |x| ≤ 1.
This completes the proof of this theorem.
8.8Applications
Chapter 9
Analytic Geometry and Polar
Coordinates
A double right-circular cone is obtained by rotating a line about a fixed axis such that the line intersects the axis and makes the same angle with the axis. The intersection point of the line and the axis is called a vertex. A conic section is the intersection of a plane and the double cone. Some of the important conic sections are the following: parabola, circle, ellipse and a hyperbola.
9.1Parabola
Definition 9.1.1 A parabola is the set of all points in the plane that are equidistant from a given point, called the focus, and a given line called the directrix. A line that passes through the focus and is perpendicular to the directrix is called the axis of the parabola. The intersection of the axis with the parabola is called the vertex.
Theorem 9.1.1 Suppose that v(h, k) is the vertex and the line x = h − p is the directrix of a parabola. Then the focus is F (h + p, k) and the axis is the horizontal line with equation y = k. The equation of the parabola is
(y − k)2 = 4p(x − h).
361
362CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES
Theorem 9.1.2 Suppose that v(h, k) is the vertex and the line y = k − p is the directrix of a parabola. Then the focus is F (h, k + p) and the axis is the vertical line with equation x = h. The equation of the parabola is
(x − h)2 = 4p(y − k).
9.2Ellipse
Definition 9.2.1 An ellipse is the locus of all points, the sum of whose distances from two fixed points, called foci, is a fixed positive constant that is greater than the distance between the foci. The midpoint of the line segment joining the two foci is called the center. The line segment through the foci and with end points on the ellipse is called the major axis. The line segment, through the center, that has end points on the ellipse and is perpendicular to the major axis is called the minor axis. The intersections of the major and minor axes with the ellipse are called the vertices.
Theorem 9.2.1 Let an ellipse have center at (h, k), foci at (h ± c, k), ends
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a, k |
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and ends of the minor axis at (h, k |
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of the major axis at ( |
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a > 0, b > 0, c > 0 and a |
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(x − h)2 |
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The length of the major axis is 2a and the length of the minor axis is 2b.
Theorem 9.2.2 Let an ellipse have center at (h, k), foci at (h, k ± c), ends
h, k |
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a and the ends of the minor axis at (h |
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b, k), |
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of the major axis at ( |
± ), 2 |
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where a > 0, b > 0, c > 0 and a |
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is |
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(x − h)2 |
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The length of the major axis is 2a and the length of the minor axis is 2b.
Remark 24 If c = 0, then a = b, foci coincide with the center and the ellipse reduces to a circle.
9.3. HYPERBOLA |
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9.3Hyperbola
Definition 9.3.1 A hyperbola is the locus of all points, the di erence of whose distances from two fixed points, called foci, is a fixed positive constant that is less than the distance between the foci. The mid point of the line segment joining the two foci is called the center. The line segment, through the foci, and with end points on the hyperbola is called the major axis. The end points of the major axis are called the vertices.
Theorem 9.3.1 Let a hyperbola |
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(h, k), |
foci at (h ± c, k), |
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vertices at (h ± a, k), where 0 < a < c, |
b = |
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c2 − a2 |
, then the equation of |
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the hyperbola is |
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(x − h)2 |
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(y − k)2 |
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Theorem 9.3.2 Let a hyperbola |
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foci at (h, k ± c), |
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vertices at (h, k ± a), where 0 < a < c, |
b = |
√ |
c2 − a2 |
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the hyperbola is |
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(y − k)2 |
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9.4Second-Degree Equations
Definition 9.4.1 The transformations
x = x0 cos θ − y0 sin θ y = x0 sin θ + y0 cos θ
and
x0 = x cos θ + y sin θ y0 = −x sin θ + y cos θ
are called rotations. The point P (x, y) has coordinates (x0, y0) in an x0y0- coordinate system obtained by rotating the xy-coordinate system by an angle
θ.
Theorem 9.4.1 Consider the equation ax2 + bxy + cy2 + dx + ey + f = 0, b 6= 0. Let cot 2θ = (a − c)/b and x0y0-coordinate system be obtained