Linear Approximations 5. The Mean Value Theorem 7 Integration 1. Two examples 2. The Fundamental Theorem of Calculus 3. Some Properties of Integrals 8 Techniques of Integration 1. Substitution 2. Powers of sine and cosine 3. Trigonometric Substitutions 4.
Integration by Parts 5. Rational Functions 6. Numerical Integration 7. Additional exercises 9 Applications of Integration 1. Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9.
Arc Length Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4. Parametric Equations 5. Calculus with Parametric Equations 11 Sequences and Series 1. Sequences 2. Series 3. The Integral Test 4. Alternating Series 5. Comparison Tests 6. Absolute Convergence 7. The Ratio and Root Tests 8. Power Series 9. Calculus with Power Series Taylor Series Taylor's Theorem Additional exercises 12 Three Dimensions 1. The Coordinate System 2. Vectors 3. The Dot Product 4. The Cross Product 5. Lines and Planes 6.
Other Coordinate Systems 13 Vector Functions 1. Space Curves 2. Calculus with vector functions 3. Arc length and curvature 4. Motion along a curve 14 Partial Differentiation 1. Functions of Several Variables 2. Limits and Continuity 3.
Degrees, Turnings, and "Bumps"
Partial Differentiation 4. The Chain Rule 5. Directional Derivatives 6. Higher order derivatives 7. Maxima and minima 8. Lagrange Multipliers 15 Multiple Integration 1. Volume and Average Height 2.
Double Integrals in Cylindrical Coordinates 3. Moment and Center of Mass 4. Surface Area 5. Triple Integrals 6. Cylindrical and Spherical Coordinates 7.
Change of Variables 16 Vector Calculus 1. Vector Fields 2.
This function is illustrated above with shown as the solid curve and as the dashed, with. Care must be taken to make sure that the correct branch of the inverse tangent function is used. As can be seen, weaves back and forth around , with crossings occurring at multiples of.
How Do We Know it is a Maximum (or Minimum)?
The curvature and tangential angle of the ellipse are given by. The entire perimeter of the ellipse is given by setting corresponding to , which is equivalent to four times the length of one of the ellipse's quadrants ,. The perimeter can be computed using the rapidly converging Gauss-Kummer series as. The error surfaces are illustrated above for these functions. The maximum and minimum distances from the focus are called the apoapsis and periapsis , and are given by. The area of an ellipse may be found by direct integration. The area can also be computed more simply by making the change of coordinates and from the elliptical region to the new region.
Then the equation becomes. The area of an arbitrary ellipse given by the quadratic equation. The area of an ellipse with semiaxes and with respect to a pedal point is. A sequence of normal and tangent vectors are plotted above for the ellipse.
Application of Maxima and Minima | Differential Calculus Review
The locus of the apex of a variable cone containing an ellipse fixed in three-space is a hyperbola through the foci of the ellipse. In addition, the locus of the apex of a cone containing that hyperbola is the original ellipse. Furthermore, the eccentricities of the ellipse and hyperbola are reciprocals. The locus of centers of a Pappus chain of circles is an ellipse. Surprisingly, the locus of the end of a garage door mounted on rollers along a vertical track but extending beyond the track is a quadrant of an ellipse Wells , p.
The envelope of the door's positions is an astroid. Abbott, P. Berger, M. Problems in Geometry.
- Maximum/Minimum Problems.
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- Anisotropic Elasticity: Theory and Applications (Oxford Engineering Science Series).
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New York: Springer-Verlag, Beyer, W. Brown, H. Five Hundred and Seven Mechanical Movements.
Related Calculus 2c-4, Examples of Maximum and Minimum
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