Solutions for Multivariable Calculus
Problem 1E:
Find dy/dx. 1. x=t1+t,y=1+tProblem 2E:
Find dy/dx. 2. x = tet, y = t + sin tProblem 3E:
Find an equation of the tangent to the curve at the point corresponding to the given value of the...Problem 4E:
Find an equation of the tangent to the curve at the point corresponding to the given value of the...Problem 5E:
Find an equation of the tangent to the curve at the point corresponding to the given value of the...Problem 6E:
Find an equation of the tangent to the curve at the point corresponding to the given value of the...Problem 7E:
Find an equation of the tangent to the curve at the given point by two methods: (a) without...Problem 12E:
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 12. x = t3 + 1, y = t2 tProblem 13E:
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 13. x = et, y = tetProblem 15E:
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 15. x = t ln t, y = t +...Problem 16E:
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? 16. x = cos t, y = sin...Problem 17E:
Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing...Problem 19E:
Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing...Problem 22E:
Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve x =...Problem 24E:
Graph the curve in a viewing rectangle that displays all the important aspects of the curve. 24. x =...Problem 26E:
Graph the curve x = 2 cos t, y = sin t + sin 2t to discover where it crosses itself. Then find...Problem 27E:
(a) Find the slope of the tangent line to the trochoid x = r d sin , y = r d cos in terms of ....Problem 28E:
(a) Find the slope of the tangent to the astroid x = a cos3, y = a sin3 in terms of . (Astroids are...Problem 29E:
At what point(s) on the curve x = 3t2 + 1, y = t3 1 does the tangent line have slope 12?Problem 30E:
Find equations of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 that pass through the point (4,...Problem 31E:
Use the parametric equations of an ellipse, x = a cos , y = b sin , 0 2, to find the area that it...Problem 34E:
Find the area of the region enclosed by the astroid x = a cos3, y = a sin3. (Astroids are explored...Problem 36E:
Let R be the region enclosed by the loop of the curve in Example 1. (a) Find the area of R. (b) If R...Problem 38E:
Set up an integral that represents the length of the curve. Then use your calculator to find the...Problem 49E:
Use Simpsons Rule with n = 6 to estimate the length of the curve x = t et, y = t + et, 6 t 6.Problem 50E:
In Exercise 10.1.43 you were asked to derive the parametric equations x = 2a cot , y = 2a sin2 for...Problem 51E:
Find the distance traveled by a particle with position (x, y) as t varies in the given time...Problem 52E:
Find the distance traveled by a particle with position (x, y) as t varies in the given time...Problem 55E:
(a) Graph the epitrochoid with equations x=11cost4cos(11t/2)y=11sint4sin(11t/2) What parameter...Problem 63E:
Find the exact area of the surface obtained by rotating the given curve about the x-axis. 63. x = a...Problem 65E:
Find the surface area generated by rotating the given curve about the y-axis. 65. x = 3t2, y = 2t3,...Problem 66E:
Find the surface area generated by rotating the given curve about the y-axis. 66. x = et t, y =...Problem 67E:
If f is continuous and f(t) 0 for a t b, show that the parametric curve x = f(t), y = g(t), a t ...Problem 69E:
The curvature at a point P of a curve is defined as =|dds| where is the angle of inclination of the...Problem 70E:
(a) Use the formula in Exercise 69(b) to find the curvature of the parabola y = x2 at the point (1,...Problem 71E:
Use the formula in Exercise 69(a) to find the curvature of the cycloid x = sin , y = 1 cos at...Problem 72E:
(a) Show that the curvature at each point of a straight line is = 0. (b) Show that the curvature at...Browse All Chapters of This Textbook
Chapter 10 - Parametric Equations And Polar CoordinatesChapter 10.1 - Curves Defined By Parametric EquationsChapter 10.2 - Calculus With Parametric CurvesChapter 10.3 - Polar CoordinatesChapter 10.4 - Areas And Lengths In Polar CoordinatesChapter 10.5 - Conic SectionsChapter 10.6 - Conic Sections In Polar CoordinatesChapter 11 - Infinite Sequences And SeriesChapter 11.1 - SequencesChapter 11.2 - Series
Chapter 11.3 - The Integral Test And Estimates Of SumsChapter 11.4 - The Comparison TestsChapter 11.5 - Alternating SeriesChapter 11.6 - Absolute Convergence And The Ratio And Root TestsChapter 11.7 - Strategy For Testing SeriesChapter 11.8 - Power SeriesChapter 11.9 - Representations Of Functions As Power SeriesChapter 11.10 - Taylor And Maclaurin SeriesChapter 11.11 - Applications Of Taylor PolynomialsChapter 12 - Vectors And The Geometry Of SpaceChapter 12.1 - Three-dimensional Coordinate SystemsChapter 12.2 - VectorsChapter 12.3 - The Dot ProductChapter 12.4 - The Cross ProductChapter 12.5 - Equations Of Lines And PlanesChapter 12.6 - Cylinders And Quadric SurfacesChapter 13 - Vector FunctionsChapter 13.1 - Vector Functions And Space CurvesChapter 13.2 - Derivatives And Integrals Of Vector FunctionsChapter 13.3 - Arc Length And CurvatureChapter 13.4 - Motion In Space: Velocity And AccelerationChapter 14 - Partial DerivativesChapter 14.1 - Functions Of Several VariablesChapter 14.2 - Limits And ContinuityChapter 14.3 - Partial DerivativesChapter 14.4 - Tangent Planes And Linear ApproximationsChapter 14.5 - The Chain RuleChapter 14.6 - Directional Derivatives And The Gradient VectorChapter 14.7 - Maximum And Minimum ValuesChapter 14.8 - Lagrange MultipliersChapter 15 - Multiple IntegralsChapter 15.1 - Double Integrals Over RectanglesChapter 15.2 - Double Integrals Over General RegionsChapter 15.3 - Double Integrals In Polar CoordinatesChapter 15.4 - Applications Of Double IntegralsChapter 15.5 - Surface AreaChapter 15.6 - Triple IntegralsChapter 15.7 - Triple Integrals In Cylindrical CoordinatesChapter 15.8 - Triple Integrals In Spherical CoordinatesChapter 15.9 - Change Of Variables In Multiple IntegralsChapter 16 - Vector CalculusChapter 16.1 - Vector FieldsChapter 16.2 - Line IntegralsChapter 16.3 - The Fundamental Theorem For Line IntegralsChapter 16.4 - Green's TheoremChapter 16.5 - Curl And DivergenceChapter 16.6 - Parametric Surfaces And Their AreasChapter 16.7 - Surface IntegralsChapter 16.8 - Stokes' TheoremChapter 16.9 - The Divergence TheoremChapter 17 - Second-order Differential EquationsChapter 17.1 - Second-order Linear EquationsChapter 17.2 - Nonhomogeneous Linear EquationsChapter 17.3 - Applications Of Second-order Differential EquationsChapter 17.4 - Series SolutionsChapter G - Complex Numbers
Book Details
Success in your calculus course starts here! James Stewart's CALCULUS texts are world-wide best-sellers for a reason: they are clear, accurate, and filled with relevant, real-world examples. With MULTIVARIABLE CALCULUS, Eighth Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an appreciation for the intrinsic beauty of the subject. His patient examples and built-in learning aids will help you build your mathematical confidence and achieve your goals in the course.
Sample Solutions for this Textbook
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