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6.15 Find the points of intersection of the parabola y = x2 + x 10 with the circle x2 + y2 = 25.
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First, plot the two graphs. g1 = Graphics[Circle[{0, 0}, 5], Axes True]; g2 = Plot[x2 + x 10, {x, 5, 5}]; Show[g1, g2, AspectRatio Automatic, PlotRange { 10, 10}]
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The parabola y = x2 + x + 1 intersects the circle x2 + y2 = 25 at four points. Now solve for the intersection points. Because of the complicated structure of the exact solution, we obtain a numerical approximation. NSolve[y x2 + x 10 && x2 + y2 25]
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4.63752, x 1.86907}, {y 2.83654, x 4.11753}, {y 4., x 3.}, {y 3.80098, x 3.24846}}
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6.16 Find the points of intersection of the limacon r = 5 4 cos and the parabola y = x2.
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First we plot both curves on the same set of axes. limacon = PolarPlot[5 4 Cos[t], {t, 0, 2 o}]; parabola = Plot[x2, {x, 3, 3}]; Show[limacon, parabola, PlotRange All]
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2 2 4 6
Equations
We convert the equation of the limacon to rectangular coordinates: r = 5 4 cos r 2 = 5r 4r cos x 2 + y2 = 5 x 2 + y2 4 x The first intersection point appears to be near (2, 2): FindRoot [{y x2, x2 + y 2 5
r = x 2 + y2 x = r cos
x2 + y 2 4x}, {x, 2}, {y, 2}]
1.53711, y 2.3627}
The second point lies near ( 3, 6): FindRoot [{y x2, x2 + y 2 5
x2 + y 2 4x}, {x, 3}, {y, 6}]
2.4552, y 6.02802}
6.17 Where does the Spiral of Archimedes, r = q, intersect the ellipse 4 x 2 + 9 y 2 = 400
SOLUTION
spiral = PolarPlot[p, {p, 0, 6 o}]; ellipse = ContourPlot[4 x2 + 9 y2 400, {x, 10, 10}, {y, 20, 20}, ContourStyle Dashing[.02]]; Show[spiral, ellipse]
5 5
The graph shows three points of intersection that appear to be near (4, 6), ( 8, 4), and ( 9, 2). To convert the polar equation to rectangular, we use the transformations r = x 2 + y 2 and = tan 1 ( y / x) . However, Newton s method is more stable if we write this as tan( x 2 + y 2 ) = y / x . FindRoot[{Tan[ x2 + y 2 ] y/x, 4x2 + 9y 2 400}, {x, 4}, {y, 6}] FindRoot[{Tan[ x2 + y 2 ] y/x, 4x2 + 9y 2 400}, {x, 8}, {y, 4}] FindRoot[{Tan[ x2 + y 2 ] y /x, 4x2 + 9y 2 400}, {x, 9}, {y, 2}]
Equations
{x {x {x
3.93476, y 6.1289} 8.04703, y 3.95785} 9.38786, y 2.29668}
6.18 Find a solution of the system of equations x+ y+z =6 sin x + cos y + tan z = 1 ex + y +
near the point (1, 2, 3).
SOLUTION
1 =5 z
FindRoot[{x + y + z 6, Sin[x] + Cos[y] + Tan[z] 1, Exp[x] + Sqrt[y] + 1/z 5}, {x, 1}, {y, 2}, {z, 3}]
1.23382, y 1.5696, z 3.19658}
C HA PTE R 7
Algebra and Trigonometry
7.1 Polynomials
Because they are so prevalent in algebra, Mathematica offers commands that are devoted exclusively to polynomials.
PolynomialQ[expression, variable] yields True if expression is a polynomial in variable, and False otherwise. Variables[polynomial] gives a list of all independent variables in polynomial. Coefficient[polynomial, form] gives the coefficient of form in polynomial. Coefficient[polynomial, form, n] gives the coefficient of form to the nth power in polynomial. CoefficientList[polynomial, variable] gives a list of the coefficients of powers of variable in polynomial, starting with the 0th power.
EXAMPLE 1
PolynomialQ[x2 + 3 x + 2, x] True PolynomialQ[x2 + 3 x + 2/x, x] False PolynomialQ[x2 + 3 x + 2/y, x] True PolynomialQ[x2 + 3 x + 2/y, y] False
EXAMPLE 2
2/y is treated as a constant with respect to x.
poly1 =(x + 1)10; poly2 = x3 5 x2 y + 3 x y2 7 y3; Variables[poly2] {x, y} Coef cient[poly1, x, 5] 252 Coef cient[poly2, x] 3 y2 Coef cient[poly2, y, 2] 3x Coef cient[poly2, x y2] 3 Coef cientList[poly1, x] {1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1}
Algebra and Trigonometry
Coef cientList[poly2, x] { 7 y3, 3 y2, 5 y, 1} Coef cientList[poly2, y] {x3, 5 x2, 3 x, 7}
Often it is convenient to write the solution of a polynomial equation as a logical expression. For example, if x2 4 = 0, then x = 2 or x = 2. Roots of polynomial equations can be expressed in this form using two specialized commands, Roots and NRoots. The solutions are given in disjunctive form separated by the symbol | | (logical or).
Roots[lhs rhs, variable] produces the solutions of a polynomial equation. NRoots[lhs rhs, variable] produces numerical approximations of the solutions of a polynomial equation.
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