Question: Write out the form of the partial
Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.
Transcribed Image Text:
(a) (r' + x)(x? – x + 3) 1 (b) x° - x .3
> Use a computer algebra system to find the Taylor polynomials Tn centered at a for n = 2, 3, 4, 5. Then graph these polynomials and f on the same screen. f(x) %3 cot x, a%3D п/4 T/4 a =
> Use a graphing device to graph the polar curve. Choose the parameter interval carefully to make sure that you produce an appropriate curve. r = cos(0/2) + cos(0/3)
> Find a power series representation for the function and determine the interval of convergence. f(x) = 1 + x
> The graph of f is shown. (a). Explain why the series is not the Taylor series of f centered at 1. (b). Explain why the series is not the Taylor series of f centered at 2. yA f 1+ 1 1.6 – 0.8(x – 1) + 0.4(x – 1) – 0.1(x – 1)3 + · .. 2.8 + 0.5(x –
> Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 0<r< 4, -m/2 s0 < m/6 -1/2 < 0 < T/6
> Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. r> 0, 7/3 s0 < 27/3
> Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 1srs2
> The Cartesian coordinates of a point are given. (i). Find polar coordinates (r, θ) of the point, where r > 0 and r (ii). Find polar coordinates (r, θ) of the point, where r (a) (3,/3, 3) (b) (1, –2)
> The Cartesian coordinates of a point are given. (i). Find polar coordinates (r, θ) of the point, where r > 0 and r (ii). Find polar coordinates (r, θ) of the point, where r (а) (2, —2) (b) (-1, 3)
> Sketch the curve with the given polar equation. r2θ = 1
> Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. (a) (-/2, 57/4) (b) (1, 57/2) (c) (2, –77/6)
> Sketch the curve with the given polar equation. r = 2 cos (3θ/2)
> Use a graphing device to graph the polar curve. Choose the parameter interval carefully to make sure that you produce an appropriate curve. r= 2 – 5 sin(6/6)
> Sketch the curve with the given polar equation. r2 = cos 4θ
> Sketch the curve with the given polar equation. r2 = 9 sin 2θ
> Sketch the curve with the given polar equation. r = 2 + sin θ
> Sketch the curve with the given polar equation. r = 1 – 2 sin θ
> Sketch the curve with the given polar equation. r = 3 cos 6θ
> Sketch the curve with the given polar equation. r = cos 5θ
> Sketch the curve with the given polar equation. r = 4 sin 3θ
> Sketch the curve with the given polar equation. r = ln θ, θ > 1
> Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. (а) (1, п) ъ) (2, -2т/3) (с) (-2, 3п/4)
> Sketch the curve with the given polar equation. r = θ, θ > 0
> Use a graphing device to graph the polar curve. Choose the parameter interval carefully to make sure that you produce an appropriate curve. r = | tan e |lcot el (valentine curve)
> Sketch the curve with the given polar equation. r = sin θ
> Sketch the curve with the given polar equation. r2 – 3r + 2 = 0
> Sketch the curve with the given polar equation. θ = -π/6
> For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a). A circle with radius 5 and center (2, 2) (b). A circle centered at the origin with ra
> For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a). A line through the origin that makes an angle of π/6 with the positive x-axis (b). A
> Find a polar equation for the curve represented by the given Cartesian equation. xy = 4
> Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > 0 and one with r (а) (1, 7п/4) (b) (-3, п/6) (с) (1, — 1)
> Find a polar equation for the curve represented by the given Cartesian equation. x2 + y2 = 2cx
> Find a polar equation for the curve represented by the given Cartesian equation. x + y = 9
> Find a polar equation for the curve represented by the given Cartesian equation. x = -y2
> Use a graphing device to graph the polar curve. Choose the parameter interval carefully to make sure that you produce an appropriate curve. r= esin e – 2 cos(40) (butterfly curve)
> Identify the curve by finding a Cartesian equation for the curve. r = tan 0 sec e
> Identify the curve by finding a Cartesian equation for the curve. r = csc e
> Identify the curve by finding a Cartesian equation for the curve. r= 2 sin e + 2 cos e
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> Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. r> 1, 7<es 27
> Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 2 <r< 3, 57/3 < 0 < 7m/3
> Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > 0 and one with r (а) (2, т/3) (b) (1, — Зӕ/4) (с) (-1, п/2)
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> Show that the curves r = a sin θ and r = a cos θ intersect at right angles.
> Evaluate the integral. f x/x – 6, dx
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. x3 > x
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. x2 < 3
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. 4 — Зх в 6
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. 2x + 7>3
> Let P (x1, y1) be a point on the ellipse x2/a2 + y2/b2 = 1 with foci F1 and F2 and let α and β be the angles between the lines PF1, PF2 and the ellipse as shown in the figure. Prove that α = β. This exp
> Sketch the region bounded by the curves. y = 4 – x2 and x – 2y = 2
> Sketch the region bounded by the curves. x + 4y = 8 and x = 2y2 - 8
> (a). Find the foci and asymptotes of the hyperbola x2 – y2 = 1 and sketch its graph. (b). Sketch the graph of y2 – x2 = 1.
> Suppose that P (x, y) is any point on the parabola with focus (0, p) and directrix y = -p. (See Figure 14 (below).) Use the definition of a parabola to show that x2 = 4py. P(x, y) F(0, p) y y=-p
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> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. t* + t? + 1 (a) x* - 1 (b) (t2 + 1)(1? + 4)?
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> Find the limit. limn→∞ ∑n i=1 2/n [ (2i/n)3 + 5 (2i/n)]
> Find the limit. limn→∞ ∑n i=1 1/n [ (i/n)3 + 1]
> Find the limit. limn→∞ ∑n i=1 1/n (i/n)2
> The region under the curve y = 1/x2 + 3x + 2 from x = 0 to x = 1 is rotated about the x-axis. Find the volume of the resulting solid.
> One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. If represents the number of female insects in a popul
> (a). Show that the area of a triangle with sides of lengths a and b and with included angle θ is A = 1/2 ab sin θ (b). Find the area of triangle ABC, correct to five decimal places, if |AB| = 10 cm | BC| = 3 cm ZABC 107°
> Graph both y = 1/ (x3 – 2x2) and an antiderivative on the same screen.
> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 2x + 1 (a) x? + 4x + 3 (b) (x + 1)°(x² + 4)²
> Use a graph of f (x) = 1/ (x2 – 2x – 3) to decide whether f20 f (x) dx is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the exact value.
> Make a substitution to express the integrand as a rational function and then evaluate the integral. cos x dx sin'x + sin x sin*x
> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx e2* + 3e* + 2
> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx 2/x + 3 + x
> Make a substitution to express the integrand as a rational function and then evaluate the integral. x, dx J9 x- 4 16
> Evaluate the integral. f 3x2 + x + 4/ x4 + 3x2 + 2, dx
> Evaluate the integral. f x – 3/ (x2 + 2x + 4)2, dx
> Evaluate the integral. f x4 + 3x2 + 1/x5 + 5x3 + 5x, dx
> (a). If is a complex-valued function of a real variable, its indefinite integral f u (x) dx is an antiderivative of u. Evaluate (b). By considering the real and imaginary parts of the integral in part (a), evaluate the real integrals (c). Compare wit
> Evaluate the integral. f dx/x (x2 + 4)2
> Evaluate the integral. f x3/x3 + 1, dx
> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. x* + 1 (a) x* + 4x 1 (b) (x² – 9)²
> Evaluate the integral. f 1/x3 – 1, dx
> Evaluate the integral. f10 x/x2 + 4x + 13, dx
> Evaluate the integral. f x + 4/x2 + 2x + 5, dx
> Evaluate the integral. f x2 + x + 1/ (x2 + 1)2, dx
> Evaluate the integral. f x3 + x2 + 2x + 1/ (x2 + 1) (x2 + 2), dx
> Evaluate the integral. f x2 – 2x – 1/ (x – 1)2 (x2 + 1), dx
> Evaluate the integral. f 10/ (x – 1) (x2 + 9), dx
> Suppose an ellipse has foci (±c, 0) and the sum of the distances from any point P (x, y) on the ellipse to the foci is 2a. Show that the coordinates of P satisfy Equation 1.
> Use the given graph of f (x) = √x to find a number δ such that if |x – 4|< 8 |V - 2|<0.4 then y4 y= Va 2.4. 2 1.6 4
> Evaluate the integral. f x2 – x + 6/x3 + 3x, dx
> Evaluate the integral. f 5x2 + 3x – 2/x3 + 2x2, dx
> Evaluate the integral. f x2 – 5x + 16/(2x + 1) (x – 2)2 dx
> Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. x? (a) x? + x – 2 (b) x? + x + 2
> Evaluate the integral. f 1/(x + 5)2 (x – 1), dx
> Evaluate the integral. f x2 + 2x – 1/x3 – x, dx
> Evaluate the integral. f11 4y2 – 7y -12/y (y + 2) (y – 3), dy
> Evaluate the integral. f10 x3 – 4x – 10/x2 – x – 6, dx
> Evaluate the integral. f43 x3 – 2x2 - 4/x3 – 2x2, dx
> Evaluate the integral. f 1/(x + a) (x + b), dx
