Question: Find the area of the region that
Find the area of the region that lies inside the first curve and outside the second curve. 
Transcribed Image Text:
r=1- sin 6, r= 1
> Show that the curve r = sin θ tan θ (called a cissoid of Diocles) has the line x = 1 as a vertical asymptote. Show also that the curve lies entirely within the vertical strip 0 < x < 1. Use these facts to help sketch the cissoid.
> Show that the polar curve r = 4 + 2 sec θ (called a conchoid) has the line x = 2 as a vertical asymptote by showing that limr→±∞ x = 2. Use this fact to help sketch the conchoid.
> The figure shows a graph of r as a function of θ in Cartesian coordinates. Use it to sketch the corresponding polar curve. TA 2- -2-
> The figure shows a graph of r as a function of θ in Cartesian coordinates. Use it to sketch the corresponding polar curve. TA 2+ 1-
> Sketch the curve with the given polar equation. r = 1 + 2 cos (θ/2)
> Sketch the curve with the given polar equation. r = 1 + 2 cos 2θ
> Use a calculator to find the length of the curve correct to four decimal places. r = 4 sin 30
> Use a calculator to find the length of the curve correct to four decimal places. r= 3 sin 20
> Find the exact length of the polar curve. r = θ, o < θ < 2π
> Find the exact length of the polar curve. r = θ2, o < θ < 2π
> Sketch the curve and find the area that it encloses. r = 2 – sin θ
> Find the exact length of the polar curve. r = e2θ, o < θ < 2π
> Find the exact length of the polar curve. r = 3 sin θ, o < θ < π/3
> Use a graph to estimate the values of θ for which the curves r = 3 + sin 5θ and r = 6 sin θ intersect. Then estimate the area that lies inside both curves.
> Sketch the curve with the given polar equation. r = 2 cos 4θ
> Find all points of intersection of the given curves. r2 = sin 2θ, r2 = cos 2θ
> Find all points of intersection of the given curves. r = sin θ, r = 2θ
> Find all points of intersection of the given curves. r = cos 3θ, r = 3θ
> Find all points of intersection of the given curves. r = 2 sin 2θ, r = 1
> Sketch the curve with the given polar equation. r = 1 – 3 cos θ
> Sketch the curve with the given polar equation. r = 2 (1- sin θ), θ > 0
> Evaluate the integral. f x – 9/(x + 5) (x – 2), dx
> Sketch the curve with the given polar equation. r = -3 cos θ
> Find the area of the region that lies inside both curves. r= sin 20, r = cos 20
> Find the area of the region that lies inside both curves. r = 1+ cos 0, r=1- cos e %3D
> Find the area of the region that lies inside both curves. r= V3 cos 0, r= sin e
> If u (x) = f (x) + ig (x) is a complex-valued function of a real variable x and the real and imaginary parts f (x) and g (x) are differentiable functions of x, then the derivative of u is defined to be u'(x) = f'(x) + ig'(x). Use this together with Equat
> Find the area of the region that lies inside the first curve and outside the second curve. r= 3 sin 0, r= 2 - sin e
> Use Euler’s formula to prove the following formulas for cos x and sin x: eir + e-ir cos x eir – e-ir sin x 2 2i
> Use De Moivre’s Theorem with n = 3 to express cos 3θ and sin 3θ in terms of cos θ and sin θ.
> Write the number in the form a + bi. e π+i
> Write the number in the form a + bi. e 2 + iπ
> Write the number in the form a + bi. e -iπ
> Write the number in the form a + bi. e iπ/3
> Write the number in the form a + bi. e 2πi
> Write the number in the form a + bi. e iπ/2
> Find the indicated roots. Sketch the roots in the complex plane. The cube roots of 1 + i
> Find the indicated roots. Sketch the roots in the complex plane. The cube roots of i
> Find the area of the region that lies inside the first curve and outside the second curve. r= 3 cos 0, r=1+ cos e
> Find the indicated roots. Sketch the roots in the complex plane. The fifth roots of 32
> Find the indicated roots. Sketch the roots in the complex plane. The eighth roots of 1
> Find the indicated power using De Moivre’s Theorem. (1 – i)8
> Find the indicated power using De Moivre’s Theorem. (2√3 + 2 i)5
> Find the indicated power using De Moivre’s Theorem. (1 – √3 i)5
> Find the indicated power using De Moivre’s Theorem. (1 + i)20
> Find polar forms for zw, z/w, and 1/z by first putting z and w into polar form. = 4(/3 + i), w = -3 – 3i
> Find polar forms for zw, z/w, and 1/z by first putting z and w into polar form. z = 2/3 – 2i, w = -1 +i
> Find polar forms for zw, z/w, and 1/z by first putting z and w into polar form. z = 4/3 – 4i, w = 8i %3D %3D
> Find polar forms for zw, z/w, and 1/z by first putting z and w into polar form. z = 3 + i, w = 1 + v3i
> Write the number in polar form with argument between 0 and 2π. 8i
> Write the number in polar form with argument between 0 and 2π. 3 + 4i
> Write the number in polar form with argument between 0 and 2π. 1 – √3 i
> Write the number in polar form with argument between 0 and 2π. -3 + 3i
> Find all solutions of the equation. z2 + 1/2 z + 1/4 = 0
> Find all solutions of the equation. z2 + x + 2 = 0
> Find all solutions of the equation. 2x2 – 2x + 1 = 0
> Find all solutions of the equation. x2 + 2x + 5 = 0
> Find all solutions of the equation. x4 = 1
> Find all solutions of the equation. 4x2 + 9 = 0
> Find the area of the region that lies inside the first curve and outside the second curve. r= 2 cos e, r = 1
> Prove the following properties of complex numbers. (a) z + w = 7 + w (b) zw m = mz (q) (c) z* = 7", where n is a positive integer [Hint: Write z = a + bi, w = c + di.]
> Find the complex conjugate and the modulus of the number. -4i
> Find the complex conjugate and the modulus of the number. -1 + 2/2 i
> Find the complex conjugate and the modulus of the number. 12 – 15i
> Evaluate the expression and write your answer in the form a + bi. V-3V-12
> Evaluate the expression and write your answer in the form a + bi. V-25
> Evaluate the expression and write your answer in the form a + bi. i100
> Evaluate the expression and write your answer in the form a + bi. i3
> Evaluate the expression and write your answer in the form a + bi. 3 4 — Зі 3.
> Evaluate the expression and write your answer in the form a + bi. 1 1+ i
> Find the area of the region enclosed by one loop of the curve. r= 2 cos e - sec e
> Evaluate the expression and write your answer in the form a + bi. 3 + 2i 1- 4i
> Evaluate the expression and write your answer in the form a + bi. 1+ 4i 3 + 2i
> Evaluate the expression and write your answer in the form a + bi. 21(를 - 1)
> Evaluate the expression and write your answer in the form a + bi. 12 + 7i
> Evaluate the expression and write your answer in the form a + bi. (1 — 21)(8 — 3і)
> Evaluate the expression and write your answer in the form a + bi. (2 + 5i)(4 – i)
> Evaluate the expression and write your answer in the form a + bi. (4 – 41) – (9 + 31)
> Evaluate the expression and write your answer in the form a + bi. (5 – 6i) + (3 + 2i)
> Evaluate the indefinite integral as an infinite series. f ex – 1/x, dx
> Evaluate the indefinite integral as an infinite series. f x cos (x3) dx
> Find the area of the region enclosed by one loop of the curve. r=1+ 2 sin e (inner loop)
> For the limit illustrate Definition 1 by finding values of that correspond to e = 0.5 and e = 0.1 e* - 1 lim - 1
> Find a power series representation for the function and determine the interval of convergence. x? f(x) = .3 a - x .3
> (a). Use the binomial series to expand 1/ √1 - x2. (b). Use part (a) to find the Maclaurin series for sin-1 x.
> Use the Maclaurin series for sin x to compute si 30 correct to five decimal places.
> Use the Maclaurin series for ex to calculate e-0.2 correct to five decimal places.
> Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? f(x) = In(1 + x²)
> If the radius of convergence of the power series ∑∞n=0cnxn is 10, what is the radius of convergence of the series ∑∞n=0ncnxn-1? Why?
> Let fn(x) = (sin nx)/n2. Show that the series ∑fn(x) converges for all values of x but the series of derivatives ∑fn'(x) diverges when x = 2nπ, an integer. For what values of x does the series ∑fn"(x) converge?
> Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? f(x) = cos(x²)
> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. sin x if x+0 f(x) = if x = 0
> Find the area of the region enclosed by one loop of the curve. r= 4 sin 30
> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. .2 f(x) = /2 + x
> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = - V4 + x? %3D
> The resistivity of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (Ωm). The resistivity of a given metal depends on the temperature according to the equation where is the temperature i
> An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and -q and are located at a distance d from each other, then the electric field E at the point P in the figure is E = q/D2 –
> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) — е* + 2е- e
> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = e + e2*
> Suppose you know that and the Taylor series of f centered at 4 converges to f (x) for all in the interval of convergence. Show that the fifth degree Taylor polynomial approximates f (5) with error less than 0.0002. (-1)*n! f®(4) 3"(n 1)
