Write the number in polar form with argument between 0 and 2π. -3 + 3i
> 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
> Find the area of the region that lies inside the first curve and outside the second curve. r=1- sin 6, r= 1
> 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
> 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)
> Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x) = sin 7x
> Use the binomial series to expand the function as a power series. State the radius of convergence. (1 – x)-/3
> Use the Alternating Series Estimation Theorem or Taylor’s Inequality to estimate the range of values of for which the given approximation is accurate to within the stated error. Check your answer graphically. x (| error |< 6 :0.01)
> Find the area of the region enclosed by one loop of the curve. r= sin 20
