Question: The resistivity of a conducting wire is
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 in 0C. There are tables that list the values of a (called the temperature coefficient) p20 and (the resistivity at 200C) for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for p (t) by its first- or second-degree Taylor polynomial at t = 20. (a). Find expressions for these linear and quadratic approximations. (b). For copper, the tables give a = 0.0039/0C and p20 = 1.7 × 10-8Ωm. Graph the resistivity of copper and the linear and quadratic approximations for -2.500C > t (c). For what values of does the linear approximation agree with the exponential expression to within one percent?
Transcribed Image Text:
p(t) = pneal-20)
> 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
> 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
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> 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
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> 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
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> 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
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> 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
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> (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
> 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
> Use the binomial series to expand the function as a power series. State the radius of convergence. 1 (1 + x)* 4
> Use the binomial series to expand the function as a power series. State the radius of convergence. + x
> Use the information from Exercise 14 to estimate sin 380 correct to five decimal places. Exercise 14: f(x) = sin x, a = "/6, n= 4, 0<x</3
> Use the information from Exercise 5 to estimate cos 800 correct to five decimal places. Exercise 5: Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. = cos x, a = T/2 1 = 7/2
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> Graph the curve and find the area that it encloses. r= 2 sin e + 3 sin 90
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> Find the Maclaurin series for f (x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn (x) → 0.] Also find the associated radius of convergence. f(x) = xe хе
> Find the Maclaurin series for f (x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn (x) → 0.] Also find the associated radius of convergence. f(x) = e5=
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = tan-'x, a = 1
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) %3D хе -2:, а %3D 0 = xe a = 0
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) In x a = 1
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. = cos x, a = T/2 1 = 7/2
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = x + e, a=0
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = 1/x, a = 2
> Graph the curve and find the area that it encloses.
> (a). Find the Taylor polynomials up to degree 3 for f (x) = 1/x centered at a = 1. Graph f and these polynomials on a common screen. (b). Evaluate f and these polynomials at x = 0.9 and 1.3. (c). Comment on how the Taylor polynomials converge to f (x).
> (a). Find the Taylor polynomials up to degree 6 for f (x) = cos x centered at a = 0. Graph f and these polynomials on a common screen. (b). Evaluate f and these polynomials at x = π/4, π/2, and π. (c). Comment on how the Taylor polynomials converge to f
> Use series to evaluate the following limit. limx→0 sin x – x/ x3
> (a). Approximate f by a Taylor polynomial with degree at the number a. (b). Graph f and Tn on a common screen. (c). Use Taylor’s Inequality to estimate the accuracy of the approximation f (X) = Tn(x) when lies in the given interval. (d)
> (a). Approximate f by a Taylor polynomial with degree at the number a. (b). Graph f and Tn on a common screen. (c). Use Taylor’s Inequality to estimate the accuracy of the approximation f (X) = Tn(x) when lies in the given interval. (d)
> Use series to approximate f10√1 + x4, dx correct to two decimal places.
> Evaluate f ex/x dx as an infinite series.
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)
> Sketch the curve and find the area that it encloses. r = 2 + cos 2θ
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. f(x)
> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, and tan-1x. x? f
> Find the Taylor series of f (x) = cos x at a = π/3.
> Find the Taylor series of f (x) = sin x at a π/6.
> Find the radius of convergence of the series ∑∞n=1 (2n)!/(n!)2 xn.
> Find the radius of convergence and interval of convergence of the series. 2"(х — 3)" Σ In + 3 00
> Find the radius of convergence and interval of convergence of the series. 2"х — 2)" A-1 (п + 2)!
> Sketch the curve and find the area that it encloses. r = 2 cos 3θ
