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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Find the function r that satisfies the given condition. r' (t) = t t² + 1 −¬¡ + te¯¹² j — - 21 √₁² + 4 =k; r(0) = i + j - 3k 3 2
Find the domains of the following vector-valued functions. r(t) = Vt + 2i + √2 - tj
Find the domains of the following vector-valued functions. r(t) = √4-fi + Vtj - V4 2 VI + t k
Evaluate the following definite integrals. In 2 (e¹i + e¹ cos(πe¹).j)dt
Find the domains of the following vector-valued functions. r(t) cos 2ti + evij evij + 1/²/201 -k t
Evaluate the following definite integrals. La+ (i + tj + 3t² k) dt
Evaluate the following definite integrals. [(₁ (sin ti + cos tj + 2tk) dt
Evaluate the following definite integrals. [,"co (6t² i + 8t³j+ 9t²k)dt
Evaluate the following definite integrals. 3 L₂(1 + 27² - 7 csc ³ (7/¹) k) dr -i CSC² dt 2t 1/2 2
Evaluate the following definite integrals. In 2 S for (041 +- 20²4 0 (e¹i+2e²¹j4e¹ k) dt
Evaluate the following definite integrals. .2 L'ªre' (1 JO te¹(i+2jk) dt
Evaluate the following definite integrals. TT/4 S™ 0 (sec²ti2 cos tj - k) dt
Let u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3). Let c be a scalar. Prove the following vector properties. |uv| ≤ |u||v|
Let u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3). Let c be a scalar. Prove the following vector properties. u • (v + w) = u•v + u• w
Let u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3). Let c be a scalar. Prove the following vector properties. u. V = V. u
Let u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3). Let c be a scalar. Prove the following vector properties. c(u v) = (cu) v = u. (cv)
Let u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3). Let c be a scalar. Prove the following vector properties. . a. Show that (u + v) • (u + v) = \u² + 2u.v + |v|². b. Show that (u + v)
Evaluate the following limits using Taylor series. lim x sin X-00 X
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = ln x, n In x, n = 2, a = 1
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = sinh 2x, n = 4, a = 0
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = cosh x, n = 3, a = In 2
Evaluate the following limits using Taylor series. lim X-0 1 + x - e* 4x²
Evaluate the following limits using Taylor series. lim x-0 2 cos 2x2 + 4x² 2 2x4
Evaluate the following limits using Taylor series. lim X-0 V1+x-1(x/2) 2 4x²
Evaluate the following limits using Taylor series. lim x-0 3 tan¹x x − 3x + x³ x5
Evaluate the following limits using Taylor series. lim x-0 3 tan x - 3x - x³ 3 x5
Evaluate the following limits using Taylor series. +² X - 16 lim x4 In (x − 3)
Evaluate the following limits using Taylor series. lim x→l x - 1 ln x
Evaluate the following limits using Taylor series. x - 2 lim x 2 In (x - 1)
Use the geometric series to determine the Maclaurin series and the interval of convergence for the following functions. 00 k=0 1 1- x' for x < 1,
Use the geometric series to determine the Maclaurin series and the interval of convergence for the following functions. 00 k=0 1 1- x' for x < 1,
Evaluate the following limits using Taylor series. lim x(e¹/x - 1) X-00
Evaluate the following limits using Taylor series. sin x − tan x lim x-0 3x³ cos x
Evaluate the following limits using Taylor series. lim x-0 (1-2x)-1/2 8x² 2
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation. f(x) = 1/x, a = 2
Evaluate the following limits using Taylor series. lim x->0+ (1 + x)2 - 4 cos √x + 3 2x²
Use the Taylor series in Table 10.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.Data from in Taylor series in Table 10.5 cos Vx
Use the geometric series to determine the Maclaurin series and the interval of convergence for the following functions. 00 k=0 1 1- x' for x < 1,
Use the geometric series to determine the Maclaurin series and the interval of convergence for the following functions. 00 k=0 1 1- x' for x < 1,
Use the Taylor series in Table 10.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.Data from in Taylor series in Table 10.5 cosh 3x
Use the geometric series to determine the Maclaurin series and the interval of convergence for the following functions. 00 k=0 1 1- x' for x < 1,
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation. f(x) = 2*, a = 1
Use the Taylor series in Table 10.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.Data from in Taylor series in Table 10.5 (1 + x4)-¹
Use the geometric series to determine the Maclaurin series and the interval of convergence for the following functions. 00 k=0 1 1- x' for x < 1,
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation. f(x) = 10¹, a = 2
Use the Taylor series in Table 10.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.Data from in Taylor series in Table 10.5 x tan¹2
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0.2 sin x² dx
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0.25 e²dx
Identify the functions represented by the following power series. 00 Σ(-1)* k=0 xk 3k
Identify the functions represented by the following power series. k=1 .2k X k
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0 0.2 V1 + x² dx x4
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0.35 5.³ 0 tan¹ x dx
Identify the functions represented by the following power series. 00 xk k-2 k(k-1)
Identify the functions represented by the following power series. Σ(-1)* k=1 kk+1 3k
Identify the functions represented by the following power series. k=2 k(k − 1)x* 3k
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Sketch the following sets of points. {(r, 0): 4 = r² ≤ 9}
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Sketch the graph of the following parabolas. Specify the location of the focus and the equation of the directrix. 12x5y 5y²
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Express the following polar coordinates in Cartesian coordinates. (4,5m)
Write an equation of the following parabolas. YA (0, 2) Directrix y = 4 4 X
Sketch the following sets of points. {(r, 0): 0 ≤ r ≤ 4,-π/2 ≤ 0 ≤ -π/3}
Consider the following parametric curves.a. Determine dy/dx in terms of t and evaluate it at the given value of t.b. Make a sketch of the curve showing the tangent line at the point corresponding to
Write an equation of the following parabolas. (-1,0). H Directrix x = -2 УА + 2 X
Sketch the following sets of points. {(r, 0): 2 ≤ r ≤ 8}
Sketch the following sets of points. {(r, 0): π/2 ≤ 0 ≤ 3π/4}
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). r 12 3 cos 0
Sketch the following sets of points. {(r, 0): 0 = 2π/3}
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = sin 2x, n = 3, a = 0
Sketch the following sets of points. {(r, 0): 1
Sketch the following sets of points. {(r, 0): 비 = 7/3}
Sketch the following sets of points. {(r, 0): r
Sketch the following sets of points. {(r, 0): r≥ 2}
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = cos x², n = 2, a = 0
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = e, n = 2, a = 0
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = ln (1+x), n = 3, a = 0
Find the nth-order Taylor polynomial for the following functions centered at the given point a. f(x) = cos x, n = 2, a = π/4
Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10-4. Although you do not need it, the exact value of the series is given in each
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. f(x) = ln (1-x) = - for -1 < x < 1,
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. f(x) = ln (1-x) = - for -1 < x < 1,
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. f(x) = ln (1-x) = - for -1 < x < 1,
Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. cos x 1x²/2; [-π/4, π/4] A
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. f(x) = ln (1-x) = - for -1 < x < 1,
Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. sin xxx³/6; [-π/4, π/4]
Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. X; [-π/6, π/6] tan x = x;
a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function. f(x) = e, a = In 2
Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. In (1 + x) = x - x²/2; [-0.2, 0.2]
Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. ex ≈ 1 + x + x²/2; [ - 1/1, 12] 22
Find the function represented by the following series and find the interval of convergence of the series. 00 k= 2k -24 X 4k
Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. V1 + x = 1 + x/2; [-0.1, 0.1]
Find the function represented by the following series and find the interval of convergence of the series. Σ(VF – 2)* k=0
Find the function represented by the following series and find the interval of convergence of the series. k=0 x2 3 1 k
Suppose you approximate sin x at the points x = -0.2, -0.1, 0.0, 0.1, and 0.2 using the Taylor polynomials p3 = x - x3/6 and p5 = x - x3/6 + x5/120. Assume that the exact value of sin x is given by a
Use the test of your choice to determine whether the following series converge. 8 00 1 k=2 k Ink
Use the test of your choice to determine whether the following series converge. 00 k=1 sin² k
Use the test of your choice to determine whether the following series converge. k=1 tan 1 k
Use the test of your choice to determine whether the following series converge. 1 1! + 4 2! + 9 16 + 3! 4!
Use the test of your choice to determine whether the following series converge. 00 Σ 100k k=2 -k

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