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 vertical line through the point (3, 3)
> Use the given graph of f (x) = x2 to find a number δ such that if |x - 1|< 8 |x² – 1|<} then y y=x² 1.5 1+ 0.5 ? 1 ?
> Use series to approximate the definite integral to within the indicated accuracy. x cos(x') dx (three decimal places)
> Evaluate the indefinite integral as an infinite series. f arctan (x2), dx
> Evaluate the indefinite integral as an infinite series. f cos x – 1/x, dx
> (a). Show that the function defined by is not equal to its Maclaurin series. (b). Graph the function in part (a) and comment on its behavior near the origin. if x + 0 f(x) : if x = 0
> (a). Expand 1/4√1 + x as a power series. (b). Use part (a) to estimate 1/4√1.1 correct to three decimal places.
> Let f (x) = ∑∞n=1xn/n2 Find the intervals of convergence for f, f', and f".
> 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) = xe¯ хе
> 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) = e* + cos x
> (a). Show that the function is a solution of the differential equation f'(x) = f (x) (b). Show that f (x) = ex. 00 f(x) = E -0 n!
> The period of a pendulum with length L that makes a maximum angle θ0 with the vertical is Where k = sin (1/2 θ0) and is the acceleration due to gravity. (In Exercise 34 in Section 5.9 we approximated this integral using Simpso
> How are the graphs of r = 1 + sin (θ – π/6) and r = 1 + sin (θ – π/3) related to the graph of r = 1 + sin θ? In general, how is the graph of r = f (θ – a) related to the graph of r = f (θ)?
> If a surveyor measures differences in elevation when making plans for a highway across a desert, corrections must be made for the curvature of the earth. (a). If R is the radius of the earth and L is the length of the highway, show that the correction is
> (a). Derive Equation 3 for Gaussian optics from Equation 1 by approximating cos ø in Equation 2 by its first-degree Taylor polynomial. (b). Show that if cos ø is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equati
> A car is moving with speed 20 m/s and acceleration 2 m/s at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled durin
> 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. arctan x = x - 3 (Jerr
> 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. cos x = 1 2 (lerror |<
> How many terms of the Maclaurin series for ln (1 + x) do you need to use to estimate ln 1.4 to within 0.001?
> Use Taylor’s Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.
> (a). Use differentiation to find a power series representation for What is the radius of convergence? (b). Use part (a) to find a power series for (c). Use part (b) to find a power series for 1 f(x) = (1 + x)? 1 f(x) = (1 + x)} 3 x? f(x) (1 + х)
> 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) = VT + x², a= 0
> 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
> 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
> Evaluate the integral. f ax/x2 – bx, dx
> 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)
> Write the sum in expanded form. ∑n-1j=0 (-i)j
> Evaluate the integral. f r2/r + 4, dr
> Solve the inequality in terms of intervals and illustrate the solution set on the real number line. x' + 3x < 4x?
> 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
> 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. (a) (r' + x)(x? – x + 3) 1 (b) x° - x .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
> Use the definition of a hyperbola to derive Equation 2 for a hyperbola with foci (±c, 0).
> Evaluate ∑ni-1 [ ∑nj-1 (i + j).
> 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)?
> Evaluate ∑ni-1 (2i + 2i).
> Find the limit. limn→∞ ∑n i=1 3/n [ (1 + 3i/n)3 - 2 (1 + 3i/n)]
> 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
