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mathematics
calculus

Questions and Answers of Calculus

Use the graphs of x = f (t) and y = g(t) to sketch the parametric curve x = f (t), y = g(t). Indicate with arrows the direction in which the curve is traced as t increases.
Graph the curve x = y - 2 sin π y.
(a) Show that the parametric equations x = x1 + (x2- x1)t, y = Yl + (y2- yt)t Where 0 ≤ t ≤ 1 describe the line segment that joins the points p1(x1, yl) and P2(x2, Y2). (b) Find parametric
Find parametric equations for the path of a particle that moves along the circle x2 + (y - 1)2 = 4in the manner described. (a) Once around clockwise, starting at (2,1) (b) Three times around
Use a graphing calculator or computer to reproduce the picture.
Compare the curves represented by the parametric equations. How do they differ? (a) x = t3, y= t2 (b) x = t6, y= t4 (c) x = e-3t, y = e-2t
Derive Equations 1 for the case π/2 < θ < π
If and are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point in the figure, using the angle as the parameter. Then eliminate the parameter
A curve, called a witch of Maria Agnesi, consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written asx = 2a cot Î¸, y
Suppose that the position of one particle at time is given byx1 = 3 sin t y1 = 2 cos t 0 ≤ t ≤ 2πAnd the position of a second particle is given byx2 = -3 + cos t y2 = 1 + sin t 0 ≤ t ≤
Investigate the family of curves defined by the parametric equations x = t2, y = t3 - ct. How does the shape change as increases? Illustrate by graphing several members of the family.
Graph several members of the family of curves with parametric equations x = t + a cos t, y = t + a sin t, where a > 0. How does the shape change as increases? For what values of does the curve have a
(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a
The curves with equations x = a sin nt, y = b cos t are called Lissajous figures. Investigate how these curves vary when a, b, and vary. (Take to be a positive integer.)
x = t sin t y = t 2 + t
Find dy/dx and d2y/dx2 for which values of is the curve concave upward? (a) x = t2 + 1, y = t 2 + t (b) x = et , y = te-t
Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. x = t3 - 3t, y = t2 - 3
Use a graph to estimate the coordinates of the rightmost point on the curve x = t -t6, y = et. Then use calculus to find the exact coordinates
Graph the curve in a viewing rectangle that displays all the important aspects of the curve. x = t4 - 2t3 - 2t2, y = t3 - t
Show that the curve x = cos t, y = sin t cos t has two tangents at(0, 0) and find their equations. Sketch the curve
(a) Find the slope of the tangent line to the trochoid x = rθ - d sin θ, y = r - d cos θ in term of θ. (b) Show that if d < r, then the trochoid does not have a vertical tangent.
At what points on the curve x = 2t3, y = 1 + 4t - t2, does the tangent line have slope 1?
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = 1 + 4t - t2, y = 2 - t3, t = 1
Use the parametric equations of an ellipse, x = a cos θ, y = b sin θ, 0 ≤ θ ≤ 2π, to find the area that it encloses.
Find the area enclosed by the x-axis and the curve x = 1 + et, y = t - t2
Find the area under one arch of the trochoid of Exercise 40 in Section 10.1 for the case d < r. In Exercise 40 Let P be a point at a distance from the center of a circle of radius r. The curve traced
Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. (a) x = t + e-t, y = t - e-t, 0 ≤ t ≤ 2. (b) x = t - 2 sin
Find the exact length of the curve. (a) x = 1 + 3t2, y = 4 + 2t3, 0 ≤ t ≤ 1 (b) x = t sin t, y = t cos t, 0 ≤ t ≤ 1
Graph the curve and find its length x = et cos t, y = et sin t, 0 ≤ t ≤ π.
Graph the curve x = sin t + sin 1.5t, y = cos t and find its length correct to four decimal places.
Use Simpson's Rule with n = 6 to estimate the length of the curve x = t - et, y = t + et, -6 ≤ t ≤ 6.
Find the distance traveled by a particle with position (x,y) as varies in the given time interval. Compare with the length of the curve. x = sin2 t, y = cos2 t, 0 ≤ t ≤ 3π.
Show that the total length of the ellipsex = a sin Î¸, y = b cos Î¸, 0 ‰¤ 0 ‰¤ 2Ï€Where is the eccentricity of the ellipse (e = c/a, where c
(a) Graph the epitrochoid with equations x = 11 cos t - 4 cos (11t / 2) y = 11 sin t - 4 sin (11t / 2) What parameter interval gives the complete curve? (b) Use your CAS to find the approximate
Set up an integral that represents the area of the surface obtained by rotating the given curve about the -axis. Then use your calculator to find the surface area correct to four decimal places. (a)
Find the exact area of the surface obtained by rotating the given curve about the -axis. x = t3, y = t2, 0 ≤ t ≤ 1
Find the surface area generated by rotating the given curve about the -axis. x = 3t2, y = 2t3, o ≤ t ≤ 5
If f' is continuous and for f' (t) ≠ 0 for a ≤ t ≤ b, show that the parametric curve x = f(t), y = g(t), a ≤ t ≤ b, can be put in the form.
The curvature at a point P of a curve is defined asWhere Î¦ is the angle of inclination of the tangent line at P, as shown in the figure. Thus the curvature is the absolute value of the
Find an equation of the tangent to the curve at the given point by two methods: (a) Without eliminating the parameter (b) By first eliminating the parameter. x = 1 + ln t, y = t2 + 2; (1, 3)
Use the formula in Exercise 69(a) to find the curvature of the cycloid x = θ - sin θ, y = 1 - cos θ at the top of one of its arches.
A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius and
Find an equation of the tangent(s) to the curve at the given point. Then graph the curve and the tangent(s). x = 6 sint, y = t2 + t; (0, 0).
Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > 0and one with r < 0. (a) (2, π/3) (b) (1, -3π/4) (c) (-1, π/2)
Identify the curve by finding a Cartesian equation for the curve. (a) r2 = 5 (b) r= 2 cos θ
Find a polar equation for the curve represented by the given Cartesian equation. (a) y = 2 (b) y = 1 + 3x (c) x2 + y2 = 2cx
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
Sketch the curve with the given polar equation by first sketching the graph of as a function of in Cartesion coordinates. (a) r = -2 sin θ (b) r = 2 (1 + cos θ)
Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. (a) x = 1 (b) (2, - 2π/3) (c) (-2, 3π/4)
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesion coordinates. (a) r = θ, θ ≥ 0 (b) r = 4 sin 3θ
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesion coordinates. (a) r = cos 4 θ (b) r = 1 - 2sin θ
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesion coordinates. (a) r2 = 9 sin 2 θ (b) r = 2 + sin 3 θ (c) r= 1 + 2cos 2θ
The figure shows a graph of r as a function of Î¸ in Cartesian coordinates. Use it to sketch the corresponding polar curve.
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 lim r → ± ∞ y = 2. Use this fact to help sketch the conchoid.
The Cartesian coordinates of a point are given. (i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (ii) Find polar coordinates (r, θ) of the point, where r > 0 and 0
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
(a) In Example 11 the graphs suggest that the limaçon r = 1 + c sin θ has an inner loop when |c| > 1. Prove that this is true, and find the values of θ that correspond to the inner loop. (b) From
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. (a) r = 2 sin θ, θ = π/6 (b) r = 1/θ, θ = π
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r = cos 2 θ, θ = π/4
Find the points on the given curve where the tangent line is horizontal or vertical. r = 3 cos θ
Show that the polar equation r = a sin θ + b cos θ, where ab ≠ 0, represents a circle, and find its center and radius.
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. (a) r = 1 + 2 sin (θ/2) (nephroid of Freeth) (b) r = e sin θ - 2
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. (a) r ≥ 1 (b) r ≥ 0, π/4 ≤ θ ≤ 3π/4
How are the graphs of r = l + sin (θ - π/6)and r = l + sin (θ - π/3)related to the graph of r = l + sin θ? In general, how is the graph of r = f (θ -a) related to the graph of r = f(θ)?
Investigate the family of curves with polar equations r = 1 + c cos θ, where c is a real number. How does the shape change as c changes?
Let P be any point (except the origin) on the curve r = f(Î¸). If Ïˆ is the angle between the tangent line at P and the radial line OP, show thatObserve that Ïˆ =
Find the area of the region that is bounded by the given curve and lies in the specified sector. (a) r = e -θ/4, π/2 ≤ θ ≤ π (b) r2 = 9 sin 2θ, r ≥ 0, 0 ≤ θ ≤ π/2
Graph the curve and find the area that it encloses. (a) r = 2 + sin 4θ (b) r = √1 + cos2(5θ)
Find the area of the region enclosed by one loop of the curve. r = 4 cos 3θ
Find the area of the region enclosed by one loop of the curve. r = sin 4θ
Find the area of the region enclosed by one loop of the curve. r = 1 + 2 sin θ (Inner loop)
Find the area of the region that lies inside the first curve and outside the second curve. (a) r = 2 cos θ, r = 1 (b) r2 = 8 cos 2θ, r = 1 + cos θ
Find the area of the region that lies inside both curves. (a) r = √3 cos θ, r = sinθ (b) r = sin 2θ, r = cos 2θ
Find the area inside the larger loop and outside the smaller loop of the limaçon r = 1/2 + cos θ.
Find all points of intersection of the given curves. (a) r = 1 + sin θ, r = 3 sin θ (b) r = 2 sin 2θ, r = 1
The points of intersection of the cardioid r = 1 + sin θ and the spiral loop r = 2θ, -π/2 ≤ θ ≤ π / 2, can't be found exactly. Use a graphing device to find the approximate values of θ at
Find the exact length of the polar curve. (a) r = 2 cos θ, 0 ≤ θ ≤ π (b) r = θ2, 0 ≤ θ ≤ 2π
Find the exact length of the curve. Use a graph to determine the parameter interval. r = cos4 (θ/4)
Find the area of the shaded region.(a)(b)
Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval. (a) One loop of the curve r = cos 2θ (b) r = sin
(a) Use Formula 10.2.6 to show that the area of the surface generated by rotating the polar curver = f (Î¸), a ‰¤ Î¸ ‰¤ b(Where f is continuous and 0
Sketch the curve and find the area that it encloses. (a) r = 2 sin θ (b) r = 3 + 2 cos θ
Find the vertex, focus, and directrix of the parabola and sketch its graph. (a) x2 = 6y (b) 2x = -y2
Find the vertices and foci of the ellipse and sketch its graph. (a) x2 / 2 + y2 / 4 = 1 (b) x2 + 9y2 = 9
Find an equation of the ellipse. Then find its foci.
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. (a) y2 / 25 - x2 / 9 = 1 (b) x2 - y2 = 100
Identify the type of conic section whose equation is given and find the vertices and foci. (a) x2 = y + 1 (b) x2 = 4y - 2y2 (c) y2 + 2y = 4x2 + 3
Find an equation for the conic that satisfies the given conditions. (a) Parabola, vertex (0, 0), focus (1,0) (b) Parabola, focus (-4,0), directix x =2 (c) Parabola, vertex (2,3), vertical axis,
Find an equation for the conic that satisfies the given conditions. (a) Ellipse, foci (0, 2), (0,6), vertices (0, 0), (0,8) (b) Ellipse, center (-1, 4), vertex (-1, 0), focus (-1, 6) (c) Hyperbola,
The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar
Find the vertex, focus, and directrix of the parabola and sketch its graph. (a) (x + 2)2 = 8(y - 3) (b) y2 + 2y + 12x + 25 = 0
In the LORAN (Long Range Navigation) radio navigation system, two radio stations located at A and B transmit simultaneous signals to a ship or an aircraft located at P. The onboard computer converts
Show that the function defined by the upper branch of the hyperbola y2 / a2 - x2 / y2 = 1 is concave upward.
Determine the type of curve represented by the equation x2 / k + y2 / k - 16 = 1 In each of the following cases: (a) k > 16, (b) 0 < k < 16, (c) k < 0. (d) Show that all the curves in parts (a)
Show that the tangent lines to the parabola x2 = 4py drawn from any point on the directrix are perpendicular.
Use parametric equations and Simpson's Rule with n = 8 to estimate the circumference of the ellipse 9x2 + 4y2 = 36.
Find the area of the region enclosed by the hyperbola x2 / a2 - y2/b2 and the vertical line through a focus.
Find the centroid of the region enclosed by the x-axis and the top half of the ellipse 9x2 + 4y2 = 36.
Let P(x1, y1) be a point on the ellipse x2/a2 + y2/b2 = 1 with foci F1 and F2 and let a and ( be the angles between the lines PF1, PF2 and the ellipse as shown in the figure. Prove that a = (. This
Find an equation of the parabola. Then find the focus and directrix.

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