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mathematics
precalculus

Questions and Answers of Precalculus

Find the area of the region that lies inside the first curve and outside the second curve.r = 4 sin θ, r = 2θ
Find the area enclosed by the loop of the strophoidr = 2 cosθ - sec θ.
Find the area of the region enclosed by one loop of the curve.r = 2 sin 5θ
Find the area of the region enclosed by one loop of the curve.r2 = 4 cos 2θ
Graph the curve and find the area that it encloses.r = 1 + 5 sin 6θ
Graph the curve and find the area that it encloses.r = √1 + cos2(5θ)
Graph the curve and find the area that it encloses.r = 3 - 2 cos 4θ
Graph the curve and find the area that it encloses.r = 2 + sin 4θ
Sketch the curve and find the area that it encloses.r = 2 - cos θ
Sketch the curve and find the area that it encloses.r = 1 - sinθ
Find the area of the shaded region. r= 2+ cos 0
Find the area of the shaded region. r?= sin 20
Find the area of the region that is bounded by the given curve and lies in the specified sector. r%3D 1/0, п/2
Find the area of the region that is bounded by the given curve and lies in the specified sector. = sin 0 + cos 0, 0 < 0 < T
Find the area of the region that is bounded by the given curve and lies in the specified sector. r%3 cos @, 0
Investigate the family of polar curves r = 1 + cosnθ where n is a positive integer. How does the shape change as n increases? What happens as n becomes large? Explain the shape for large n by
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve.r = 2 + cos(9θ/4)
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve.r = 1 + cos999 θ(Pac-Man curve)
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve.r = |tan||cot θ| (valentine curve)
Find the points on the given curve where the tangent line is horizontal or vertical.r = 1 + cosθ
Find the points on the given curve where the tangent line is horizontal or vertical.r = 1 - sin θ
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r = 1 + 2 cos 0, 0 = 1/3
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. cos(®/3), Ө — п т
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r%3 2 + sin 3ө, ө — п/4
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r%3D 2 сos 0, ө — п/3
The figure shows a graph of r as a function of θ in Cartesian coordinates. Use it to sketch the corresponding polar curve. 2-
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r = cos(θ/3)
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r = sin (θ/2)
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r2θ = 1
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r = 2 + sin 3θ
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r2 = cos 4θ
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r = 1 + 5 sinθ
Sketch the curve with the given polar equation by first sketching the graph of r as a function of in Cartesian coordinates.r = 1 + 3 cosθ
Identify the curve by finding a Cartesian equation for the curve.r2cos 2θ = 1
Identify the curve by finding a Cartesian equation for the curve.r − 4 sec θ
Find a formula for the distance between the points with polar coordinates (r1, θ1) and (r2, θ2)
(a) Use the formula in Exercise 69(b) to find the curvature of the parabola y = x2 at the point (1, 1).(b) At what point does this parabola have maximum curvature?
Use Formula 1 to derive Formula 6 from Formula 8.2.5 for the case in which the curve can be represented in the form y = F(x), a < x < b.
Find the exact area of the surface obtained by rotating the given curve about the x-axis. |х 3а сcos'0, y %3D a sin'®, 0 < 0 < T/2 0< 0< m/2 x = a coS
Find the exact area of the surface obtained by rotating the given curve about the x-axis. 1
Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. x =
Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. y =
Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. х
Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. x =
(a) Graph the epitrochoid with equationsx = 11 cos t - 4 cos(11t/2)y = 11 sin t - 4 sins(11t/2)What parameter interval gives the complete curve?(b) Use your CAS to find the approximate length of this
Find the total length of the astroid x = a cos3, y = a sin3, where a . 0.
Find the length of the loop of the curve x = 3t - t3, y − 3t2.
Graph the curve and find its exact length. T/4
Graph the curve and find its exact length. x = e' cos t, y = e' sin t, 0 < t< T
Find the exact length of the curve. x = 3 cos t - cos 3t, y = 3 sin t – sin 3t, 0
Find the exact length of the curve. x = t sin , y = t cos , 0
Find the exact length of the curve. x = e – t, y = 4e2, 0
Find the exact length of the curve. x = 1 + 3t, y = 4 + 2t°, 0
Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. x = t + Vt, y = t – /t, 0 < t < 1
Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. x = t? – t, 1
Find the area of the region enclosed by the astroid x = a cos3θ, y = a sin3θ. (Astroids are explored in the Laboratory Project on page 649.) У a х
Find the area enclosed by the x-axis and the curvex = t3 + 1, y = 2t - t2.
Find the area enclosed by the curve x = t2 - 2t, y = √t and the y-axis.
At what point(s) on the curve x = 3t2 + 1, y = t3 - 1 does the tangent line have slope 1/2?
Graph the curve x = -2 cos t, y = sin t + sin 2t to discover where it crosses itself. Then find equations of both tangents at that point.
Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve x = t4 - 2t, y = t + t4.Then find the exact coordinates.
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. х 3D cos 0, у 3 cos 30 y = cos 30
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. sin e x = e sine, y = ecose cos e y = ecos6
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. 3t, y = t – 3t² x = t3 –
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? y = sin 2t, 0
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward?x = t - ln t, y = t + ln t
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? х 3D ? + 1, у%3D е' — 1
Find dy/dx and d2y/dx2. For which values of t is the curve concave upward? = t² – t x = t° + 1, y
Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. x = t? – t, y = t² + t + 1; (0, 3)
Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. y = t² + t; (0, 2) x = sin mt,
Find an equation of the tangent to the curve at the given point by two methods:(a) Without eliminating the parameter and (b) By first eliminating the parameter.
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = e' sin mt, y = e2t; t = 0 %3D
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = t cos t, y = t sin t; t =
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. Vt, y = t? – 2t; t= 4
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = t3 + 1, y = t + t; t= -1
Find dy/dx.x = tet, y = t + sin t
Find dy/dx. y = I +t х х 1 +t'
Investigate the family of curves defined by the parametric equations x = cos t, y = sin t - sin ct, where c > 0. Start by letting c be a positive integer and see what happens to the shape as c
Graph several members of the family of curves x = sin t + sin nt, y = cos t + cos nt, where n is a positive integer. What features do the curves have in common? What happens as n increases?
If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle θ as the parameter. The line segment AB is
Compare the curves represented by the parametric equations. How do they differ?(a)(b)(c) -2 х%3 т у%3Dt? У -2t х3 е', у—е
Use a graphing calculator or computer to reproduce the picture. ул 4- х 2.
Graph the curves y = x3 - 4x and x = y3 - 4y and find their points of intersection correct to one decimal place.
Describe the motion of a particle with position (x, y) as t varies in the given interval.x = sin t, y = cos 2t, -2π < t < 2π
Describe the motion of a particle with position (x, y) as t varies in the given interval.x = 2 + sin t, y = 1 + 3 cos t, π/2 < t < 2π
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. |х 3D
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x = sinh
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = √t +
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = t2, y =
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = et, y =
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. y — csc
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. cos 0, y =
(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 Cartesian
(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 Cartesian
(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 Cartesian
(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 Cartesian
(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 Cartesian
(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 Cartesian
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.x = 1 sin t, y = cos t, -π < t < π
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. .3 .2 x = t° + t, y = t² + 2, -2 < t< 2 %3D

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