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mathematics
calculus

Questions and Answers of Calculus

At what points on the curve x = t3 + 4t, y = 6t2 is the tangent parallel to the line with equations x = – 7t, y = 12t – 5?
Find equations of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 that pass through the point (4, 3).
Use the parametric equations of an ellipse, x = a cos θ, y = b sin θ, < θ < 2π, to find the area that it encloses.
Find the area bounded by the curve x = t – 1/t, y = t + 1/t 2.5.
Find the area bounded by the curve x = cos t, y = et, 0 < t < π/2, and the lines y = 1 and x = 0.
Find the area of the region enclosed by the asteroid x = a cos3θ, y = a sin3θ. (Asteroids are explored in the Laboratory Project on page 659.)
Find the area under one arch of the trochoid of Exercise 38 in Section 10.1 for the case d < r.
Let R be the region enclosed by the loop of the curve in Example 1.(a) Find the area of R.(b) If R is rotated about the -axis, find the volume of the resulting solid.(c) Find the centroid of R.
Set up, but do not evaluate, an integral that represents the length of the cur
Use Simpson’s Rule with n = 6 to estimate the length of the curve x = t – et, y = t + et, – 6 < t < 6.
In Exercise 41 in Section 10.1 you were asked to derive the parametric equations x = 2a cot θ, y = 2 a sin2 θ for the curve called the witch of Maria Agnesi. Use Simpson’s Rule with n = 4
Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. Compare with the length of the curve.
Show that the total length of the ellipse x = a sin θ, y = b cos θ a > b > 0, is where is the eccentricity of the ellipse (e = c/a, where c = √a2 €“ b2).
Find the total length of the asteroid x = a cos3θ, y = a sin3θ, where a > 0.
(a) Graph the epitrochoid with equations x = 11 cos t – 3 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
A curve called Cornu€™s spiral is defined by the parametric equations where and are theFresnel functions that were introduced in Chapter 5(a) Graph this curve. What happens as
Set up, but do not evaluate, an integral that represents the area of the surface obtained by rotating the given curve about the x-axis.
Find the area of the surface obtained by rotating the given curve about the -axis.
Graph the curve x = 2 cos θ – cos 2θ y = 2 sin θ – sin 2θ. If this curve is rotated about the -axis, find the area of the resulting surface. (Use your graph to help find the
If the curve x = t + t3 y = t – 1/t2 1 < t < 2 is rotated about the -axis, use your calculator to estimate the area of the resulting surface to three decimal places.
If the arc of the curve in Exercise 50 is rotated about the x-axis, estimate the area of the resulting surface usingSimpson’s Rule with n = 4.
Find the surface area generated by rotating the given curve about the y-axis.
If f’ is continuous and 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 y = F(x).
Use Formula 2 to derive Formula 7 from Formula 8.2.5 for the case in which the curve can be represented in the form y = F(x), a < x < b.
(a) For a parametric curve x = x (t), y = y (t), derive the formula k = | xy €“ xy| / [x2 + y2]3/2 where the dots indicate derivatives with respect to x = dx/dt, so x = dx/dt.(b) Br
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) Show that the curvature at each point of a straight line is k = 0.(b) Show that the curvature at each point of a circle of radius is k = 1/r.
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 involutes of the circle. If the circle has radius and
A cow is tied to a silo with radius by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow.
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
Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.
Find the distance between the points with polar coordinates (1, π/6) and (3, 3π/4).
Find a formula for the distance between the points with polar coordinates (π1, θ1) and (π2, θ2)
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. 27. (a) A line through the origin
The figure shows the graph of as a function of θ 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 t→±∞ y = – 1. Use this fact to help sketch the
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
Sketch the curve (x2 + y2)3 = 4x2y2.
Show that the curve r = 2 – csc θ (also a conchoid) has the line y = – 1as a horizontal asymptote by showing that lim t→±∞ y = – 1. Use this fact to help sketch the conchoid.
(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
Match the polar equations with the graphs labeled I€“VI. Give reasons for your choices. (Don€™t use a graphing device.)
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ.
Find the points on the given curve where the tangent line is horizontal or vertical.
Show that the polar equation r = a sin θ + b cos θ, where ab ≠ 0, represents a circle, and find its center and radius.
Show that the curves r = a sin θ and r = a cos θ intersect at right angles.
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve.
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 – α)
Use a graph to estimate the -coordinate of the highest points on the curve r = sin 2θ. Then use calculus to find the exact value.
(a) Investigate the family of curves defined by the polar equations r = sin n θ, where n is a positive integer. How is the number of loops related to n? (b) What happens if the equation in part
A family of curves is given by the equations r = 1 + c sin nθ, where is a real number and is a positive integer. How does the graph change as increases? How does it change as changes? Illustrate
A family of curves has polar equations Investigate how the graph changes as the number changes. In particular, you should identify the transitional values of for which the basic shape of the curve
The astronomer Giovanni Cassini (1625–1712) studied the family of curves with polar equations r4 – 2c2r2 cos 2θ + c4 – a4 = 0 where and are positive real numbers. These curves are called
Let P be any point (except the origin) on the curve r = f(θ). If ψ is the angle between the tangent line at P and the radial line OP, show that.
(a) Use Exercise 81 to show that the angle between the tangent line and the radial line is ψ = π/4 at every point on the curve r = e6. (b) Illustrate part (a) by graphing the curve and the
Find the vertex, focus, and directrix of the parabola and sketch its graph.
Find an equation of the parabola. Then find the focus and directrix.
Find an equation of the ellipse. Then find its foci.
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
A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm.(a) Find an equation of the parabola.(b) Find the diameter of
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
Use the definition of a hyperbola to derive Equation 6 for a hyperbola with foci (±c, 0) and vertices (±a, 0).
Show that the function defined by the upper branch of the hyperbola y2/b2 – x2/b2 = 1 is concave upward.
Find an equation for the ellipse with foci (1, 0) and (– 1, – 1) and major axis of length 4.
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, and (c) k < 0.(d) Show that all the curves in parts (a) and
(a) Show that the equation of the tangent line to the parabola y2 = 4px at the point (x0, y0) can be written as y0y = 2p(x + x0)(b) What is the -intercept of this tangent line? Use this fact to draw
Use Simpson’s Rule with n = 10 to estimate the length of the ellipse x2 + 4y2 = 4.
The planet Pluto travels in an elliptical orbit around the Sun (at one focus). The length of the major axis is 1.18 X 1010 km and the length of the minor axis is 1.14 X 1010 km. Use Simpson’s Rule
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 in the figure. Prove that α
Let P(x1, y1) be a point on the hyperbola 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 hyperbola as shown in the
Write a polar equation of a conic with the focus at the origin and the given data.
(a) Find the eccentricity,(b) Identify the conic,(c) Give an equation of the directrix, and(d) Sketch the conic.
(a) Find the eccentricity and directrix of the conic r = 1 / (4 – 3 cos θ) and graph the conic and its directrix. (b) If this conic is rotated counterclockwise about the origin through an
Graph the parabola r = 5 / (2 + 2 sin θ) and its directrix. Also graph the curve obtained by rotating this parabola about its focus through an angle π/6.
Graph the conics r = e / (1 – e cos θ) with e = 0.4, 0.6, 0.8, and 1.0 on a common screen. How does the value of affect the shape of the curve?
(a) Graph the conics r = ed / (1 + e sin θ) for e = 1 and various values of d. How does the value of affect the shape of the conic? (b) Graph these conics for d = 1 and various values of e. How
Show that a conic with focus at the origin, eccentricity e, and directrix x = – d has polar equation r = ed / 1 – e cos θ.
Show that a conic with focus at the origin, eccentricity e, and directrix y = d has polar equation r = ed / 1 + e sin θ.
Show that a conic with focus at the origin, eccentricity e, and directrix y = €“ d has polar equation
Show that the parabolas r = c / (1 + cos θ) and r = c / (1 – cos θ) intersect at right angles.
(a) Show that the polar equation of an ellipse with directrix x = €“ d can be written in the form(b) Find an approximate polar equation for the elliptical orbit of Earth around the Sun (at
(a) The planets move around the Sun in elliptical orbits with the Sun at one focus. The positions of a planet that are closest to and farthest from the Sun are called its perihelion and aphelion,
The orbit of Halley’s Comet last seen in 1986 and due to return in 2062 is an ellipse with eccentricity 0.97 and one focus at the Sun. The length of its major axis is 36.18 AU. [An astronomical
The Hale-Bopp comet, discovered in 1995, has an elliptical orbit with eccentricity 0.9951 and the length of the major axis is 356.5 AU. Find a polar equation for the orbit of this comet. How close to
The planet Mercury travels in an elliptical orbit with eccentricity. Its 0.206 minimum distance from the Sun 4.6 X 107 is km. Use the results of Exercise 26(a) to find its maximum distance from the
The distance from the planet Pluto to the Sun is 4.43 X 109 km at perihelion and 7.37 X 109 km at aphelion. Use Exercise 26 to find the eccentricity of Pluto’s orbit.
Using the data from Exercise 29, find the distance traveled by the planet Mercury during one complete orbit around the Sun. (If your calculator or computer algebra system evaluates definite
Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.
Write three different sets of parametric equations for the curve y = √x.
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 increases.
The curve with polar equation r = (sin θ) / θ is called a cochleoid. Use a graph of as a function of θ in Cartesian coordinates to sketch the cochleoid by hand. Then graph it with a
Graph the ellipse r = 2 / (4 – 3 cos θ) and its directrix. Also graph the ellipse obtained by rotation about the origin through an angle 2π/3.
Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.
Use a graph to estimate the coordinates of the lowest point on the curve x = t3 – 3t, y = t2 + t +1. Then use calculus to find the exact coordinates.
Find the area of the surface obtained by rotating the given curve about the -axis.
The curves defined by the parametric equations are called strophoids(from a Greek word meaning €œto turn or twist€). Investigate how these curves vary as varies.
A family of curves has polar equations ra = | sin 2 θ | where a is a positive number. Investigate how the curves change as a changes.
Find the foci and vertices and sketch the graph.
Find an equation of the parabola with focus (0, 6) and directrix y = 2.
Find an equation of the hyperbola with foci (0, ±5) and vertices (0, ±2).

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