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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Evaluate the series two ways.a. Use a telescoping series argument.b. Use a geometric series argument after first simplifying 4 3k 3k+1 |k=1 4 3k 3k+1°
Evaluate the series two ways.a. Use a telescoping series argument.b. Use a geometric series argument after first simplifying 1 2k+1 2k k=1 2k 2k+1°
Evaluate each series or state that it diverges. In((k + 1)k¯|) =2(In k) In (k + 1) |k=2 8.
Evaluate each series or state that it diverges. _k TT k+1 k=1e iek 8.
Evaluate each series or state that it diverges. (-2)* 3k+1 k=1 8.
Evaluate each series or state that it diverges. (sin-'(1/k) – sin-(1/(k + 1)), k=1
Determine whether the following statements are true and give an explanation or counterexample.a. is a convergent geometric series.b. If a is a real number andconverges, thenconverges.c. If the series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim S,
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim S,
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim S,
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim S,
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges.where a
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges.where p
Consider positive real numbers x and y. Notice that 43 < 34, while 32 > 23, and 42 = 24. Describe the regions in the first quadrant of the xy-plane in which xy > yx and xy < yx. Find a
Prove that the equationsx = a cos t + b sin t, y = c cos t + d sin t,where a, b, c, and d are real numbers, describe a circle of radius R provided a2 + c2 = b2 + d2 = R2 and ab + cd = 0.
Assume a curve is given by the parametric equations x = f(t) and y = g(t), where f and g are twice differentiable. Use the Chain Rule to show that
Explain and carry out a method for graphing the curve x = 1 + cos2 y - sin2 y using parametric equations and a graphing utility.
A projectile launched from the ground with an initial speed of 20 m/s and a launch angle u follows a trajectory approximated byx = (20 cos θ)t, y = -4.9t2 + (20 sin θ)t,where x and y are the
A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given byx = 100t, y =
A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given byx = 80t, y = -4.9t2 + 3000, for t ≥ 0,where
Use the equations in Exercise 102 to plot the paths of the following moons in our solar system.a. Each year our moon revolves around Earth about n = 13.4 times, and the distance from the Sun to Earth
An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the
A general hypocycloid is described by the equationsUse a graphing utility to explore the dependence of the curve on the parameters a and b. (a – b)t (a – b) cos t + b cos (a – b)t y = (a – b)
An epitrochoid is the path of a point on a circle of radius b as it rolls on the outside of a circle of radius a. It is described by the equationsUse a graphing utility to explore the dependence of
A trochoid is the path followed by a point b units from the center of a wheel of radius a as the wheel rolls along the x-axis. Its parametric description is x = at - b sin t, y = a - b cos t. Choose
A family of curves called hyperbolas (discussed in Section 10.4) has the parametric equations x = a tan t, y = b sec t, for -π < t < π and |t| ≠ π/2, where a and b are nonzero real
The Lamé curve described by where a, b, and n are positive real numbers, is a generalization of an ellipse.a. Express this equation in parametric form (four pairs of equations are needed).b. Graph
Consider the following Lissajous curves. Graph the curve and estimate the coordinates of the points on the curve at which there is (a) A horizontal tangent line.(b) A vertical tangent line.x =
Consider the following Lissajous curves. Graph the curve and estimate the coordinates of the points on the curve at which there is (a) A horizontal tangent line.(b) A vertical tangent line.x =
Find real numbers a and b such that equations A and B describe the same curve.A: x = t + t3, y = 3 + t2; -2 ≤ t ≤ 2B: x = t1/3 + t, y = 3 + t2/3; a ≤ t ≤ b
Find real numbers a and b such that equations A and B describe the same curve.A: x = 10 sin t, y = 10 cos t; 0 ≤ t ≤ 2πB: x = 10 sin 3t, y = 10 cos 3t; a ≤ t ≤ b
Find all the points at which the following curves have the given slope.x = 2 + √t, y = 2 - 4t; slope = -8
Find all the points at which the following curves have the given slope.x = t + 1/t, y = t - 1/t; slope = 1
Find all the points at which the following curves have the given slope.x = 2 cos t, y = 8 sin t; slope = -1
Find all the points at which the following curves have the given slope.x = 4 cos t, y = 4 sin t; slope = 1/2
Eliminate the parameter to express the following parametric equations as a single equation in x and y.x = a sinn t, y = b cosn t, where a and b are real numbers and n is a positive integer
Eliminate the parameter to express the following parametric equations as a single equation in x and y.x = tan t, y = sec2 t - 1
Eliminate the parameter to express the following parametric equations as a single equation in x and y.x = √t + 1, y = 1/t + 1
Eliminate the parameter to express the following parametric equations as a single equation in x and y.x = t, y = √4 - t2
Eliminate the parameter to express the following parametric equations as a single equation in x and y.x = sin 8t, y = 2 cos 8t
Eliminate the parameter to express the following parametric equations as a single equation in x and y.x = 2 sin 8t, y = 2 cos 8t
Which of the following parametric equations describe the same curve?a. x = 2t2, y = 4 + t; -4 ≤ t ≤ 4.b. x = 2t4, y = 4 + t2; -2 ≤ t ≤ 2.c. x = 2t2/3, y = 4 + t1/3; -64 ≤ t ≤ 64.
Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.a. x = 1 + s, y = 2s and x = 1 + 2t,
Find parametric equations (not unique) of the following ellipses. Graph the ellipse and find a description in terms of x and y.An ellipse centered at (0, -4) with major and minor axes of lengths 10
Find parametric equations (not unique) of the following ellipses. Graph the ellipse and find a description in terms of x and y.An ellipse centered at (-2, -3) with major and minor axes of lengths 30
Find parametric equations (not unique) of the following ellipses. Graph the ellipse and find a description in terms of x and y.An ellipse centered at the origin with major and minor axes of lengths
Find parametric equations (not unique) of the following ellipses. Graph the ellipse and find a description in terms of x and y.An ellipse centered at the origin with major axis of length 6 on the
An ellipse is generated by the parametric equations x = a cos t, y = b sin t. If 0 < a < b, then the long axis (or major axis) lies on the y-axis and the short axis (or minor axis) lies on the
An ellipse is generated by the parametric equations x = a cos t, y = b sin t. If 0 < a < b, then the long axis (or major axis) lies on the y-axis and the short axis (or minor axis) lies on the
Match equations a–d with graphs A–D. Explain your reasoning.a. x = t2 - 2, y = t3 - tb. x = cos (t + sin 50t), y = sin (t + cos 50t)c. x = t + cos 2t, y = t - sin 4td. x = 2 cos t + cos 20t, y =
Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.The upper half of the parabola x = y2, originating at (0, 0).
Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.The lower half of the circle centered at (-2, 2) with radius 6, oriented in
Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.The line that passes through the points (1, 1) and (3, 5), oriented in the
Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.The left half of the parabola y = x2 + 1, originating at (0, 1).
Find an equation of the line tangent to the curve at the point corresponding to the given value of t.x = cos t + t sin t, y = sin t - t cos t; t = π/4
Find an equation of the line tangent to the curve at the point corresponding to the given value of t.x = et, y = ln (t + 1); t = 0
Find an equation of the line tangent to the curve at the point corresponding to the given value of t.x = t2 - 1, y = t3 + t; t = 2
Find an equation of the line tangent to the curve at the point corresponding to the given value of t.x = sin t, y = cos t; t = π/4
Determine whether the following statements are true and give an explanation or counterexample.a. The equations x = -cos t, y = -sin t, for 0 ≤ t ≤ 2π, generate a circle in the clockwise
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
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
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
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
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
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
Consider the family of curvesPlot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.a = 7, b = 4, c = 1. 1 sin at ) cos ( t- sin bt х — sin bt sin at ) sin ( t + 2 +
Consider the family of curvesPlot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.a = 18, b = 18, c = 7. 1 sin at ) cos ( t- sin bt х — sin bt sin at ) sin ( t + 2 +
Consider the family of curvesPlot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.a = 6, b = 12, c = 3. 1 sin at ) cos ( t- sin bt х — sin bt sin at ) sin ( t + 2 +
Consider the family of curvesPlot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.a = b = 5, c = 2. 1 sin at ) cos ( t- sin bt х — sin bt sin at ) sin ( t + 2 +
Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 2 sin' t 2 sin 2t, y |x = cos t cos ||
Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. a² – b² cos a² – b² х — sin' t; a = 4 and b =
Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest.x = cos t + t sin t, y = sin t - t cos t
Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. 312 1 + t° 3t 3 » Y = .3
Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest.x = 2 cot t, y = 1 - cos 2t
Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest.x = t cos t, y = t sin t; t ≥ 0
Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies.The
Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies.The
Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies.The
Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies.The
Find a parametric description of the line segment from the point P to the point Q. Solutions are not unique.P(8, 2), Q(-2, -3)
Find a parametric description of the line segment from the point P to the point Q. Solutions are not unique.P(-1, -3), Q(6, -16)
Find a parametric description of the line segment from the point P to the point Q. Solutions are not unique.P(1, 3), Q(-2, 6)
Find a parametric description of the line segment from the point P to the point Q. Solutions are not unique.P(0, 0), Q(2, 8)
Find the slope of each line and a point on the line. Then graph the line.x = 1 + 2t/3, y = -4 - 5t/2
Find the slope of each line and a point on the line. Then graph the line.x = 8 + 2t, y = 1
Find the slope of each line and a point on the line. Then graph the line.x = 4 - 3t, y = -2 + 6t
Find the slope of each line and a point on the line. Then graph the line.x = 3 + t, y = 1 - t
Find parametric equations that describe the circular path of the following objects. For Exercises 33–35, assume (x, y) denotes the position of the object relative to the origin at the center of the
Find parametric equations that describe the circular path of the following objects. For Exercises 33–35, assume (x, y) denotes the position of the object relative to the origin at the center of the
Find parametric equations that describe the circular path of the following objects. For Exercises 33–35, assume (x, y) denotes the position of the object relative to the origin at the center of the
Find parametric equations that describe the circular path of the following objects. For Exercises 33–35, assume (x, y) denotes the position of the object relative to the origin at the center of the
Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique.A circle
Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique.A circle
Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique.A circle
Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique.A circle
Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique.A circle
Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of x and y. Answers are not unique.A circle
Eliminate the parameter to find a description of the following circles or circular arcs in terms of x and y. Give the center and radius, and indicate the positive orientation.x = 1 - 3 sin 4πt, y =
Eliminate the parameter to find a description of the following circles or circular arcs in terms of x and y. Give the center and radius, and indicate the positive orientation.x = -7 cos 2t, y = -7
Eliminate the parameter to find a description of the following circles or circular arcs in terms of x and y. Give the center and radius, and indicate the positive orientation.x = 2 sin t - 3, y = 2

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