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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 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 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(-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(8, 2), Q(-2, -3)

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 segment of the parabola y = 2x2 - 4, where -1 ≤ x ≤ 5.

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 complete curve x = y3 - 3y.

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 piecewise linear path from P(-2, 3) to Q(2, -3) to R(3, 5), using parameter values 0 ≤ t ≤ 2.

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 path consisting of the line segment from (-4, 4) to (0, 8), followed by the segment of the parabola y = 8 - 2x2 from (0, 8) to (2, 0), using parameter values 0 ≤ t ≤ 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 = t cos t, y = t sin t; t ≥ 0

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.

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.

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest.

Consider the family of curves

Plot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.

a = b = 5, c = 2.

Consider the family of curves

Plot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.

a = 6, b = 12, c = 3.

Consider the family of curves

Plot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.

a = 18, b = 18, c = 7.

Consider the family of curves

Plot the curve for the given values of a, b, and c with 0 ≤ t ≤ 2π.

a = 7, b = 4, c = 1.

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 the given value of t.

x = 2 + 4t, y = 4 - 8t; t = 2

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 the given value of t.

x = 3 sin t, y = 3 cos t; t = π/2

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 the given value of t.

x = cos t, y = 8 sin t; t = π/2

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 the given value of t.

x = 2t, y = t3; t = -1

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 the given value of t.

x = t + 1/t, y = t - 1/t; t = 1

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 the given value of t.

x = √t, y = 2t; t = 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 direction.

b. An object following the parametric curve x = 2 cos 2πt, y = 2 sin 2πt circles the origin once every 1 time unit.

c. The parametric equations x = t, y = t2, for t ≥ 0, describe the complete parabola y = x2.

d. The parametric equations x = cos t, y = sin t, for -π/2 ≤ t ≤ π/2, describe a semicircle.

e. There are two points on the curve x = -4 cos t, y = sin t, for 0 ≤ t ≤ 2π, at which there is a vertical tangent line.

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

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 = 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 = cos t + t sin t, y = sin t - t cos t; t = π/4

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 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 direction of increasing x.

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 the counterclockwise direction.

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).

Match equations a–d with graphs A–D. Explain your reasoning.

a. x = t2 - 2, y = t3 - t
b. x = cos (t + sin 50t), y = sin (t + cos 50t)
c. x = t + cos 2t, y = t - sin 4t
d. x = 2 cos t + cos 20t, y = 2 sin t + sin 20t

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 x-axis. If 0 < b < a, the axes are reversed. The lengths of the axes in the x- and y-directions are 2a and 2b, respectively. Sketch the graph of the following ellipses. Specify an interval in t over which the entire curve is generated.

x = 4 cos t, y = 9 sin t.

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 x-axis. If 0 < b < a, the axes are reversed. The lengths of the axes in the x- and y-directions are 2a and 2b, respectively. Sketch the graph of the following ellipses. Specify an interval in t over which the entire curve is generated.

x = 12 sin 2t, y = 3 cos 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 the origin with major axis of length 6 on the x-axis and minor axis of length 3 on the y-axis, generated counterclockwise.

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 12 and 2, on the x- and y-axes, respectively, generated clockwise.

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 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise.

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 and 3, parallel to the x- and y-axes, respectively, generated clockwise.

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, y = 3t.

b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t.

c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t.

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.

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

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 = t, y = √4 - t2

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 = tan t, y = sec2 t - 1

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

Find all the points at which the following curves have the given slope.

x = 4 cos t, y = 4 sin t; slope = 1/2

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 = t + 1/t, y = t - 1/t; slope = 1

Find all the points at which the following curves have the given slope.

x = 2 + √t, y = 2 - 4t; slope = -8

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 ≤ 2π
B: x = 10 sin 3t, y = 10 cos 3t; a ≤ t ≤ b

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 ≤ 2
B: x = t1/3 + t, y = 3 + t2/3; a ≤ t ≤ b

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 = sin 2t, y = 2 sin t; 0 ≤ t ≤ 2π

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 = sin 4t, y = sin 3t; 0 ≤ t ≤ 2π

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 the curve for a = 4 and b = 2, for various values of n. 

c. Describe how the curves change as n increases.

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 numbers. Graph the hyperbola with a = b = 1. Indicate clearly the direction in which the curve is generated as t increases from t = -π to t = π.

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 specific values of a and b, and use a graphing utility to plot different trochoids. In particular, explore the difference between the cases a > b and 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 equations

Use a graphing utility to explore the dependence of the curve on the parameters a, b, and c.

A general hypocycloid is described by the equations

Use a graphing utility to explore the dependence of the curve on the parameters a and b.

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 parametric equations

x(θ) = a cos θ + cos nθ, y(θ) = a sin θ + sin nθ. 

The distance from the moon to the planet is taken to be 1, the distance from the planet to the Sun is a, and n is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of n produce loops for a fixed value of a.

a. a = 4, n = 3.

b. a = 4, n = 4.

c. a = 4, n = 5.

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 is approximately a = 389.2 times the distance from Earth to our moon.

b. Plot a graph of the path of Callisto (one of Jupiter’s moons) that corresponds to values of a = 727.5 and n = 259.6. Plot a small portion of the graph to see the detailed behavior of the orbit.

c. Plot a graph of the path of Io (another of Jupiter’s moons) that corresponds to values of a = 1846.2 and n = 2448.8. Plot a small portion of the path of Io to see the loops in its orbit. 

Data from Exercise 102

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 parametric equations

x(θ) = a cos θ + cos nθ, y(θ) = a sin θ + sin nθ. 

The distance from the moon to the planet is taken to be 1, the distance from the planet to the Sun is a, and n is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of n produce loops for a fixed value of a.

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 by

x = 80t, y = -4.9t2 + 3000, for t ≥ 0,

where the origin is the point on the ground directly beneath the plane at the moment of the release. Graph the trajectory of the packet and find the coordinates of the point where the packet lands.

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 by

x = 100t, y = -4.9t2 + 4000, for t ≥ 0,

where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

A projectile launched from the ground with an initial speed of 20 m/s and a launch angle u follows a trajectory approximated by

x = (20 cos θ)t, y = -4.9t2 + (20 sin θ)t,

where x and y are the horizontal and vertical positions of the projectile relative to the launch point (0, 0).

a. Graph the trajectory for various values of u in the range 0 < θ < π/2.

b. Based on your observations, what value of θ gives the greatest range (the horizontal distance between the launch and landing points)?

Explain and carry out a method for graphing the curve x = 1 + cos2 y - sin2 y using parametric equations and a graphing utility.

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

Prove that the equations

x = 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.

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 parametric description of the curve that separates the two regions.

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 is a positive integer.

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 is a positive integer.

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.

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.

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.

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.

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 converges and |a| < |b|, then the series converges.

d. Viewed as a function of r, the series 1 + r2 + r3 + · · · takes on all values in the interval (1/2, ∞).

e. Viewed as a function of r, the seriestakes on all values in the interval  (-1/2, ∞).

Evaluate each series or state that it diverges.

Evaluate each series or state that it diverges.

Evaluate each series or state that it diverges.

Evaluate each series or state that it diverges.

Evaluate the series two ways.

a. Use a telescoping series argument.

b. Use a geometric series argument after first simplifying

Evaluate the series two ways.

a. Use a telescoping series argument.

b. Use a geometric series argument after first simplifying 

The Greek philosopher Zeno of Elea (who lived about 450 b.c.) invented many paradoxes, the most famous of which tells of a race between the swift warrior Achilles and a tortoise. Zeno argued 

The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead. 

In other words, by giving the tortoise a head start, Achilles will never overtake the tortoise because every time Achilles reaches the point where the tortoise was, the tortoise has moved ahead. Resolve this paradox by assuming that Achilles gives the tortoise a 1-mi head start and runs 5 mi/hr to the tortoise’s 1 mi/hr. How far does Achilles run before he overtakes the tortoise, and how long does it take?

The Greeks solved several calculus problems almost 2000 years before the discovery of calculus. One example is Archimedes’ calculation of the area of the region R bounded by a segment of a parabola, which he did using the “method of exhaustion.” As shown in the figure, the idea was to fill R with an infinite sequence of triangles. Archimedes began with an isosceles triangle inscribed in the parabola, with area A1, and proceeded in stages, with the number of new triangles doubling at each stage. He was able to show (the key to the solution) that at each stage, the area of a new triangle is 1/8 of the area of a triangle at the previous stage; for example, A2 = 1/8 A1, and so forth. Show, as Archimedes did, that the area of R is 4/3 times the area of A1.

a. Evaluate the series

b. For what values of a does the series

converge, and in those cases, what is its value?

Suppose you take out a home mortgage for $180,000 at a monthly interest rate of 0.5%. If you make payments of $1000/month, after how many months will the loan balance be zero? Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer using infinite series.

Suppose you borrow $20,000 for a new car at a monthly interest rate of 0.75%. If you make payments of $600/month, after how many months will the loan balance be zero? Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer using infinite series.

A fishery manager knows that her fish population naturally increases at a rate of 1.5% per month. At the end of each month, 120 fish are harvested. Let Fn be the fish population after the nth month, where F0 = 4000 fish. Assume that this process continues indefinitely. Use infinite series to find the long-term (steady-state) population of the fish.

Suppose that you take 200 mg of an antibiotic every 6 hr. The half-life of the drug is 6 hr (the time it takes for half of the drug to be eliminated from your blood). Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood.

In 1978, in an effort to reduce population growth, China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more young boys than girls. To solve this problem, some people have suggested replacing the one-child policy with a one-son policy: A family may have children until a boy is born. Suppose that the one-son policy were implemented and that natural birth rates remained the same (half boys and half girls). Using geometric series, compare the total number of children under the two policies.

An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction p of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is ultimately transmitted by the window? Assume the amount of incoming light is 1.

Suppose a rubber ball, when dropped from a given height, returns to a fraction p of that height. In the absence of air resistance, a ball dropped from a height h requires √2h/g seconds to fall to the ground, where g ≈ 9.8 m/s2 is the acceleration due to gravity. The time taken to bounce up to a given height equals the time to fall from that height to the ground. How long does it take a ball dropped from 10 m to come to rest?

Imagine that the government of a small community decides to give a total of $W, distributed equally, to all its citizens. Suppose that each month each citizen saves a fraction p of his or her new wealth and spends the remaining 1 - p in the community. Assume no money leaves or enters the community, and all the spent money is redistributed throughout the community. 

a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of $W increased (in terms of p)? Economists refer to this increase in the investment as the multiplier effect. 

b. Evaluate the limits p→0 and p→1, and interpret their meanings.

The fractal called the snowflake island (or Koch island) is constructed as follows: Let I0 be an equilateral triangle with sides of length 1. The figure I1 is obtained by replacing the middle third of each side of I0 with a new outward equilateral triangle with sides of length 1/3 (see figure). The process is repeated where In + 1 is obtained by replacing the middle third of each side of In with a new outward equilateral triangle with sides of length 1/3n + 1. The limiting figure as n→∞ is called the snowflake island.

a. Let Ln be the perimeter of In. Show that

b. Let An be the area of In. Find It exists!

a. Consider the number 0.555555 . . ., which can be viewed as the series Evaluate the geometric series to obtain a rational value of 0.555555 . .

b. Consider the number 0.54545454 . . ., which can be represented by the series Evaluate the geometric series to obtain a rational value of the number. 

c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length p, say, n1, n2 . . . , np, where n1, . . . ,np are integers between 0 and 9. Explain how to use geometric series to
obtain a rational form for

d. Try the method of part (c) on the number

e. Prove that

If we know thatthen what can we say about

Is it true that if the terms of a series of positive terms decrease to zero, then the series converges? Explain using an example.

Can the Integral Test be used to determine whether a series diverges?

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