Stewart 9e Section 10.2: Calculus with Parametric Curves

1 Find dy/dx · Level 1

Find \(\dfrac{d y}{d x}\). \(x = t/(1+t), y = \sqrt{1+t}\)

2 Find dy/dx · Level 1

Find \(\dfrac{d y}{d x}\). \(x = t e^t, y = t + \sin t\)

3 Tangent Line · Level 2

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. \(x = t^3 + 1, y = t^4 + t; t = -1\)

4 Tangent Line · Level 2

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. \(x = \sqrt{t}, y = t^2 - 2 t; t = 4\)

5 Tangent Line · Level 2

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. \(x = t \cos t, y = t \sin t; t = \pi\)

6 Tangent Line · Level 2

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. \(x = e^t \sin(\pi t), y = e^{2 t}; t = 0\)

7 Tangent Line - Two Methods · Level 2

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. \(x = 1 + \ln t, y = t^2 + 2; (1, 3)\)

8 Tangent Line - Two Methods · Level 2

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. \(x = 1 + \sqrt{t}, y = e^{t^2}; (2, e)\)

9 Tangent Line and Graph · Level 2

Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. \(x = t^2 - t, y = t^2 + t + 1; (0, 3)\)

10 Tangent Line and Graph · Level 2

Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. \(x = \sin(\pi t), y = t^2 + t; (0, 2)\)

11 Concavity · Level 3

Find \(\dfrac{d y}{d x}\) and \(\dfrac{d^2 y}{d x^2}\). For which values of \(t\) is the curve concave upward? \(x = t^2 + 1, y = t^2 + t\)

12 Concavity · Level 3

Find \(\dfrac{d y}{d x}\) and \(\dfrac{d^2 y}{d x^2}\). For which values of \(t\) is the curve concave upward? \(x = t^3 + 1, y = t^2 - t\)

13 Concavity · Level 3

Find \(\dfrac{d y}{d x}\) and \(\dfrac{d^2 y}{d x^2}\). For which values of \(t\) is the curve concave upward? \(x = e^t, y = t e^{-t}\)

14 Concavity · Level 3

Find \(\dfrac{d y}{d x}\) and \(\dfrac{d^2 y}{d x^2}\). For which values of \(t\) is the curve concave upward? \(x = t^2 + 1, y = e^t - 1\)

15 Concavity · Level 3

Find \(\dfrac{d y}{d x}\) and \(\dfrac{d^2 y}{d x^2}\). For which values of \(t\) is the curve concave upward? \(x = t - \ln t, y = t + \ln t\)

16 Concavity · Level 3

Find \(\dfrac{d y}{d x}\) and \(\dfrac{d^2 y}{d x^2}\). For which values of \(t\) is the curve concave upward? \(x = \cos t, y = \sin 2 t, 0 < t < \pi\)

17 Horizontal/Vertical Tangents · Level 3

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. \(x = t^3 - 3 t, y = t^2 - 3\)

18 Horizontal/Vertical Tangents · Level 3

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. \(x = t^3 - 3 t, y = t^3 - 3 t^2\)

19 Horizontal/Vertical Tangents · Level 3

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. \(x = \cos \theta, y = \cos 3 \theta\)

20 Horizontal/Vertical Tangents · Level 3

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. \(x = e^{\sin \theta}, y = e^{\cos \theta}\)

21 Extreme Points · Level 3

Use a graph to estimate the coordinates of the rightmost point on the curve \(x = t - t^6, y = e^t\). Then use calculus to find the exact coordinates.

22 Extreme Points · Level 3

Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve \(x = t^4 - 2 t, y = t + t^4\). Then find the exact coordinates.

23 Graphing · Level 2

Graph the curve in a viewing rectangle that displays all the important aspects of the curve. \(x = t^4 - 2 t^3 - 2 t^2, y = t^3 - t\)

24 Graphing · Level 2

Graph the curve in a viewing rectangle that displays all the important aspects of the curve. \(x = t^4 + 4 t^3 - 8 t^2, y = 2 t^2 - t\)

25 Multiple Tangents at a Point · Level 3

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.

26 Self-Intersection · Level 3

Graph the curve \(x = -2 \cos t, y = \sin t + \sin 2 t\) to discover where it crosses itself. Then find equations of both tangents at that point.

27 Trochoid · Level 3

(a) Find the slope of the tangent line to the trochoid \(x = r \theta - d \sin \theta, y = r - d \cos \theta\) in terms of \(\theta\). (See Exercise 10.1.40.)

Answer for 27(a)

(b) Show that if \(d < r\), then the trochoid does not have a vertical tangent.

Answer for 27(b)

Enter your answer directly below each part above.

28 Astroid · Level 3

(a) Find the slope of the tangent to the astroid \(x = a \cos^3 \theta, y = a \sin^3 \theta\) in terms of \(\theta\). (Astroids are explored in the Laboratory Project on page 649.)

Answer for 28(a)

(b) At what points is the tangent horizontal or vertical?

Answer for 28(b)

(c) At what points does the tangent have slope \(1\) or \(-1\)?

Answer for 28(c)

Enter your answer directly below each part above.

29 Tangent Slope at Specific Value · Level 3

At what point(s) on the curve \(x = 3 t^2 + 1, y = t^3 - 1\) does the tangent line have slope \(\dfrac{1}{2}\)?

30 Tangent Through External Point · Level 4

Find equations of the tangents to the curve \(x = 3 t^2 + 1, y = 2 t^3 + 1\) that pass through the point \((4, 3)\).

31 Area of Ellipse · Level 3

Use the parametric equations of an ellipse, \(x = a \cos \theta, y = b \sin \theta, 0 \leq \theta \leq 2 \pi\), to find the area that it encloses.

32 Area Enclosed · Level 3

Find the area enclosed by the curve \(x = t^2 - 2 t, y = \sqrt{t}\) and the \(y\)-axis.

33 Area Under Curve · Level 3

Find the area enclosed by the \(x\)-axis and the curve \(x = t^3 + 1, y = 2 t - t^2\).

34 Area of Astroid · Level 3

Find the area of the region enclosed by the astroid \(x = a \cos^3 \theta, y = a \sin^3 \theta\). (Astroids are explored in the Laboratory Project on page 649.)

35 Area Under Trochoid · Level 3

Find the area under one arch of the trochoid of Exercise 10.1.40 for the case \(d < r\).

36 Loop - Area, Volume, Centroid · Level 4

Let \(cal(R)\) be the region enclosed by the loop of the curve in Example 1.

(a) Find the area of \(cal(R)\).

Answer for 36(a)

(b) If \(cal(R)\) is rotated about the \(x\)-axis, find the volume of the resulting solid.

Answer for 36(b)

(c) Find the centroid of \(cal(R)\).

Answer for 36(c)

Enter your answer directly below each part above.

37 Arc Length - Numerical · Level 2

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. \(x = t + e^{-t}, y = t - e^{-t}, 0 \leq t \leq 2\)

38 Arc Length - Numerical · Level 2

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. \(x = t^2 - t, y = t^4, 1 \leq t \leq 4\)

39 Arc Length - Numerical · Level 2

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. \(x = t - 2 \sin t, y = 1 - 2 \cos t, 0 \leq t \leq 4 \pi\)

40 Arc Length - Numerical · Level 2

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. \(x = t + \sqrt{t}, y = t - \sqrt{t}, 0 \leq t \leq 1\)

41 Exact Arc Length · Level 3

Find the exact length of the curve. \(x = 1 + 3 t^2, y = 4 + 2 t^3, 0 \leq t \leq 1\)

42 Exact Arc Length · Level 3

Find the exact length of the curve. \(x = e^t - t, y = 4 e^{\dfrac{t}{2}}, 0 \leq t \leq 2\)

43 Exact Arc Length · Level 3

Find the exact length of the curve. \(x = t \sin t, y = t \cos t, 0 \leq t \leq 1\)

44 Exact Arc Length · Level 3

Find the exact length of the curve. \(x = 3 \cos t - \cos 3 t, y = 3 \sin t - \sin 3 t, 0 \leq t \leq \pi\)

45 Exact Arc Length · Level 3

Graph the curve and find its exact length. \(x = e^t \cos t, y = e^t \sin t, 0 \leq t \leq \pi\)

46 Exact Arc Length · Level 3

Graph the curve and find its exact length. \(x = \cos t + \ln(\tan\left(\dfrac{t}{2}\right)), y = \sin t, \dfrac{\pi}{4} \leq t \leq 3 \dfrac{\pi}{4}\)

47 Arc Length - Numerical · Level 3

Graph the curve \(x = \sin t + \sin 1.5 t, y = \cos t\) and find its length correct to four decimal places.

48 Length of Loop · Level 3

Find the length of the loop of the curve \(x = 3 t - t^3, y = 3 t^2\).

49 Simpson's Rule · Level 3

Use Simpson's Rule with \(n = 6\) to estimate the length of the curve \(x = t - e^t, y = t + e^t, -6 \leq t \leq 6\).

50 Witch of Agnesi - Simpson · Level 3

In Exercise 10.1.43 you were asked to derive the parametric equations \(x = 2 a \cot \theta, y = 2 a \sin^2 \theta\) for the curve called the witch of Maria Agnesi. Use Simpson's Rule with \(n = 4\) to estimate the length of the arc of this curve given by \(\dfrac{\pi}{4} \leq \theta \leq \dfrac{\pi}{2}\).

51 Distance Traveled · Level 3

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. \(x = \sin^2 t, y = \cos^2 t, 0 \leq t \leq 3 \pi\)

52 Distance Traveled · Level 3

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. \(x = \cos^2 t, y = \cos t, 0 \leq t \leq 4 \pi\)

53 Length of Ellipse · Level 4

Show that the total length of the ellipse \(x = a \sin \theta, y = b \cos \theta, a > b > 0\), is \( L = 4 a \displaystyle\int_{0}^{\dfrac{\pi}{2}} \sqrt{1 - e^2 \sin^2 \theta} d \theta \) where \(e\) is the eccentricity of the ellipse \(\left(e = \dfrac{c}{a}\right)\), where \(c = \sqrt{a^2 - b^2}\).

54 Length of Astroid · Level 3

Find the total length of the astroid \(x = a \cos^3 \theta, y = a \sin^3 \theta\), where \(a > 0\).

55 Epitrochoid · Level 4

(a) Graph the epitrochoid with equations \( x = 11 \cos t - 4 \cos\left(11 \dfrac{t}{2}\right) \) \( y = 11 \sin t - 4 \sin\left(11 \dfrac{t}{2}\right) \) What parameter interval gives the complete curve?

Answer for 55(a)

(b) Use your CAS to find the approximate length of this curve.

Answer for 55(b)

Enter your answer directly below each part above.

56 Cornu's Spiral · Level 4

A curve called Cornu's spiral is defined by the parametric equations \( x = C(t) = \displaystyle\int_{0}^{t} \cos(\pi u^2/2) d u \) \( y = S(t) = \displaystyle\int_{0}^{t} \sin(\pi u^2/2) d u \) where \(C\) and \(S\) are the Fresnel functions that were introduced in Chapter 5.

(a) Graph this curve. What happens as \(t \rightarrow \infty\) and as \(t \rightarrow -\infty\)?

Answer for 56(a)

(b) Find the length of Cornu's spiral from the origin to the point with parameter value \(t\).

Answer for 56(b)

Enter your answer directly below each part above.

57 Surface Area - Numerical · Level 3

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. \(x = t \sin t, y = t \cos t, 0 \leq t \leq \dfrac{\pi}{2}\)

58 Surface Area - Numerical · Level 3

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. \(x = \sin t, y = \sin 2 t, 0 \leq t \leq \dfrac{\pi}{2}\)

59 Surface Area - Numerical · Level 3

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. \(x = t + e^t, y = e^{-t}, 0 \leq t \leq 1\)

60 Surface Area - Numerical · Level 3

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. \(x = t^2 - t^3, y = t + t^4, 0 \leq t \leq 1\)

61 Exact Surface Area · Level 3

Find the exact area of the surface obtained by rotating the given curve about the \(x\)-axis. \(x = t^3, y = t^2, 0 \leq t \leq 1\)

62 Exact Surface Area · Level 3

Find the exact area of the surface obtained by rotating the given curve about the \(x\)-axis. \(x = 2 t^2 + \dfrac{1}{t}, y = 8 \sqrt{t}, 1 \leq t \leq 3\)

63 Exact Surface Area - Astroid · Level 3

Find the exact area of the surface obtained by rotating the given curve about the \(x\)-axis. \(x = a \cos^3 \theta, y = a \sin^3 \theta, 0 \leq \theta \leq \dfrac{\pi}{2}\)

64 Surface Area - Rotated Curve · Level 4

Graph the curve \( x = 2 \cos \theta - \cos 2 \theta, y = 2 \sin \theta - \sin 2 \theta \) If this curve is rotated about the \(x\)-axis, find the exact area of the resulting surface. (Use your graph to help find the correct parameter interval.)

65 Surface Area About y-axis · Level 3

Find the surface area generated by rotating the given curve about the \(y\)-axis. \(x = 3 t^2, y = 2 t^3, 0 \leq t \leq 5\)

66 Theory - Inverse Function · Level 4

If \(f'\) is continuous and \(f'(t) \neq 0\) for \(a \leq t \leq b\), show that the parametric curve \(x = f(t), y = g(t), a \leq t \leq b\), can be put in the form \(y = F(x)\). [Hint: Show that \(f^{-1}\) exists.]

67 Theory - Surface Area Derivation · Level 4

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 \(y = F(x), a \leq x \leq b\).

68 Curvature Formula · Level 5

The curvature at a point \(P\) of a curve is defined as \( \kappa = |\dfrac{d \phi}{d s}| \) where \(\phi\) 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 rate of change of \(\phi\) with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at \(P\) and will be studied in greater detail in Chapter 13.

(a) For a parametric curve \(x = x(t), y = y(t)\), derive the formula \( \kappa = |\cdot(x) \cdot.double(y) - \cdot.double(x) \cdot(y)| / (\cdot(x)^2 + \cdot(y)^2)^{\dfrac{3}{2}} \) where the dots indicate derivatives with respect to \(t\), so \(\cdot(x) = \dfrac{d x}{d t}\). [Hint: Use \(\phi = \arctan\left(\dfrac{d y}{d x}\right)\) and Formula 2 to find \(\dfrac{d \phi}{d t}\). Then use the Chain Rule to find \(\dfrac{d \phi}{d s}\).]

Answer for 68(a)

(b) By regarding a curve \(y = f(x)\) as the parametric curve \(x = x, y = f(x)\), with parameter \(x\), show that the formula in part (a) becomes \( \kappa = |\dfrac{d^2 y}{d x^2}| / (1 + \left(\dfrac{d y}{d x}\right)^2)^{\dfrac{3}{2}} \)

Answer for 68(b)

Enter your answer directly below each part above.

69 Curvature of Parabola · Level 3

(a) Use the formula in Exercise 69(b) to find the curvature of the parabola \(y = x^2\) at the point \((1, 1)\).

Answer for 69(a)

(b) At what point does this parabola have maximum curvature?

Answer for 69(b)

Enter your answer directly below each part above.

70 Curvature of Cycloid · Level 3

Use the formula in Exercise 69(a) to find the curvature of the cycloid \(x = \theta - \sin \theta, y = 1 - \cos \theta\) at the top of one of its arches.

71 Curvature of Line and Circle · Level 3

(a) Show that the curvature at each point of a straight line is \(\kappa = 0\).

Answer for 71(a)

(b) Show that the curvature at each point of a circle of radius \(r\) is \(\kappa = \dfrac{1}{r}\).

Answer for 71(b)

Enter your answer directly below each part above.

72 Involute of Circle · Level 4

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 \(r\) and center \(O\) and the initial position of \(P\) is \((r, 0)\), and if the parameter \(\theta\) is chosen as in the figure, show that parametric equations of the involute are \( x = r(\cos \theta + \theta \sin \theta), y = r(\sin \theta - \theta \cos \theta) \)

73 Cow Grazing Area · Level 5

A cow is tied to a silo with radius \(r\) by a rope just long enough to reach the opposite side of the silo. Find the grazing area available for the cow.

74 Example - Arc Length of Cycloid · Level 3

Find the length of one arch of the cycloid \(x = r(\theta - \sin \theta), y = r(1 - \cos \theta)\).

75 Example - Surface Area of Sphere · Level 3

Show that the surface area of a sphere of radius \(r\) is \(4 \pi r^2\).

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