Stewart 9e Section 10.2: Calculus with Parametric Curves

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Stewart 9e Section 10.2: Calculus with Parametric Curves 0/75
1 Differentiation: Composite, Implicit, and Inverse Functions · Level 1
Find \(\dfrac{d y}{d x}\). \(x = t/(1+t), y = \sqrt{1+t}\)
2 Differentiation: Composite, Implicit, and Inverse Functions · Level 1
Find \(\dfrac{d y}{d x}\). \(x = t e^t, y = t + \sin t\)
3 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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.)
(b) Show that if \(d < r\), then the trochoid does not have a vertical tangent.

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28 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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.)
(b) At what points is the tangent horizontal or vertical?
(c) At what points does the tangent have slope \(1\) or \(-1\)?

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29 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Applications of Integration · Level 3
Find the area enclosed by the curve \(x = t^2 - 2 t, y = \sqrt{t}\) and the \(y\)-axis.
33 Applications of Integration · Level 3
Find the area enclosed by the \(x\)-axis and the curve \(x = t^3 + 1, y = 2 t - t^2\).
34 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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.)
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35 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the area under one arch of the trochoid of Exercise 10.1.40 for the case \(d < r\).
36 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Let \(cal(R)\) be the region enclosed by the loop of the curve in Example 1.
(a) Find the area of \(cal(R)\).
(b) If \(cal(R)\) is rotated about the \(x\)-axis, find the volume of the resulting solid.
(c) Find the centroid of \(cal(R)\).

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37 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · Level 3
Find the exact length of the curve. \(x = t \sin t, y = t \cos t, 0 \leq t \leq 1\)
44 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the length of the loop of the curve \(x = 3 t - t^3, y = 3 t^2\).
49 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Applications of Integration · 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 Applications of Integration · Level 3
Find the total length of the astroid \(x = a \cos^3 \theta, y = a \sin^3 \theta\), where \(a > 0\).
55 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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?
(b) Use your CAS to find the approximate length of this curve.

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56 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 \(t \rightarrow -\infty\)?
(b) Find the length of Cornu's spiral from the origin to the point with parameter value \(t\).

Enter your answer directly below each part above.

57 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Applications of Integration · 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 Applications of Integration · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Applications of Integration · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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.
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(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 \(\dfrac{d \phi}{d s}\).]
(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 \( \kappa = |\dfrac{d^2 y}{d x^2}| / (1 + \left(\dfrac{d y}{d x}\right)^2)^{\dfrac{3}{2}} \)

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69 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
(a) Use the formula in Exercise 69(b) to find the curvature of the parabola \(y = x^2\) at the point \((1, 1)\).
(b) At what point does this parabola have maximum curvature?

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70 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
(a) Show that the curvature at each point of a straight line is \(\kappa = 0\).
(b) Show that the curvature at each point of a circle of radius \(r\) is \(\kappa = \dfrac{1}{r}\).

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72 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · 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 \( x = r(\cos \theta + \theta \sin \theta), y = r(\sin \theta - \theta \cos \theta) \)
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73 Analytical Applications of Differentiation · 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.
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74 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the length of one arch of the cycloid \(x = r(\theta - \sin \theta), y = r(1 - \cos \theta)\).
75 Applications of Integration · Level 3
Show that the surface area of a sphere of radius \(r\) is \(4 \pi r^2\).

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