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mathematics
calculus early transcendentals 9th

Questions and Answers of Calculus Early Transcendentals 9th

Find the derivative of the vector function.r(t) = a + t b + t2c
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The projection of the curve r(t) = (cos 2t,
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = t2i + t4j + t6k
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The vector equations r(t) = (t, 2t, t + 1)
Draw the projection of the curve onto the given plane.r(t) = (t2, t3, t−3), yz-plane
Find the unit tangent vector T(t) at the given point on the curve.r(t) = (t3 + 1, 3t − 5, 4/t), (2, −2, 4)
(a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature.r(t) = (t, t2, 4)
Find a vector equation and parametric equations for the line segment that joins P to Q.P(−2, 1, 0), Q(5, 2, −3)
(a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature. r(t) = (1, t, įr*)
Find a vector equation and parametric equations for the line segment that joins P to Q.P(0, 0, 0), Q(−7, 4, 6)
A disk of radius 1 is rotating in the counterclockwise direction at a constant angular speed ω. A particle starts at the center of the disk and moves toward the edge along a fixed radius so that its
(a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature. r(t) = (1. 4r°, r²)
Find a vector equation and parametric equations for the line segment that joins P to Q.P(3.5, −1.4, 2.1), Q(1.8, 0.3, 2.1)
In designing transfer curves to connect sections of straight railroad tracks, it’s important to realize that the acceleration of the train should be continuous so that the reactive force exerted by
Find the point on the curve r(t) = (2 cos t, 2 sin t, et), 0 ≤ t ≤ π, where the tangent line is parallel to the plane √3 x + y = 1.
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen.x
Find an equation of the plane that contains the curve with the given vector equation.r(t) = (t, 4, t2)
Find an equation of the plane that contains the curve with the given vector equation.r(t) = (t, t2, t)
Find an equation of the plane that contains the curve with the given vector equation.r(t) = (sin t, cos t, − cos t)
Find an equation of the plane that contains the curve with the given vector equation.r(t) = (2t, sin t, t + 1)
Evaluate the integral. (ti - tj+ 3t° k) dt Jo
(a) Is the curvature of the curve C shown in the figure greater at P or at Q? Explain.(b) Estimate the curvature at P and at Q by sketching the osculating circles at those points. y A
Evaluate the integral. ( (21/2 i + (t + 1) Vi k) dt
Evaluate the integral. 1 i + j+ k dt t + 1 t? + 1 12 + 1
Evaluate the integral. (/4 (sec t tan ti + t cos 2t j + sin? 2t cos 2t k) dt
Use a computer algebra system to compute the curvature function κ(t). Then graph the space curve and its curvature function. Comment on how the curvature reflects the shape of the curve.r(t) = (t
Evaluate the integral. cos ti j+ sec?t k) dt +
Graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.r(t) = (tet, e−t, t)
The position function of a spacecraft isand the coordinates of a space station are s6, 4, 9d. The captain wants the craft to coast into the space station. When should the engines be turned off? 4
The curvature of a plane parametric curve x = f(t), y = g(t) is given bywhere the dots indicate derivatives with respect to t.Use Theorem 10 to prove the given formula for curvature. | iÿ – jä| к
The curvature of a plane parametric curve x = f(t), y = g(t) is given bywhere the dots indicate derivatives with respect to t.Find the curvature of the curve x = t2, y = t3. | iÿ – jä| к
The curvature of a plane parametric curve x = f(t), y = g(t) is given bywhere the dots indicate derivatives with respect to t.Find the curvature of the curve x = a cos ωt, y = b sin ωt. | iÿ –
The curvature of a plane parametric curve x = f(t), y = g(t) is given bywhere the dots indicate derivatives with respect to t.Find the curvature of the curve x = et cos t, y = et sin t. | iÿ –
If r(t) = a cos ωt + b sin ωt, where a and b are constant vectors, show that r(t) × r'(t) = ωa × b.
(a) Graph the curve with parametric equations(b) Show that the curve lies on the hyperboloid of one sheet 144x2 + 144y2 − 25z2 = 100. x = % sin 8t – sin 18t 27 26 y = - cos 18t 27 26 cos 8t + 39
If r(t) ≠ 0, show that 1 -r(t) • r'(t). d |r()|= Tr()| dt
If u(t) = r(t) · [r'(t) × r"(t)], show thatu'(t) = r(t) · [r'(t) × r"'(t)]
Use Formula 14 to find the torsion at the given value of t.r(t) = (sin t, 3t, cos t ), t = π/2
(a) Show that dB/ds is perpendicular to B.(b) Show that dB/ds is perpendicular to T.(c) Deduce from parts (a) and (b) that dB/ds is parallel to N.
Use Formula 14 to find the torsion at the given value of t. r(t) = (r, 21, 1), t- 1 2t,
Use Theorem 15 to find the torsion of the given curve at a general point and at the point corresponding to t = 0.r(t) = (et, e–t, t)
Use Theorem 15 to find the torsion of the given curve at a general point and at the point corresponding to t = 0.r(t) = (cos t, sin t, sin t)
A space curve C given by r(t) = (x(t), y(t), z(t)) is called planar if it lies in a plane.(a) Show that C is planar if and only if there exist scalars a, b, c, and d, not all zero, such that ax(t) +
Calculate the given quantity ifa = i + j − 2kb = 3i − 2j + kc = j − 5k(a) 2a + 3b(b)|b|(c) a · b(d) a × b(e)|b × c|(f) a · (b × c)(g) c × c(h) a × (b × c)(i) compa b(j) proja b(k) The
Describe and sketch the surface.y2 + 9z2 = 9
Using the vectors shown in the figure, write each sum or difference as a single vector.(a) AB(vector) + BC(vector)(b) CD(vector) + DB(vector)(c) DB(vector) − AB(vector)(d) DC(vector) + CA(vector) +
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3, |u × v| =
Find a vector equation and parametric equations for the line.The line through the point (6, 0, −2) and parallel to the linex = 4 − 3t y = −1 + 4t z = 6 + 5t
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3, u × v = v ×
Find a · b.a = (4, 1, 1/4), b = (6, −3, −8)
Suppose v1 and v2 are vectors with |v1| = 2, |v2| = 3, and v1 · v2 = 5. Let v3 = projv1 v2, v4 = projv2 v3, v5 = projv3 v4, and so on. Compute n=1 Vn
Copy the vectors in the figure and use them to draw the following vectors.(a) u + v(b) u − v(c) 2u(d) −1/2 v(e) 3u + v(f) v − 2u u V
Prove the specified property of cross productsProperty 1: a × b = –b × a
Find an equation of the plane.The plane through the point (3, 2, 1) and with normal vector 5i + 4j + 6k
Determine whether the given vectors are orthogonal, parallel, or neither.(a) a = (9, 3), b = (−2, 6)(b) a = (4, 5, −2), b = (3, −1, 5)(c) a = −8i + 12j + 4k,
Prove the specified property of cross productsProperty 2: (ca) × b = c(a × b) = a × (cb)
Find an equation of the plane.The plane through the point (−3, 4, 2) and with normal vector (6, 1, −1)
Find an equation of the sphere with center (−1, 4, 5) that just touches (at only one point) the(a) xy-plane(b) yz-plane(c) xz-plane.
Prove the specified property of cross productsProperty 3: a × (b + c) = a × b + a × c
Find an equation of the plane.The plane through the point (5, −2, 4) and perpendicular to the vector −i + 2j + 3k
Which coordinate plane is closest to the point (7, 3, 8)? Find an equation of the sphere with center (7, 3, 8) that just touches (at one point) that coordinate plane.
Find an equation of the plane.The plane through the origin and perpendicular to the linex = 1 − 8t y = −1 − 7t z = 4 + 2t
Describe in words the region of R3 represented by the equation(s) or inequalities.z = −2
Find an equation of the plane.The plane through the point (1, 3, −1) and perpendicular to the line x + 3 z - 1 4 -y =
Describe in words the region of R3 represented by the equation(s) or inequalities.x = 3
Identify and sketch the graph of each surface.x = 3
Find an equation of the plane.The plane through the point (9, −4, −5) and parallel to the plane z = 2x − 3y.
Describe in words the region of R3 represented by the equation(s) or inequalities.y ≥ 1
The initial point of a vector v in V2 is the origin and the terminal point is in quadrant II. If v makes an angle 5π/6 with the positive x-axis and |v| = 4, find v in component form.
Identify and sketch the graph of each surface.x = z
Find an equation of the plane.The plane through the point (2.1, 1.7, −0.9) and parallel to the plane 2x − y + 3z = 1
Find the acute angle between the lines. Use degrees rounded to one decimal place.y = 4 − 3x, y = 3x + 2
Describe in words the region of R3 represented by the equation(s) or inequalities.x < 4
Identify and sketch the graph of each surface.y = z2
Find the acute angle between the lines. Use degrees rounded to one decimal place.5x − y = 8, x + 3y = 15
Find an equation of the plane.The plane that contains the line x = 1 + t, y = 2 − t, z = 4 − 3t and is parallel to the plane 5x + 2y + z = 1
Describe in words the region of R3 represented by the equation(s) or inequalities.−1 ≤ x ≤ 2
Identify and sketch the graph of each surface.x2 = y2 + 4z2
Describe in words the region of R3 represented by the equation(s) or inequalities.z = y
Identify and sketch the graph of each surface.4x − y + 2z = 4
Describe in words the region of R3 represented by the equation(s) or inequalities.x2 + y2 = 4, z = −1
Identify and sketch the graph of each surface.−4x2 + y2 − 4z2 = 4
Find the direction cosines and direction angles of the vector.(4, 1, 8)
Reduce the equation to one of the standard forms, classify the surface, and sketch it.4x2 − y + 2z2 = 0
Describe in words the region of R3 represented by the equation(s) or inequalities.x2 + y2 = 4
Identify and sketch the graph of each surface.y2 + z2 = 1 + x2
Find an equation of the plane.The plane through the points (3, 0, −1), (−2, −2, 3), and (7, 1, −4)
Find the direction cosines and direction angles of the vector.(−6, 2, 9)
Describe in words the region of R3 represented by the equation(s) or inequalities.y2 + z2 ≤ 25
Identify and sketch the graph of each surface.4x2 + 4y2 − 8y + z2 = 0
Find the direction cosines and direction angles of the vector.3i − j − 2k
Describe in words the region of R3 represented by the equation(s) or inequalities.x2 + z2 ≤ 25, 0 ≤ y ≤ 2
Identify and sketch the graph of each surface.x = y2 + z2 − 2y − 4z + 5
Find the direction cosines and direction angles of the vector.−0.7i + 1.2j − 0.8k
Describe in words the region of R3 represented by the equation(s) or inequalities.x2 + y2 + z2 = 4
Find the direction cosines and direction angles of the vector.(c, c, c), where c > 0
Describe in words the region of R3 represented by the equation(s) or inequalities.x2 + y2 + z2 ≤ 4
A rower wants to row her kayak across a channel that is 1400 ft wide and land at a point 800 ft upstream from her starting point. She can row (in still water) at 7 ft/s and the current in the channel
Describe in words the region of R3 represented by the equation(s) or inequalities.1 ≤ x2 + y2 + z2 ≤ 5
A pilot is steering a plane in the direction N 45° W at an airspeed (speed in still air) of 180 mi/h. A wind is blowing in the direction S30° E at a speed of 35mi/h. Find the true course and the

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