(a) What does the equation y = x2 represent as a curve in R2? circle parabola hyperbola ellipse line (b) What does it represent as a surface in R3? parabolic cylinder ellipsoid cone hyperboloid elliptic paraboloid (c) What does the equation z = y2 represent? parabolic cylinder elliptic paraboloid cone ellipsoid hyperboloid
Describe the surface. x2 + z2 = 3 sphere ellipsoid hyperboloid circular cylinder elliptic cylinder hyperbolic cylinder parabolic cylinder elliptic paraboloid Sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) (Write an equation for the cross section at y = −3 using x and z.) (Write an equation for the cross section at y = 0 using x and z.) (Write an equation for the cross section at y = 3 using x and z.)
Describe the surface. y2 + 3z2 = 3 cone ellipsoid hyperboloid elliptic cylinder hyperbolic cylinder parabolic cylinder elliptic paraboloid hyperbolic paraboloid Sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) (Write an equation for the cross section at x = −3 using y and z.) (Write an equation for the cross section at x = 0 using y and z.) (Write an equation for the cross section at x = 3 using y and z.)
(a) Find and identify the traces of the quadric surface x2 + y2 − z2 = 64 given the plane. x = k Find the trace. Identify the trace. circle ellipse hyperbola parabola y = k Find the trace. Identify the trace. circle ellipse hyperbola parabola z = k Find the trace. Identify the trace. circle ellipse hyperbola parabola Describe the surface from one of the graphs in the table. ellipsoid elliptic paraboloid hyperbolic paraboloid cone hyperboloid of one sheet hyperboloid of two sheets (b) If we change the equation in part (a) to x2 − y2 + z2 = 64, how is the graph affected? The graph is rotated so that its axis is the x-axis. The graph is rotated so that its axis is the y-axis. The graph is rotated so that its axis is the z-axis The graph is shifted one unit in the negative y-direction. The graph is shifted one unit in the positive y-direction. (c) What if we change the equation in part (a) to x2 + y2 + 2y − z2 = 63? The graph is rotated so that its axis is the x-axis. The graph is rotated so that its axis is the y-axis. The graph is rotated so that its axis is the z-axis The graph is shifted one unit in the negative y-direction. The graph is shifted one unit in the positive y-direction.
Use traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) x = y2 + 3z2 (Write an equation for the cross section at z = 0 using x and y.) (Write an equation for the cross section at y = 0 using x and z.) (Write an equation for the cross section at x = −3 using y and z.) (Write an equation for the cross section at x = 0 using y and z.) (Write an equation for the cross section at x = 3 using y and z.) Identify the surface. elliptic cone parabolic cylinder hyperbolic paraboloid elliptic paraboloid elliptic cylinder hyperboloid of two sheets ellipsoid hyperboloid of one sheet
Use traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) 16x2 + 9y2 + 9z2 = 144 (Write an equation for the cross section at z = 0 using x and y.) (Write an equation for the cross section at y = 0 using x and z.) (Write an equation for the cross section at x = 0 using y and z.) Identify the surface. elliptic cone parabolic cylinder hyperboloid of two sheets ellipsoid elliptic paraboloid hyperbolic paraboloid hyperboloid of one sheet elliptic cylinder
Use traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) 7x2 + y + 7z2 = 0 (Write an equation for the cross section at z = 0 using x and y.) (Write an equation for the cross section at y = −7 using x and z.) (Write an equation for the cross section at y = 7 using x and z.) (Write an equation for the cross section at x = 0 using y and z.) Identify the surface. hyperboloid of one sheet hyperbolic paraboloid ellipsoid hyperboloid of two sheets elliptic cone elliptic cylinder parabolic cylinder elliptic paraboloid
Find the domain of the vector function. (Enter your answer using interval notation.) r(t) = ⟨ln(t+3), t 16−t2, 2t⟩
Find each of the following limits. lim t→0 e −6t = lim t→0 t2 sin2(t) = lim t→0 sin(4t) = Find the limit of the given vector function. lim t→0 (e−6ti + t2 sin2(t)j + sin(4t)k)
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = ⟨t2−3, t⟩
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = e−ti + etj
At what points does the curve r(t) = ti + (3t − t2)k intersect the paraboloid z = x2 + y2? (smaller t-value) (x, y, z) = ( ) (larger t-value) (x, y, z) = ( )
Example Video Example Describe the curve defined by the vector function r(t) = ⟨2+t, 3+4t, −5+2t⟩ Solution The corresponding parametric equations are x = , y = 3+4t, z = which we recognize from the equations x = x0+at, y = y0+bt, z = z0+ct as parametric equations of a line passing through the point (2, 3, −5) and parallel to the vector ⟨1, 4, 2⟩. Alternatively, we could observe that the function can be written as r = r0+tv, where r0 = ⟨2, 3, −5⟩ and v = , and this is the vector equation of a line as given by the equation r = r0+tv
Consider the following vector equation. r(t) = ⟨3t−4, t2+2⟩ (a) Find r′(t). r′(t) = (b) Sketch the plane curve together with the position vector r(t) and the tangent vector r′(t) for the given value of t = −1.
Consider the following vector equation. r(t) = 4sin(t)i − 3cos(t)j (a) Find r′(t). r′(t) = (b) Sketch the plane curve together with the position vector r(t) and the tangent vector r′(t) for the given value of t = 3π/4.
Find the derivative, r′(t), of the vector function. r(t) = ⟨e−t, 4t−t3, ln(t)⟩ r′(t) =
Find the derivative, r′(t), of the vector function. r(t) = t4i + cos(t5)j + sin2(t)k r′(t) =
Find the derivative, r′(t), of the vector function. r(t) = 1 6+ti + t 6+tj + t2 6+tk r′(t) =
Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = ⟨t2 − 3t, 1 + 4t, 1 3 t3 + 1 2 t2⟩, t = 4 T(4) =
Consider the following vector function. r(t) = ⟨t2 − 2t, 1 + 3t, 13 t3 + 12 t2⟩ Find each of the following. r′(t) = r′(2) = |r′(2)| = Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = ⟨t2 − 2t, 1 + 3t, 13 t3 + 12 t2⟩, t = 2 T(2) =
Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = ⟨tan−1(t), 2e6t, 12tet⟩, t = 0 T(0) =
Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = cos(t)i + 8tj + 3sin(2t)k, t = 0 T(0) =
Consider the following vector function. r(t) = cos(t)i + 8tj + 3sin(2t)k Find each of the following. r′(t) = r′(0) = |r′(0)| = Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = cos(t)i + 8tj + 3sin(2t)k, t = 0 T(0) =
Find the unit tangent vector T(t) at the given point on the curve. r(t) = ⟨t3 + 1, 6t − 9, 4/t⟩ (2, −3, 4)