Question: Find an equation of the plane. The

> Show that the lines with symmetric equations x = y = z and x + 1 = y/2 = z/3 are skew, and find the distance between these lines.

> Where does the normal line to the parabola y = x2 - 1 at the point (-1,0) intersect the parabola a second time? Illustrate with a sketch.

> Find equations of the planes that are parallel to the plane x + 2y - 2z = 1 and two units away from it.

> Show that the distance between the parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is //

> Find the distance between the given parallel planes. 6z = 4y - 2x, 9z = 1 - 3x + 6y

> Which of the following four lines are parallel? Are any of them identical? Li: x= 1 + 6t, y=1 – 3t, z= 12t + 5 L2: x= 1 + 2t, y= t, z =1 + 4t L3: 2x – 2 = 4 – 4y = z + 1 L4: r = (3, 1, 5) + t(4, 2, 8)

> Which of the following four planes are parallel? Are any of them identical? Pi: 3x + 6у — 32 — 6 Р:: 4х — 12у + 82 3D 5 Р:: 9у — 1 + 3х + 62 Р:: z — х + 2у — 2

> Find parametric equations for the line through the point (0, 1, 2) that is perpendicular to the line x = 1 + t, y = 1 - t, z = 2t and intersects this line.

> Find parametric equations for the line through the point (0, 1, 2) that is perpendicular to the line x = 1 + t, y = 1 - t, z = 2t and intersects this line.

> (a) Find the point at which the given lines intersect: (b). Find an equation of the plane that contains these lines. r = (1, 1, 0) + 1(1, –1, 2) r= (2,0, 2) + s(-1, 1,0)

> Find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c.

> Differentiate the function. f (x) = x ln x - x

> At what point on the y = 1 + 2ex – 3x curve is the tangent line parallel to the line ?Illustrate by 3x – y = 5 graphing the curve and both lines.

> Find dy/dx by implicit differentiation. x4 + x2y2 + y3 = 5

> Find an equation for the plane consisting of all points that are equidistant from the points (2, 5, 5) and (-6, 3, 1).

> Find an equation for the plane consisting of all points that are equidistant from the points (1, 0, -2) and (3, 4, 0).

> Find symmetric equations for the line of intersection of the planes. z = 2x - y - 5, z = 4x + 3y - 5

> Find symmetric equations for the line of intersection of the planes. 5x - 2y - 2z = 1, 4x + y + z = 6

> (a). Find parametric equations for the line of intersection of the planes and (b). find the angle between the planes. 3x - 2y + z = 1, 2x + y - 3z = 3

> (a). Find parametric equations for the line of intersection of the planes and (b). find the angle between the planes. x + y + z = 1, x + 2y + 2z = 1

> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x + 2y - z = 2, 2x - 2y + z = 1

> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x + 4y - 3z = 1, 23x + 6y + 7z = 0

> Find direction numbers for the line of intersection of the planes x + y + z = 1 and x + z = 0.

> Find equations of both lines that are tangent to the curve y = x3 – 3x2 + 3x -3 and are parallel to the line 3x – y = 15.

> Where does the line through (-3, 1, 0) and (-1, 5, 6) intersect the plane 2x + y - z = -2?

> Differentiate. y = secθ  tanθ 

> Find the point at which the line intersects the given plane. x = t - 1, y = 1 + 2t, z = 3 - t; 3x - y + 2z = 5

> Find the point at which the line intersects the given plane. x = 2 - 2t, y = 3t, z = 1 + t; x + 2y – z = 7

> Use intercepts to help sketch the plane. 6x + 5y - 3z = 15

> Use intercepts to help sketch the plane. 6x - 3y + 4z = 6

> Use intercepts to help sketch the plane. 3x + y + 2z = 6

> Use intercepts to help sketch the plane. 2x + 5y + z = 10

> Find an equation of the plane. The plane that passes through the line of intersection of the planes x - z = 1 and y + 2z = 3 and is perpendicular to the plane x + y - 2z = 1

> Find an equation of the plane. The plane that passes through the point (1, 5, 1) and is perpendicular to the planes 2x + y - 2z = 2 and x + 3z = 4

> Find an equation of the tangent line to the curve y = x4 + 1 that is parallel to the line 32x – y = 15.

> Find an equation of the plane. The plane that passes through the points (0, -2, 5) and (-1, 3, 1) and is perpendicular to the plane 2z = 5x + 4y

> Find an equation of the plane. The plane that passes through the point (3, 1, 4) and contains the line of intersection of the planes x + 2y + 3z = 1 and 2x - y + z = -3

> Explain why the natural logarithmic function y = ln x is used much more frequently in calculus than the other logarithmic functions y = logb x.

> Find an equation of the plane. The plane that passes through the point (6, -1, 3) and contains the line with symmetric equations x/3 = y + 4 = z/2.

> Find an equation of the plane. The plane that passes through the point (3, 5, -1) and contains the line x = 4 - t, y = 2t - 1, z = -3t.

> Find an equation of the plane. The plane through the points (3, 0, -1), (-2, -2, 3), and (7, 1, -4)

> Find an equation of the plane. The plane through the points (2, 1, 2), (3, -8, 6), and (-2, -3, 1)

> Find an equation of the plane. The plane through the origin and the points (3, -2, 1) and (1, 1, 1)

> Find an equation of the plane. The plane through the points (0, 1, 1), (1, 0, 1), and (1, 1, 0)

> 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

> Find an equation of the plane. The plane through the point (3, -2, 8) and parallel to the plane z = x + y

> Find an equation of the plane. The plane through the point (2, 0, 1) and perpendicular to the line x = 3t, y = 2 - t, z = 3 + 4t

> Find an equation of the plane. The plane through the point (-1, 1 2 , 3) and with normal vector i + 4j + k

> Find an equation of the plane. The plane through the point (5, 3, 5) and with normal vector 2i + j – k.

> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. у — 1 y z - 2 L: 1 -1 3 х — 2 L2: у — 3 y -2 7 || 2.

> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. х — 2 L1: у — 3 z - 1 1 -2 -3 х — 3 у+ 4 z - 2 y 3 L2: 1 -7 ||

> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. L1: x = 3 + 2t, y = 4 - t, z = 1 + 3t L2: x = 1 + 4s, y = 3 - 2s, z = 4 + 5s

> Find parametric equations for the line segment from (-2, 18, 31) to (11, -4, 48).

> (a) The curve y = x / (1 + x2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3, 0.3). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

> Find a vector equation for the line segment from (6, -1, 9) to (7, 6, 0).

> (a). Find parametric equations for the line through (2, 4, 6) that is perpendicular to the plane x - y + 3z = 7 (b). In what points does this line intersect the coordinate planes?

> Find parametric equations and symmetric equations for the line. The line of intersection of the planes x + -y + 3z = 1 and x - y + z = 1

> Find parametric equations and symmetric equations for the line. The line through (2, 1, 0) and perpendicular to both i + j and j + k

> Find parametric equations and symmetric equations for the line. The line through the points (-8, 1, 4) and (3, -2, 4)

> Two tanks are participating in a battle simulation. Tank A is at point (325, 810, 561) and tank B is positioned at point (765, 675, 599). (a). Find parametric equations for the line of sight between the tanks. (b). If we divide the line of sight into 5

> Let L1 be the line through the points (1, 2, 6) and (2, 4, 8). Let L2 be the line of intersection of the planes P1 and P2, where P1 is the plane x - y + 2z + 1 = 0 and P2 is the plane through the points (3, 2, -1), (0, 0, 1), and (1, 2, 1). Calculate the

> Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2.

> Let L1 be the line through the origin and the point (2, 0, -1). Let L2 be the line through the points (1, -1, 1) and (4, 1, 3). Find the distance between L1 and L2.

> If v1, v2, and v3 are non-coplanar vectors, let (These vectors occur in the study of crystallography. Vectors of the form n1v1 + n2v2 + n3v3, where each ni is an integer, form a lattice for a crystal. Vectors written similarly in terms of k1, k2, and k

> Suppose that a ± 0. (a). If a ∙ b = a ∙ c, does it follow that b = c? (b). If a × b = a × c, does it follow that b = c? (c). If a ∙ b = a ∙ c and a × b = a × c, does it follow that b = c?

> Prove that a.c b. c (а Х Ь) . (с х d) - a·d b.d

> Use Exercise 50 to prove that a × (b × c) + b × (c × a) + c × (a × b) = 0 Exercise 50: Prove Property 6 of cross products, that is, a × (b × c) = (a ∙ c) b – (a ∙ b )c

> Prove Property 6 of cross products, that is, a × (b × c) = (a ∙ c) b – (a ∙ b )c

> Prove that (a – b) × (a + b) = 2 (a × b).

> If a + b + c = 0, show that a × b = b × c = c × a

> For what value of x does the graph f(x) = ex – 2x of have a horizontal tangent?

> (a). Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance d from P to the plane is (b). Use the formula in part (a) to find the distance from the point P (2, 1, 4) to the plane through the points Q (1, 0,

> Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy -plane. Find the maximum and minimum values of the length of the vector u × v. In what direction does u × v point?

> (a). A horizontal force of 20 lb is applied to the handle of a gearshift lever as shown. Find the magnitude of the torque about the pivot point P. (b). Find the magnitude of the torque about P if the same force is applied at the elbow Q of the lever.

> A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P. 60 N 70° T10 P

> (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (1, 0, 1), Q (-2, 1, 3), R (4, 2, 5)

> Prove the property of cross products (Theorem 11). Property 4: (a + b) × c = a × c + b × c

> Find a formula for f (n)(x) if f (x) = ln(x - 1).

> The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are |a | = 3 and |b | = 2. (a). Find |a × b |. (b). Use the right-hand rule to decide whether the components of a × b are positiv

> Find |u × v | and determine whether u × v is directed into the page or out of the page. |v| =5 45° |u| = 4

> Find the work done by a force F = 8 i - 6 j + 9k that moves an object from the point (0, 10, 8) to the point (6, 12, 20) along a straight line. The distance is measured in meters and the force in newtons.

> (a) How is the number e defined? (b) Use a calculator to estimate the values of the limits correct to two decimal places. What can you conclude about the value of e? 2.7h – 1 lim 2.8h – 1 lim and h h

> Reduce the equation to one of the standard forms, classify the surface, and sketch it. x2 + y2 - 2x - 6y - z + 10 = 0

> Small birds like finches alternate between flapping their wings and keeping them folded while gliding (see Figure 1). In this project we analyze this phenomenon and try to determine how frequently a bird should flap its wings. Some of the principles are

> If g(x) = x/ex, find g(n)(x).

> Use logarithmic differentiation to find the derivative of the function. y = x cos x

> If f (x) = x2 / (1 + x), find f “(1).

> Use logarithmic differentiation to find the derivative of the function. y = xx

> (a) If f (x) = (x2 – 1) ex, find f ‘(x) and f ‘’(x). (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f ‘, and f ‘’.

> (a) If f (x) = (x2 – 1) / (x2 + 1), find f ‘(x) and f ‘’(x). (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f ‘, and f ‘’.

> Differentiate. y = 1 t 3 + 2 t 2 – 1

> Differentiate the function. B(y) = cy-6

> Differentiate. y = x2 + 1 / x3 + 1

> Differentiate. f (z) = (1 – ez)(z + ez)

> Find dy/dx by implicit differentiation. 2x2 + xy - y2 = 2

> Differentiate. J(v) = (v3 - 2v)(v-4 + v-2)

> Differentiate the function. g(x) = x2 (1 -2x)

> Sketch the parabolas y = x2 and y = x2 - 2x + 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?

> If c > ½, how may points through the point (0,c) are normal lines to the parabola y = x2? What if c ≤ ½ ?

> Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y = x2. Where do these lines intersect?