Vector Calculus Questions and Answers

Consider the following functions. f₁(x) = x, f₂(x) = x², f3(x) = 5x - 4x² g(x) = c1f1(x) + c2f2(x) + c3f3(x) Solve for C1, C2, and C3 so that g(x) = 0 on the interval (-∞, ∞). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.) {C1, C2, C3} Determine whether f1, f2, f3 are linearly independent on the interval (-∞, ∞). linearly dependent linearly independent

For the polynomial below, - 1 is a zero. g(x)=x³ + 5x² + x - 3 Express g (x) as a product of linear factors.

Divide. (-2x4-12x³-4x² +15x+18) ÷ (-2x²+4) Write your answer in the following form: Quotient +

Suppose you deposit $5000 at 3% interest compounded continously. Find the average value of you account during the first 4 years.

Graph the parabola. y=-3x²-24x-42 Plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. Then click on the graph-a- function button.

All edges of a cubs are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters? (Hint:V = s³)

Find parametric equations for the line through the point (0, 2, 2) that is parallel to the plane x + y + z = 3 and perpendicular to the line x = 1 + t, y = 2-t, z = 2t. (Use the parameter t.) (x(t), y(t), z(t)) = __________

(a) Show that the lines L₁ : x = -2 2t; y = 1 + 4t; z 2 t and L₂ :x=2-4t; y = 3 + 8t; z = 1+ 2t are parallel. (b) Find the equation of the plane P₁that contains these lines. (c) Find the equation of the plane P₂ that is perpendicularto P₁ and contains the points (-2, 1, 2) and (2, 3, 1). What is the equation of the linethat is the intersection of the planes P₁ and P₂?-

For each types of vector spaces given above, give a set of vectors that is linearly dependent. Show your work. (a)M 2x2

Show that the curve with parametric equations x = t cos(t), y = t sin(t), z = t lies on the cone z² = x² + y². suppose a curve is defined by the parametric equations x = t cos(t), y = t sin(t), z = t; then x²=______x² + y² =_____and z^2=____ thus, x² + y²=(____) (cos²(t)(cos² (t) + sin²(t)) = z², so the curve lies on the cone z² = x² + y².,

Sketch the curve with the given vector equation by finding the following points. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) r(t) = t² i + t^4j+t^6 k r(-2) (x, y, z) = r(-1) (x, y, z) = r(0) (x, y, z)= r(1) (x, y, z)= r(2) (x, y, z)=

Consider the following vector equation.r(t) = (3t − 4, t² +2) a) Find r'(t)

Describe the curve defined by the vector function r(t) = (4 + t, 1 + 3t, -4 + 5t). Solution The corresponding parametric equations are x=___, y = 1 + 3t, z =____ which we recognize from the equations x = xo + at, y = Yo + bt, z = Zo + ct as parametric equations of a line passing through the point (4, 1, -4) and parallel to the vector (1, 3, 5). Alternatively, we could observe that the function can be written as r = ro + tv, where ro = (4, 1, -4) and v =_______ and this is the vector equation of a line as given by the equation r=ro + tv.

Try to sketch by hand the curve of intersection of the parabolic cylinder y = x² and the top half of the ellipsoid x² + 3y^2 + 3z^2 = 9. Then find parametric equations for this curve. (x(t), y(t), z(t)) =_________for -1.3 ≤ts 1.3 Use these equations to graph the curve. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.)

Find a vector function, r(t), that represents the curve of intersection of the two surfaces. the cone z = √x² + y² and the plane z = 7+ y

Find the domain of the vector function. (Enter your answer in interval notation.) r(t)=(t-2/t+2)i+ sin(t)j + In(16 - t²) k

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) =

(1) Determine the linear transformation that rotates the complex plane through theta about z (2) Determine a linear transformation that maps |z - 1|= 1 onto |z - 2i |= 2.

Let f(x) = 4x+6. Find f=1¹(2)2(Simplify your answer.)

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

Find a vector equation and parametric equations for the line. (Use the parameter t.) the line through the point (7, 0, -4) and parallel to the line x = 4 - 4t, y = -1 + 2t, z = 6 + 8t r(t) = (x(t), y(t), z(t)) =