Calculus 3

Test File

Spring 1993

 

Test #1

 

1.)        Let u = 2i - 3j + 5k, v = i + j - k and w = 3i + k.  Find the following, if possible.  If it is not possible, state why it is not possible.

            a.)        2u - 3v

            b.)        an equation for the plane through the point (1, 0, -1) with w as its normal vector

            c.)        a unit vector in the same direction as v

            d.)        u × v

            e.)        u × (v × w)

            f.)         two non-zero vectors that are orthogonal to both v and w

            g.)        a set of parametric equations for the line through (1, 2, 3) with v as its direction vector

            h.)        symmetric form for the equations for the line on the point (3, 0, -1) with w as its direction vector

            i.)         area of the triangle with u and w as adjacent sides

            j.)         the angle between v and w

            k.)        the projection of u in the direction of v

2.)        Prove that for any two vectors, u and v, that u × v = v × u.

3.)        Graph the following equations.  All equations are in three dimensional space.

            a.)  x + y = 1

            b.)  18φ2 - 9πφ + π2 = 0 (spherical coordinates)

4.)        Convert the point (1, _2, 2) from rectangular coordinates into cylindrical coordinates and into spherical coordinates.

5.)        For each of the graphs below, select an equation from the list of equations, that could give the graph.  Also choose the name of the surface from the list of names.  Use the appropriate letter to indicate your choice.

 

            equations -     a.)        x2 + y2 + z2 = 1                       b.)        x2 - y2 - z2 = 1

                                    c.)        y2 - x2 = z                                d.)        x2 - y2 = z

                                    e.)        y2 = z                                       f.)         2x + 3y + 2z = 6

                                    g.)        x2 + y2 = z                               h.)        x2 + y2 - z2 = 1

                                    i.)         25y2 + z2 = 25                        j.)         x2 + y2 - z2 = 0

            names -          A.)  paraboloid                                  B.)  cylinder

                                    C.)  hyperboloid of two sheets         D.)  ellipsoid

                                    D.)  hyperboloid of one sheet          E.)  cone

                                    F.)  hyperbolic paraboloid                G.)  thing-a-ma-jig    

                                    H.)  hyperbolic paraboloid                I.)  doo-hickey

6.)        The dot product is a             not a                .

7.)        Prove that ½cv½ = ½c½ ½v½.

8.)        Given the points A(1, 0, 2), B(4, 5, 0), C(0, -3, 5), D(-2, 0, 7) and E(0, -1, 7) do the following:

            a.)        Find the symmetric and parametric forms for the equations for the line connecting A and B.

            b.)        Find the midpoint, P, of the line segment AB and the plane on P that is perpendicular to AB.

            c.)        Find the angle between DE and the positive x-axis.

            d.)        Find two unit vectors orthogonal to the plane determine by C, D and E.  Find an equation of the plane.

            e.)        How far is the point A from the plane in part d.)

            f.)         Find the point of intersection between the line in part a.) and the plane in part d.).

            g.)        Find the distance between the line joining A and B and the line joining C and E.

            h.)        Find the area of the parallelogram  with adjacent sides AE and AB.

            i.)         Find the volume of the parallelepiped determined by AE, AB and AC.

9.)        Find the plane containing the point (2, 0, 1) and perpendicular to the planes 2x - 4y - z - 7 = 0 and x - y + z - 1 = 0.

10.)     For the points A(1, 0, 2), B(4, 5, 0), C(0, -3, 5),     D(-2, 0, 7), E(0, -1, 7), find the point M where the line through A and B passes through the plane determined by C, D and E.

11.)     Identify the surface and graph:

            a.)        x2 + y2 + z2 = 1                       b.)        x2 - y2 - z2 = 1

            c.)        y2 - x2 = z                                d.)        x2 - y2 = z

            e.)        y2 = z                                       f.)         2x + 3y + 2z = 6

            g.)        x2 + y2 = z                               h.)        x2 + y2 - z2 = 1

            i.)         25y2 + z2 = 25                        j.)         x2 + y2 - z2 = 0

12.)     Given vectors u = 2i - 3j + k and v = i - 3j find the following:

            a.)        2u + 3v                                   b.)        a unit vector in the direction of u

            c.)        ½u½                                         d.)        ½2138u½

            e.)        the angle between u and v

13.)     Find all cross products among i, j and k.  Find a rule for determining these cross products.

14.)     Prove that u ´ v = -v ´ u.

 

Test #2

 

1.)        Find the domain of f(x,y,z) = z ln(x+y) - xesin z. 

2.)        For values of k = -1, 0, 1 draw :

            a.)        level curves for f(x,y) = x + y.

            b.)        level surfaces for f(x,y,z) = x2 - y2 - z2.

3.)        Using the definition of limit, prove that

                        lim(x,y)®(0,0) (4x2 + 4y2) = 0.

4.)        Determine if the following limits exist.

            a.)        lim(x,y)®(1,3) (x2 - y2)                

            b.)       

            c.)       

5.)        Find fx, fy, fxx, fyx, fxy and fyy if f(x,y) = xy - sin(xy).

6.)        Let f(x,y) be defined as follows:

 

            a.)        Is f continuous at (0, 0)?

            b.)        If f is not continuous at (0, 0), determine whether the discontinuity is removable or non-removable.  If f is continuous at (0, 0), determine whether or not it is differentiable there.

7.)        Let r(t) = ln(3t) i + cot(2t) j + e2t k.  Set up, BUT DO NOT EVALUATE, the integral for the arc length of the curve from t = 0 to t = 2π.

8.)        A projectile is fired with an initial speed of 1500 feet/sec at an angle of elevation of 60°.  How far had it gone when it hit the ground?

9.)        The equation of motion of a particle is given by r(t) = sin 2t i + cos 2t j + t k.  For t = π/2, find the following:

            a.)        principal unit tangent vector

            b.)        principal unit normal vector

            c.)        tangential component of acceleration

            d.)        normal component of acceleration.

10.)     If y = sec x find the equation of the circle of curvature at the point (0, 1).

11.)     Find the domain of f(x,y,u,v,w) = w ln(x-y) - xuev/w.

12.)     For values of k = -1, 0, 1 draw :

            a.)        level curves for f(x,y) = xy.

            b.)        level surfaces for f(x,y,z) = x2 + y2 - z2.

13.)     Using the definition of limit, prove that

                        lim(x,y)®(3,2) (3x - 4y) = 1.

14.)     Determine if the following limits exist.

            a.)        im(x,y)®(0,0) (x2 + y2)                 b.)       

            c.)       

            d.)       

            e.)       

15.)     Find fx, fy, fxx, fyx, fxy and fyy if f(x,y) = x2y + x sin(2y).

16.)     Let f(x,y) be defined as follows:

 

 

            a.)  Is f continuous at (0, 0)?

            b.)  Is f differentiable at (0, 0)?

17.)     Define :

            a.)        lim(x,y)®(a,b) f(x,y) = L               b.)        f(x,y)/y

18.)     Let r(t) = 2 sin(3t) i + 2 cos(3t) j + 4t k.

            a.)        Find the arc length of the curve from t = 0 to t = 2π.

            b.)        Find the integral of r(t) from t = 0 to t = 2π.

            c.)        Graph the path of the particle from t = 0 to t = 2π.

19.)     Let r(t) = (sin t)/t i + t1/2 j + cos t k. 

            a.)        Find the domain of r(t).

            b.)        Find where r(t) is continuous.

            c.)        If r(t) is discontinuous anywhere, determine whether the discontinuities are removable.

            d.)        Find limt®0 r(t).

20.)     A projectile is fired with an initial speed of 1500 feet/sec at an angle of elevation of 60°.  Find the velocity at time t and the maximum altitude.

21.)     The equation of motion of a particle is given by r(t) = 2t i + ln t j + t1/2 k.  For t = 1, find the velocity, speed, acceleration, principal unit tangent vector, principal unit normal vector, tangential component of acceleration, and normal component of acceleration.

22.)     Find the curvature and the equation of the circle of curvature for the curve given parametrically by x = t and y = 1/t at the point (1, 1).

23.)     Let r(t) = t2 i + (3t3 + 4t) j.  Find the time(s), when the velocity and acceleration vectors are orthogonal.

24.)     A barrel has the shape of an ellipsoid with equal pieces cut from the ends by planes perpendicular to the center axis of the barrel.  The height of the barrel is 12, the midsection radius is 5 and the end radii are both 3.  Find the volume of the barrel.

25.)     Prove that Dt[r(t) × u(t)] = r(t) × u¢(t) + r¢(t) × u(t).

26.)     If y = xex find the equation of the circle of curvature at       the point (0, 0).

 

Test #3

 

1.)        Find the area bounded by r = sin 2Θ.

2.)        Set up, BUT DO NOT EVALUATE a double integral that gives the area of the ellipse  x2/a2 + y2/b2= 1.

3.)        Use the total differential to approximate .

4.)        Let f(x, y) = 8x2 - 24xy + y2.  Find the absolute extrema of f on the triangular region bounded with vertices at (0, 0), (2, 3) and (0, 3).

5.)        For the following integral, switch the order of integration and then evaluate the resulting integral.

6.)        Find the volume of the solid bounded by x2 + y2 = 16, x = z, x = 0, y = 0, z = 0.

7.)        Suppose w = u2 cos v, u = xy + 2, and v = cos x.  Use the chain rule to find w/x and w/y.

8.)        Given that xz - xy = yz + 3, where z = f(x, y), find zx and zy.

9.)        Find the tangent plane to the curve y = x2 at the point (0, 1, 5).

10.)     Find the directional derivative of f(x, y, z) = xy2z3 at the point (1, 0, 1) in the direction of the vector u = 2i - j + 2k.

11.)     Use the definition of differentiability to show that f(x, y) = x2y - 2xy3 is continuous.

12.)     Use the total differential to approximate Ö[(1.01)3(0.97)].

13.)     Suppose w = u2 sin v, u = xy + 2, and v = cos x.  Use two different methods to find w/x and w/y.

14.)     Given that xyz - xy + xz = yz + 3, where z = f(x, y), find zx and zy.

15.)     Find the volume of the solid bounded by x2 + y2 = 16, x = z, x = 0, y = 0, z = 0.

16.)     Evaluate

17.)     Find the area bounded by x - y + 1 = 0, 7x - y - 17 and 2x + y + 2 = 0.

18.)     Find the area bounded by r = 3 + 2 sin Θ.

19.)     Evaluate

20.)     Use double integrals to derive the formulas for the following.

            a.)        volume of a cube with side length a.

            b.)        a circle of radius r.

            c.)        a right circular cylinder with radius r and height h.

21.)     Find the area of the region bounded by y = 1, y = x + 6 and above the parabola  y = x2.

22.)     Find the equations of the tangent plane and normal line to xyz - 4xz3 + y3 = 10 at the point (-1, 2, 1).

23.)     Let f(x, y) = x2 - 7xy + y2.  Let P be the point (1, -1).  Find the directional derivative at P in the direction of a = 2i + j.  Find a unit vector in the direction of greatest decrease of f at P.  Find the rate of change in the direction of greatest decrease.

24.)     Let f(x, y) = 8x2 - 24xy + y2. 

            a.)  Find all local extrema.

            b.)  Find the absolute extrema of f on the region bounded by y = x2 and y = x + 5.

25.)     Prove that the sphere given by x2 + y2 + z2 = 4 and the cone given by x2 + y2 - z2 = 0 are orthogonal at all points of iintersection.  Note - The surfaces F(x, y, z) = 0 and G(x, y, z) = 0 are said to be orthogonal at the point of intersection, P, if the normal vectors to the surfaces at P are orthogonal.

26.)     Let f(x, y) = x2 + (y - 1)2. 

            a.)        Find all local extrema.

            b.)        Find the absolute extrema of f on the triangle with vertices at the points      (0, 0), (4, 0) and (4, 4).

27.)     Find the points on the surface xyz = 1 that are closest to the origin.  Hint - the distance is minimized wherever the square of the distance is minimized.

 

Test #4

 

1.)        Find the center of mass of the region bounded by y = cos x, y = 0, x = -π/2 and x = π/2 if the density is ρ(x) = ½x½.

2.)        Set up, BUT DO NOT SOLVE, the integrals necessary to find the z-coordinate of the center of mass of the solid bounded

            by y = 0, x = 0, z = 0 and x + 2y + 3z = 6, where the density is ρ(x, y, z) = x.      

3.)        Find the area of the surface given by f(x, y) = x + y over the region bounded by x = 0, y = 0 and x + y = 2.

4.)        Find the center of mass of the solid in the first octant bounded by x2 + y2 + z2 = 9 if the density is constant.

5.)        Use a change of variables to find the volume of the solid below the surface         z = (x - y)ln(x+y) and above the region in the xy-plane bounded by y = x, y = x - 2, x + y = 0 and

            x + y = 1.  SET UP THE INTEGRAL BUT DO NOT SOLVE

6.)        Determine whether the vector field F(x, y) = 2xyi + x2j is conservative.

7.)        Find the divergence of the vector field F(x, y, z) = 6x2i - xy2j.

8.)        Evaluate the line integral over the specified path.

,  C : r(t) = 4ti + 3tj,  0 £ t £ 2

9.)        Evaluate the given line integral using the Fundamental Theorem of Line Integrals.

,  C : smooth curve from (0, 0) to (3, 8)

            Note : If you think it might help, f(x,y) = xy is a potential function for F(x, y) = yi + xj.

10.)     Use Green's Theorem to evaluate the line integral  where C is the boundary of the region lying between the graphs of y = 0 and y = 4 - x2.

11.)     Find the center of mass of the region bounded by y = sec x and y = 4 if the density is ρ(x) = ½x½.

12.)     Find the center of mass of the solid bounded by y = 0, x = 0, and x + 2y + 3z = 6, where the density is ρ(x, y, z) = xyz.

13.)     Find the area of the surface given by f(x, y) = x3 + y over the region bounded by x = 0, y = 0 and x + y = 2.

14.)     Find the center of mass of the solid bounded by x2 + y2 + z2 = 9 if the density is given by ρ(x, y, z) = (x2 + y2 + z2)3.

15.)     Find the center of mass of a right circular cylinder of height h and radius r if the density at a point is the sum of the distances A and B where A is the distance from the point to the base of the cylinder and B is the distance from the point to the axis of the cylinder.

16.)     Use a change of variables to find the volume of the solid below the surface         z = (x - y)e(x+y) and above the region in the xy-plane bounded by y = x, y = x - 2,   x + y = 0 and x + y = 1.

 

Final Exam

 

1.)        For u = i + 2j + 3k and v = i - k, find the following.

            a.)        u - 2v  b.)        u × v     c.)        ½ u ½

            d.)        u ´ v   e.)        v ´ v

2.)        Find equations for the line on the origin with direction

            vector i + 2j + 3k.

3.)        Graph each of the following.

            a.)        z = x2 + y2

            b.)        z2 = x2 + y2

            c.)        z = x2 - y2

4.)        Let r(t) = cos t i + sin t j - t k be the equation for the motion of a particle in space.  Find the following.

            a.)        r¢(t)                  b.)        T(t)                  c.)        lim t®3π/2 r(t)

     d.)                  e.)        ½ v(t) ½

5.)        Find the following limit if it exists.

                        lim(x,y)®(0,0)   x2y 

                                       x2 + y2

6.)        Let f(x, y, z) = x cos(yz).  Find the following.

            a.)        Ñf

            b.)        fxx

            c.)        directional derivative of f at (1, 0, 1) in the direction of 3j + 4k.

7.)        Use the chain rule to find dw/dt if w = x2 + y2, x = et and y = e-t.                           

8.)        Find the normal line to the surface z2 + x2 - y2 = 0 at the point (3, 5, 4). 

9.)        Let f(x, y) = 2x2 + 2xy + y2 + 4x - 3.

            a.)        Find all relative extrema.

            b.)        Find the absolute extrema in the square with vertices (0, 0), (0, -4), (4, -4) and (4, 0).

10.)     Switch the order of integration.       

11.)     Find the volume of the solid in the first octant bounded by x=1, y=1 and z + y = 1.

12.)     Change to cylindrical coordinates and evaluate.

13.)     Find the volume inside x2 + y2 = 4 and outside y2 + x2 - z2 = 0.