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Midterm Exam I, Calculus III, Sample A
- (10 points) Show that the 4 points P 1 = (0, 0 , 0), P 2 = (2, 3 , 0), P 3 = (1, − 1 , 1), P 4 = (1, 4 , −1) are coplanar (they lie on the same plane), and find the equation of the plane that contains them.
- (10 points) Find the equation of the plane that is equidistant from the points (3, 2 , 1) and (− 3 , − 2 , −1) (that is, every point on the plane has the same distance from the two given points).
- (6 points) Find the vector projection of b onto a if a = 〈 4 , 2 , 0 〉 and b = 〈 1 , 1 , 1 〉.
- (12 points) Consider the curve r(t) =
2 cos t i + sin t j + sin t k.
(a) (8 points) Find the unit tangent vector function T(t) and the unit normal vector function N(t). (b) (4 points) Compute the curvature κ.
- (10 points) Find the length of the curve with parametric equation:
r(t) = 〈et, et^ sin t, et^ cos t〉,
between the points (1, 0 , 1) and (e^2 π^ , 0 , e^2 π^ ).
- (12 points) A spaceship is traveling with acceleration a(t) = 〈et, t, sin 2t〉.
At t = 0, the spaceship was at the origin, r(0) = 〈 0 , 0 , 0 〈, and had initial velocity v(0) = 〈 1 , 0 , 0 〉. Find the position of the spaceship at t = π.
- (10 points) Write the equation of the tangent line to the curve with parametric equation r(t) = 〈
t, 1 , t^4 〉,
at the point (1, 1 , 1).
- (12 points) Using cylindrical coordinates, find the parametric equations of the curve that is the intersection of the cylinder x^2 + y^2 = 4 and the cone z =
x^2 + y^2. (This problem refers to the material not covered before midterm 1.)
- (6 points) Let f (x, y) = sin(x^2 + y^2 ) + arcsin(y^2 ). Calculate: ∂^2 f ∂x∂y
(This problem refers to the material not covered before midterm 1.)
- (12 points) Show that the following limit does not exist:
lim (x,y)→(0,0)
7 x^2 y(x − y) x^4 + y^4 Justify your answer. (This problem refers to the material not covered before midterm 1.)
Midterm Exam I, Calculus III, Sample B
- (6 Points) Find the center and radius of the following sphere x^2 + y^2 + z^2 − 6 x + 4z − 3 = 0.
- (6 Points) Write each combination of vectors as a single vector
(a)
AB +
BC
(b)
AC −
BC
(c)
AD +
DB +
BA.
- (6 Points) Find the cosine of the angle between the vectors a = 〈 1 , 2 , 3 〉 and b = 〈 2 , 0 , − 1 〉.
- (7 Points) Find the area the triangle with vertices P = (2, 1 , 7), Q = (1, 1 , 5), R = (2, − 1 , 1).
- (5 Points) Show that the line
x = 3 + t, y = 1 + 2t, z = 1 − 2 t
is parallel to the plane 2 x + 3y + 4z = 5.
- (6 Points) Find a vector parallel to the line of intersection for the two planes
x + 2y + 3z = 0 and x − 3 y + 2z = 0.
- (6 Points) Find cosine of the angle between intersection planes
2 x + y + z = 0 and 3 x − y + 2z = 0.
- (6 Points) Match the following equations with their graphs.
- (8 Points) Find the length of the curve r(t) = 〈t^2 , 2 t, ln t〉 from the point (1, 2 , 0) to the point (e^2 , 2 e, 1).
- (8 Points) Consider a particle whose acceleration is given by
a(t) = 〈t, t^2 , cos 2t〉
with initial velocity v(0) = 〈 1 , 0 , 1 〉. Find the velocity of the particle as a function of time.
Midterm Exam I, Calculus III, Sample C
- (10 Points) Find an equation of the sphere that has center at (1, − 2 , −5) and passes through the origin.
- (6 Points) Find a vector in the opposite direction as v = (− 5 , 3 , 7), and has length 6.
- Let A = (1, 0 , 0 , 0), B = (1, 2 , 2), C = (3, 0 , 1).
(a) (6 Points) Find the area of triangle ABC (b) (6 Points) Find the equation of the plane passing through A, B, and C (Write the answer in the form ax + by + cz = d).
- (a) (6 Points) Find parametric equations of the line which passes through the point (1, 1 , 1) and is perpendicular to the plane 2x + y + z = 0. (b) (6 Points) Find the equation of the plane passing through the point (1, 1 , 1) and is parallel to the plane 2x + y + z = 0.
- (3 Points) The rulling (lines on a cylinder) of the cylinder defined by equation y = z^2 is perpendicular to (check one)
xy-plane yz-plane xz-plane.
- (5 Points) Change the following cylindrical equation to a rectangular equation, and identify the surface. r^2 + z^2 − 2 r sin θ = 0. (This problem refers to the material not covered before midterm 1.)
- (4 Points) The point (− 1 , 1 ,
- is given in rectangular coordinates. Find the cylindrical and spherical coordinates for this point. (This problem refers to the material not covered before midterm 1.)
- A certain surface is defined by equation (y − z)^2 + x^2 = 1
(a) (4 Points) Find and sketch the traces on the horizontal planes z = − 1 , 0 , 1 (b) (4 Points) Which one of the figures best match with this equations? Circle the surface.
Midterm Exam I, Calculus III, Sample D
- (10 Points) Find an equation for the plane consisting of all points that are equidistant from P = (− 1 , 2 , −3) and Q = (4, − 2 , 2).
- ( 8 Points) Find all values of x such that the vectors 〈x, − 3 , x〉 and 〈− 5 , 2 , x〉 are orthogonal.
- 8 Points for each part) For each of the following pairs of planes P 1 , P 2 , determine whether P 1 and P 2 are parallel or intersect. If the planes are parallel, explain and find the distance between them; if the planes intersect, find the line of intersection.
P 1 : x + 2y − 4 z = 2 P 2 : − 2 x − 4 y + 8z = 1
P 1 : x + 2y − 4 z = 2 P 2 : 2 x + y + z = 1
- (a) (6 Points) Sketch a graph of the surface
x^2 +
y^2 4
− z = − 1.
Find and label the points at which the surface intersects the x-axis. (b) (6 Points) Find the equation of the curve of intersection C of the surface in part (a) and the plane y = −2.
- Let C be the space curve traced by the vector-valued function
r(t) = 4 cos ti + 4 sin tj + 3tk.
(a) (8 Points) Find the equation of the line tangent to the C at the point (4, 0 , 0). (b) (8 Points) Find the curvature κ(t) at the point (− 4 , 0 , 3 π). (c) (8 Points) Find the unit tangent vector T(t), normal vector N(t), and binormal vector B(t) at the point (− 4 , 0 , 3 π). (d) (8 Points) Find the length of the curve from (4, 0 , 0) to (4, 0 , 6 π).
- Let v(t) = 〈 4 t − 3 , 3 et, 4 cos 2t〉 be the velocity vector of a particle at time t.
(a) (6 Points) Find the acceleration vector a(t). (b) (6 Points) Find the position vector r(t) if the particle has has initial position r(0) = 〈 0 , 1 , 2 〉.
- (10 Points) A particle moves with position function
r(t) = 〈t^2 + 1, t, 3 t − 1 〉.
Find the tangential and normal components of acceleration at the point (2, 1 , 2).
