Engr 233 - Applied Advanced Calculus Test 1
Midterm Questions And Answers 2026
This test has 3 problems, each problem is worth 20 points, and you have 60 minutes to complete
this test. To receive full credit, you must show your work.
Problem 1. The position of a particle at time t is described by the function →r (t) = ⟨ 23t3, t, t2⟩.
(a) Compute the velocity and acceleration of the particle at time t.
(b) What is the total distance travelled by the particle between the times t = 1 and t = 3
(c) Find the curvature of →r (t).
Problem 2. Consider the function f (x, y) = 9 − x2 − y2.
(a) Find the equation for the plane that is tangent to the graph of f at the point (3, 2, −4).
(b) Find the vector equation of the line that passes through the point (0, 0, 9) and travels in the
normal direction to the graph of f at (0, 0, 9).
(c) Find the point of intersection between the plane computed in part (a) and the line computed
in part (b).
(d) ∂f ∂f
If x = u2 − uv and y = euv , use the chain rule to compute
2
and .
∂u ∂v
Problem 3. Let F→ be the vector field given by F→ (x, y, z) = ⟨xz2 + 3yx2, xy sin(z − 1), 2y − z⟩.
(a) Compute curl(F→ ).
(b) Compute div(F→ ).
(c) Find the rate of change of div(F→ ) in the direction →v = ⟨2, −6, −3⟩ at the point (3, 2, 1).
Midterm Questions And Answers 2026
This test has 3 problems, each problem is worth 20 points, and you have 60 minutes to complete
this test. To receive full credit, you must show your work.
Problem 1. The position of a particle at time t is described by the function →r (t) = ⟨ 23t3, t, t2⟩.
(a) Compute the velocity and acceleration of the particle at time t.
(b) What is the total distance travelled by the particle between the times t = 1 and t = 3
(c) Find the curvature of →r (t).
Problem 2. Consider the function f (x, y) = 9 − x2 − y2.
(a) Find the equation for the plane that is tangent to the graph of f at the point (3, 2, −4).
(b) Find the vector equation of the line that passes through the point (0, 0, 9) and travels in the
normal direction to the graph of f at (0, 0, 9).
(c) Find the point of intersection between the plane computed in part (a) and the line computed
in part (b).
(d) ∂f ∂f
If x = u2 − uv and y = euv , use the chain rule to compute
2
and .
∂u ∂v
Problem 3. Let F→ be the vector field given by F→ (x, y, z) = ⟨xz2 + 3yx2, xy sin(z − 1), 2y − z⟩.
(a) Compute curl(F→ ).
(b) Compute div(F→ ).
(c) Find the rate of change of div(F→ ) in the direction →v = ⟨2, −6, −3⟩ at the point (3, 2, 1).
