Calculus III Exam #1
Chapter 11
Spring 2014
Name ________________________________
Show all of your work on this paper. Solutions without correct supporting work will not earn credit. The best 10 of the
11 problems will count towards your grade.
1. Given 𝒖 = ⟨2, −3,1⟩ and 𝒗 = 2𝒋 − 5𝒌, find
a.
|𝒖| = ________________
b. 𝒖 ∙ 𝒗 = _________________
c. A unit vector in the opposite direction of u = __________________________
2. Given 𝒖 = ⟨2, −3,1⟩ and 𝒗 = 2𝒋 − 5𝒌. Find the following.
a. 𝑢 × 𝑣 = ______________
b.
angle between u and v to the nearest tenth of a degree = ________________
3. Given 𝒖 = ⟨2, −3,1⟩ and 𝒗 = 2𝒋 − 5𝒌.
a.
Find 𝑝𝑟𝑜𝑗→ 𝒖
𝑣
b. Find the area of the triangle formed by using vectors u and v as two adjacent sides of the triangle.
4.
5. Find parametric equations for the line through the points (1,0,-1) and (-1,4,1).
6. Find the length of the curve 𝒓(𝑡) = (t) 𝒊 + (2⁄3)𝑡
3⁄
2 𝒌,
0≤𝑡≤8
7. Given 𝒓(𝑡) = (6𝑠𝑖𝑛2𝑡)𝒊 + (6𝑐𝑜𝑠2𝑡)𝒋 + 5𝑡𝒌 find the following:
8.
a.
The unit tangent vector T
b.
Curvature = ________________________
a.
Evaluate:
b.
Evaluate:
lim (cos 2𝑡 𝒊 − 4 sin 𝑡 𝒋 +
𝑡→𝜋/2
2𝑡
𝜋
∫ (cos 2𝑡 𝒊 − 4 sin 𝑡 𝒋 +
2𝑡
𝜋
𝒌) = ________________________
𝒌) 𝑑𝑡 = ____________________________
9. A force F = ⟨𝟑, 𝟑, 𝟐⟩ (N) moves an object from P(1, 1, 0) to Q (6, 6, 0) along a linear path. What is the work done
by the force?
10. A projectile is fired over horizontal ground from the origin with an initial speed of 50 m/s. What firing angles will
produce a range of 250m?
11. Match the equations with its corresponding graph.
Equation a matches with graph ____________
Equation b matches with graph ____________
Equation c matches with graph ____________
Equation d matches with graph ____________
Equation e matches with graph ____________
Equation f matches with graph ____________