# Calculus III Exam #1 Chapter 11 Spring 2014 ```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
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 &times; 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 ____________
```