Download Calculus Maximus: Worksheet 7.1 - Introduction to Parametric and Vector Calculus and more Summaries Vector Analysis in PDF only on Docsity!
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Worksheet 7. 1 —Intro to Parametric & Vector Calculus
Show all work. No calculator unless explicitly stated.
Short Answer
- If
2
x t 1 and
3 t
y e , find
dy
dx
- If a particle moves in the xy plane so that at any time t! 0 , its position vector is
2 2
ln t 5 t ,3 t ,
find its velocity vector at time t 2.
- A particle moves in the xy plane so that at any time t , its coordinates are given by
5
x t 1 ,
4 3
y 3 t 2 t. Find its acceleration vector at t 1.
- If a particle moves in the xy plane so that at time t , its position vector is
2
sin 3 ,
t t
§ S·
, find the
velocity vector at time
t
S
- A particle moves on the curve y ln x so that its x - component has velocity x t c( )^ t 1 for t t 0. At
time t 0 , the particle is at the point 1, 0. Find the position of the particle at time t 1.
- A particle moves in the (^) xy plane in such a way that its velocity vector is
3
1 t t ,. If the position
vector at t 0 is 5, 0^ , find the position of the particle at t 2.
- (Calculator Permitted) A particle moves in the xy plane so that the position of the particle is given by
x t ( ) 5 t 3sin t and y t ( ) 8 t 1 cos t. Find the velocity vector at the time when the particle’s
horizontal position is x 25.
Free Response:
- The position of a particle at any time t t 0 is given by
2
x t ( ) t 3 and
3
y t t.
(a) Find the magnitude of the velocity vector at time t 5.
(b) Find the total distance traveled by the particle from t 0 to t 5.
(c) Find
dy
dx
as a function of x.
- Point P x y , moves in the xy plane in such away that
dx
dt t
and 2
dy
t
dt
for t t 0.
(a) Find the coordinates of P in terms of t when t 1 , x ln 2, and y 0.
(b) Write an equation expressing y in terms of x.
(c) Find the average rate of change of y with respect to x as t varies from 0 to 4.
(d) Find the instantaneous rate of change of y with respect to x when t 1.
Multiple Choice:
- A parametric curve is defined by x sin t and y csc t for 0
t
S
. This curve is
(A) increasing & concave up (B) increasing & concave down (C) decreasing & concave up
(D) decreasing & concave down (E) decreasing with a point of inflection
- The parametric curve defined by x ln t , y t for t! 0 is identical to the graph of the function
(A) y ln x for all real x (B) y ln x for x! 0 (C)
x
y e for all real x
(D)
x
y e for x! 0 (E) ln
x
y e for x! 0
- The position of a particle in the xy plane is given by
2
x t 1 and y ln 2 t 3 for all t t 0. The
acceleration vector of the particle is
(A)
t
t
(B)
2
t
t
(C)
2
2 t 3
(D)
2
2 t 3
(E)
2
2 t 3
