Discussion 5 Worksheet
Vector-valued functions and partial derivatives
Date: 9/10/2021
MATH 53 Multivariable Calculus
1 Integrals of Vector Functions
1. Evaluate R2
1hln(t)
t, eโtidt.
2. Suppose that ~
r00(t) = h6t, sin(t)iand it is known that ~
r0(0) = h1,โ1iand ~r(0) = h0,1i. Find
a formula for ~r(t).
2 Vector Function Basics
(a) Find the limit
lim
tโ0๎eโ3t,t2
sin2t,cos 2t๎.
(b) Find the limit
lim
tโโ ๎1 + t2
1โt2,arctan t, 1โeโ2t
t๎.
(c) Find a vector equation and parametric equations for the line segment that joins (2,0,0) to
(6,2,โ2).
(d) Find a vector equation and parametric equations for the line segment that joins (1,5,6) to
(3,1,8).
(e) Find a vector function that represents the curve of the intersection of the cone z=px2+y2
and z= 1 + y.
(f) Suppose the trajectories of two particles are given by
r1(t) = ht2,7tโ12, t2ir2(t) = h4tโ3, t2,5tโ6i
for tโฅ0. Do the particles collide?
3 Challenge: Vector Orthogonality
(a) Show that if |r(t)|=c(a constant), then r0(t) is orthogonal to r(t) for all t.
4 Graphs of Multivariable Functions
Sketch the graph of the function.
(a) f(x, y) = y;
(b) f(x, y) = 10 โ4xโ5y;
1
