1. Vector-Valued Functions
Start with Week 3 - Wednesday Survey
Math 200
Week 3 - Wednesday
Vector-Valued Functions

2. Main Questions for Today
Math 200
Main Questions for Today
What’s a vector valued function?
How do we find the domain of a vector valued function?
How do we compute the derivative of a vector-valued function?
Do the derivative rules from calc 1 (e.g. the product rule) still apply?
How do we integrate vector-valued functions?

3. Vector-Valued Functions
Math 200
Vector-Valued Functions
A vector-valued functions has scalars as input and vectors as output.
Domain: Scalars
Range: Vectors
E.g.
r(t) = <3t, t2, t3+1>
r(t) = <cos(t), sin(t), t>
r(t) = <ln(t), 1, t2>

4. Math 200
It’s perfectly fine to just think of this as a different way of writing parametric equations
…And you can visualize the curve as being drawn out by the vectors you get when you plug in various t values

5. Domain Domain: the set of allowable inputs for a given function.
Math 200
Domain
Domain: the set of allowable inputs for a given function.
Natural vs. Given Domain
Sometimes, we’re given the domain for a function.
e.g. r(t) = <cos(t),sin(t),1>, 0<t<π/4
Other times, we aren’t given the domain and we have to figure out what domain makes sense (or is a natural choice)
e.g. r(t) = <ln(t), t + 1, et>
Looking at the components one at a time, we can say that the domain is t>0

6. Math 200
Domain Examples
Determine the (natural) domain for each of the following vector valued functions. The easiest way to do so may be to figure out what needs to be omitted for each component.

7. Math 200
Derivatives
Differentiating vector-valued functions is actually pretty straightforward: we just differentiate each component
E.g. r(t) = <3t2, ln(t), 4>
r’(t) = <6t, 1/t, 0>
For practice’s sake, compute the derivatives of the following vector-valued functions:
r(t) = <sin(t), t3, 1 - t> r’(t) = <cos(t), 3t2, -1>
r(t) = <2tcos(t), t2, t-1> r’(t) = <2cos(t) - 2tsin(t), 2t, 1>

8. Math 200
Tangent vectors
In calc 1, we use the derivative to find, among other things, the slope of the tangent line to the original function.
E.g. If I want the slope of the line tangent to f(x) = x2 at x=- 1, I first find f’ and then plug in x=-1.
f’(x) = 2x
f’(-1) = -2
So, the slope of the tangent line to f at x = -1 is -2
Now, we have a vector-valued function as the derivative, so what do we do with that.

9. Math 200
Tangent vectors cont.
Consider the function r(t) = <cos(t), sin(t), t>.
Taking the derivative we get…
r’(t) = <-sin(t), cos(t), 1>
Let’s plug in a value for t, say π/2:
r’(π/2) = <-1, 0, 1>
What does this vector tell us?
At t = π/2, we’re at r(π/2) = <0, 1, π/2>
Let’s plot r’(π/2) = <-1, 0, 1> at (0, 1, π/2)

10. The vector r’(π/2) is tangent to the curve at r(π/2)
Math 200
The vector r’(π/2) is tangent to the curve at r(π/2)
Now, find parametric equations for the line tangent to r at t=π/2
Point: (0, 1, π/2)
Vector: <-1, 0, 1>

11. Math 200
Example
Consider the vector-valued function r(t) = <3t-1,t2,ln(t-1)>
Find parametric equations for the line tangent to r at the point (5,4,0)
Express the line as a vector-valued function

12. We just integrate term by term
Math 200
Integration
Since differentiation works with vector-valued functions, it stands to reason that anti-differentiation also works, and in the same way…
We just integrate term by term
The interpretation isn’t quite the same as in calc 2…
It’s not area under the curve anymore
The interpretation is context-dependent

13. Math 200
Integration example
Suppose v(t)=<4t,et> is the velocity vector for some particle moving along the plane.
If the particle’s initial position (t=0) is (2,3), find its position function r(t)