SUNY Geneseo Department of Mathematics

Derivatives of Vector Valued Functions

Monday, September 24

Math 223 01
Fall 2018
Prof. Doug Baldwin

Misc

Extra credit colloquium write-ups are due any time before our final exam.

First parts of section 3.2

Find the derivative of 〈 1/t, tan(2t), √(sin t + cos t), e^t/(2t+1) 〉

The key idea is to differentiate each component of the vector. That plus a little chance to review differentiation produces...

Differentiation rules applied to components of a vector

Find the second derivative of 〈 s² - 3s + 1/s, sin(2s) 〉

Higher-order derivatives work exactly as you would expect, just take derivatives multiple times.

Components of a vector differentiated twice

What’s the velocity of a particle moving along the path 〈 sin t, cos t, t 〉?

Velocity is the derivative of position

Spiral with r(t) = (sin(t),cos(t),t) and v(t) = (cos(t),-sin(t),1)

Find the unit tangent vector to that path

The derivative is also tangent to the curve at all points. The unit tangent vector is just the derivative scaled to have length 1.

Scale (cos(t),-sin(t),1) to length 1 by dividing by sqrt(2)

Is that really a dot product in property vi of derivatives ( d/dt [r(f(t))] = r’(f(t)) • f’(t) )?

No, it has to be a scalar product just from the fact that the argument to r has to be a scalar.

You can also convince yourself of this by checking the derivation of that property:

Differentiate a vector valued function by differentiating its components.

2nd derivatives work as expected.

Velocity, tangent, unit tangent vector.

Integrals of vector valued functions.

Read the “Integrals of Vector-Valued Functions” subsection of section 3.2.