Consider the vector field S(u)= (u), ,u ,e^ u , , (u)

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Consider the vector field

\mathbf S(u)=\langle \sin(u),\,u\,e^{u},\,\ln(u)\rangle.

The scalar parameter

u

is implicitly defined as a differentiable function of time

t

by the equation

u^2+\ln(u)=t.

Assume

u>0

for all relevant

t

. (a) Show that

\dfrac{du}{dt}=\dfrac{1}{2u+\tfrac{1}{u}}

. (b) Derive an explicit expression for

\dfrac{d\mathbf S}{dt}

in terms of

u

and

t

. (c) Evaluate

\dfrac{d\mathbf S}{dt}

at the instant when

u=1

. (d) Discuss whether the direction of

\dfrac{d\mathbf S}{dt}

u=1

is more influenced by the first, second, or third component of

\mathbf S

. Support your reasoning quantitatively.