Question Type 8: Finding the time of change in both directions for a given object Exercises

For the parametric trajectory given by

find the times when the object changes direction in the $x$-direction.

For the same trajectory

find the time when the object changes direction in the $y$-direction.

Find the time(s) when the trajectory $x(t)=3+t(9-t^2),\quad y(t)=4+t(11+t)$ has a horizontal tangent.

Determine the interval(s) of $t$ for which the object moves to the right (i.e. $dx/dt>0$) given

Determine the interval(s) of $t$ for which the object moves upward (i.e. $dy/dt>0$) given

Find the time(s) when the same trajectory has a vertical tangent.

Show that the object never comes to rest (i.e. its speed never becomes zero) for

Find the local minimum of $y(t)$ by determining the time at which $y(t)$ attains an extremum and identifying whether it is a minimum, given $y(t)=4+t(11+t).$

Solve for the time(s) $t$ when the object crosses the $x$-axis (i.e. $y(t)=0$) for $y(t)=4+11t+t^2.$

Find the local maximum of $x(t)$ by determining the time(s) at which $x(t)$ attains an extremum and identifying which is a maximum, given $x(t)=3+t(9-t^2).$

Determine whether there is any time $t$ such that the tangent to the trajectory $x(t)=3+t(9-t^2),\quad y(t)=4+t(11+t)$ has slope $1$, i.e. $\frac{dy}{dx}=1.$

Find the time at which the $x$-coordinate equals 10, given $x(t)=3+9t-t^3.$