The domain of is all real numbers.
or Substituting this into gives
At the point the curvature is equal to 4. Therefore, the radius of the osculating circle is
A graph of this function appears next:
The vertex of this parabola is located at the point Furthermore, the center of the osculating circle is directly above the vertex. Therefore, the coordinates of the center are The equation of the osculating circle is
The units for velocity and speed are feet per second, and the units for acceleration are feet per second squared.
Section 3.1 Exercises
a. b. c. Yes, the limit as t approaches is equal to d.
a. b. c. Yes
The limit does not exist because the limit of as t approaches infinity does not exist.
where k is an integer
where n is an integer
All t such that
a variation of the cube-root function
a circle centered at with radius 3, and a counterclockwise orientation
Find a vector-valued function that traces out the given curve in the indicated direction.
For left to right, where t increases
One possibility is By increasing the coefficient of t in the third component, the number of turning points will increase.
Section 3.2 Exercises
- Undefined or infinite
To show orthogonality, note that
The last statement implies that the velocity and acceleration are perpendicular or orthogonal.
Section 3.3 Exercises
Arc-length function: r as a parameter of s:
The maximum value of the curvature occurs at
The curvature approaches zero.
The curvature is decreasing over this interval.
Section 3.4 Exercises
The range is approximately 886.29 m.