SUMMARY
The tangent line at t=0 for the parametric curve defined by the equations x=8sin(5t)-3, y=8cos(5t)+3, and z=3t-4 is derived using the derivatives of the parametric equations. The derivatives at t=0 yield the values x'(0)=40, y'(0)=0, and z'(0)=3, resulting in the displacement vector (40, 0, 3). The parametric equation for the tangent line can be expressed as r(t) = (x(0), y(0), z(0)) + h(40, 0, 3), where (x(0), y(0), z(0)) is the position vector at t=0.
PREREQUISITES
- Understanding of parametric equations
- Knowledge of derivatives and their applications in vector calculus
- Familiarity with the concept of tangent lines in three-dimensional space
- Basic proficiency in evaluating trigonometric functions
NEXT STEPS
- Study the derivation of parametric equations for curves in three dimensions
- Learn about vector calculus, focusing on tangent vectors and their applications
- Explore the use of derivatives in physics, particularly in motion along curves
- Investigate the graphical representation of parametric curves and their tangents
USEFUL FOR
Students studying calculus, particularly those focusing on parametric equations and vector calculus, as well as educators teaching these concepts in mathematics or physics courses.