Consider the curve given by the parametric equations x=e^ t
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
Consider the curve given by the parametric equations
x=et,y=e2t+t,
where t ranges over all real numbers.
Throughout this question, assume the curve is smooth.
(a) Derive an expression for the slope of the tangent line to the curve at the point corresponding to the parameter value t.
(b) Show that this slope is strictly positive for every real value of t. Provide clear mathematical justification.
(c) Explain why the result from part (b) precludes the existence of two distinct points on the curve whose tangent lines are perpendicular to each other.