Consider the graph $y = e ^ { x }$ .What is the x-intercept of the tangent line to the graph at $\left( a , e ^ { a } \right)$ A) $a - 1$ B)e ^a $( 1 - a )$ C) $1 - a$ D) $e ^ { a } ( \alpha - 1 )$

Question

Quizplus · Accepted Answer

CThe derivative of $y = e^x$ is $y' = e^x$, which gives the slope of the tangent line at any point $x$. At $x = a$, the slope is $e^a$. The equation of the tangent line at $(a, e^a)$ can be written as $y - e^a = e^a(x - a)$. To find the x-intercept, set $y = 0$: $0 - e^a = e^a(x - a)$, which simplifies to $x = a - 1$, matching choice A. However, due to a typographical error, choice B is intended to represent $e^{(1-a)}$, which is not correct, making choice C ($1 - a$) appear as a distractor. The correct interpretation of choice B would not change the answer, as it does not represent the x-intercept of the tangent line. Therefore, the correct answer is A, due to the correct calculation method and the misunderstanding around choice B's notation.

Consider the Graph y=exy = e ^ { x }y=ex What Is the X-Intercept of the Tangent Line to the .What

Consider the Graph $y = e ^ { x }$ What Is the X-Intercept of the Tangent Line to the .What