Question

question 1
let $f(x)$ be a function and let $f(a)$ be the derivative of $f$ at $x = a$. which of the following statements are true?
i. $f(a)$ is the slope of the line tangent to the graph $y = f(x)$ at $x = a$
ii. $f(a)$ is the instantaneous rate of change of $f(x)$ at $x = a$
\bigcirc ii only
\bigcirc i and ii
\bigcirc neither statement is true

Explanation:

• For Statement I: By the definition of the derivative, the derivative of a function \( f(x) \) at a point \( x = a \), \( f'(a) \), represents the slope of the tangent line to the graph of \( y = f(x) \) at \( x = a \). This is a fundamental concept in calculus related to the geometric interpretation of the derivative.
• For Statement II: The derivative \( f'(a) \) also represents the instantaneous rate of change of the function \( f(x) \) at \( x = a \). The instantaneous rate of change is the limit of the average rate of change as the interval over which we calculate the average rate shrinks to zero, and this limit is precisely the derivative. So both statements I and II are true.

Answer:

I and II