Question

question 1
0.5 pts
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 )
ii only
i and ii
neither statement is true
i only

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 \( x = a \). This is because the derivative measures how the function changes at an exact point (instantaneous) as opposed to an average rate of change over an interval. Since both statements I and II are correct based on the definitions of the derivative, the correct option is "I and II".

Answer:

I and II (the option with "I and II" text, as per the multiple - choice options in the question)